FLUID MECHANICS AND PHYSICS

Transcription

Jean de Climont
FLUID
MECHANICS
AND
PHYSICS
Editions d'Assailly
2
Jean de Climont
FLUID
MECHANICS
AND
PHYSICS
Editions d'Assailly
3
ISBN 978 2 9024 2524 2
© Editions d'Assailly, 2014, 2016
4
SUMMARY
N°
Title
page
6
1.
The implementation of the kinetic momentum
theorem to the friction within fluids
2.
Physical modelling of volume waves within
free surface liquids
57
3.
Swell physical modelling
63
4.
Ocean Tides
80
5.
Implementation of the Hamilton’s principle
in fluid mechanics
89
6.
Polarization of the K coronal layer of the
Sun within the Galaxy plane
112
7.
The consequences of Pr. Allais' second
option of his third hypothesis to explain the
results of his analysis of Miller's
interferometer measurements
152
8.
The inversion of the electron magnetic
property and its implication.
157
9. Gravitation zonal effects.
174
10. Electron beams magnetic field
179
11. Electron intrinsic magnetic field is not a
187
dipole, the Rowland effect.
5
1
The implementation
of the kinetic momentum theorem
to the friction
within fluids
A common solution to both fluid mechanics
anomalies:
swelling of the surface of whirl-well flows
and
laminar separation point
of the flow around a cylinder
Configurations
of the flow
around a cylinder
March 1995
6
7
Abstract:
The free surface of whirl-well flows is above the theoretical level.
The azimutal angle of separation of the laminar flow around a cylinder is about 82°. Neither
the speed profiles, nor the various asumptions proposed up to now, allow for such an angle
below 90° which is the angle where the differential pressure sign changes.
Using the kinetic momentum theorem, the Fluid Mechanics equations in cylindrical
coordinates have been directly established. They provide a complementary term related to
the Coriolis acceleration. However, they mainly allow for a new approach of friction within
fluids, leading to a common solution to both non conformances, complying with experiments.
In addition, a full and accurate description of the various configurations of the flow around
a cylinder has been obtained.
14
Abstract.
00 INTRODUCTION
01 Objectives
02 Principles
10 COMMON PART OF THE FLUID MECHANICS EQUATIONS
11 Cylindrical coordinate equations
12 Analysis of the complementary term
13. Moments of the friction forces
14. New expression of frictions
20 IMPLEMENTATION TO THE WHIRL-WELL FLOW
21 Relative speeds within the fluid
22 Modified equations
23 Boundary conditions
24 Results
25 Friction effect on the bottom
26 Lowering of the air pressure as a result of the sweeping by the vortex
30 IMPLEMENTATION TO THE FLOW AROUND A CYLINDER
34 Results: the flow configurations
35 Equation of the boundary layer
36 Results: the separation points
40 IMPLEMENTATION TO THE MOTION OF SUPERFLUID HELIUM IN A
ROTATING CYLINDER
42 Equation of the parietal sheet
43 Equation within the fluid
44 Results
50 CODING
51 Equations of the whirl-well flow (CODES PTAVEVO, PTSANVO et PTFRORA.BAS)
52 Equations of the flow around a cylinder (CODE CYLINDRE.BAS,CYCLTU1.BAS)
53 Equations of the superfluid helium (CODE HELIUM.BAS)
54 Polhausen and modified Polhausen methods(CODES POLHA50,LAMNNAP et LAM1NAP.BAS)
60 REFERENCES
15
00 INTRODUCTION
01 Objectives
During the last two decades, the visco and plasto-elastic models allow for major
improvements in the knowledge of material behaviour. In the meantime, fluid mechanics
remains fully experimental. There is still a large gap between theoretical solutions and
experimental results.
At the beginning of the twentieth century, Fluids mechanics was considered by high level
scientists as a lumpen-science. No improvements of the modelling have been proposed
which could be compared to those implemented in various technical areas. Conversely, huge
investments and increasing testing expenses are made in order to balance a lack of
knowledge hidden by subtle calculations. Unexpected vibrations, unforeseen separations,
untimely cavitations and overflows occurring at random are still the dread of engineers.
Two other kinds of difficulties are prevailing in the present approach of friction within
viscous fluid flows.
These flows are always producing vortices. When the flow is symmetrical, two opposite
vortices are generated. Thus, it seems that the flow is complying with the kinetic moment
conservation principle. It is not. Half a cylinder lying on a plane generates only one line of
vortex, fully developed, although the flow is disturbed by the plane. Furthermore, such a
global approach of the kinetic moment conservation principle hides the time factor. The
wake vortices cannot appear just when the flow passes the edge. The kinetic moment shall
already be in the fluid before. Even less could they be their own reciprocal cause of
existence.
The second kind of difficulties is related to the accumulation of energy within the boundary
layer. The part of this energy not dissipated within the boundary layer, is dissipated within
the wake, but the form under which this energy is accumulated is not identified. Even so,
this energy is not taken into account. The eddy form shown by the wake surely gives the
solution.
This report is devoted to the kinetic friction within fluids. The word "kinetic" is used there
as usual, although not fully logic" to designate the phenomena related to the moments in the
frame of the kinetic moment theorem as opposed to the dynamic effects studied in the frame
of the momentum theorem
16
02 Principles
The mass conservation equation is not changed.
Only permanent and incompressible plane flows are envisaged.
The moments only exist for rotating flows, or part of flows. Thus only one equation is
involved. The other ones are obtained by projection of the inertia and forces on the axis of
the plane rotating flow and on the direction of the origin within the plane.
The applicable equations have been established in each case, nevertheless, the inertial
moment and pressure effect moment are common to all cases; they have been set up in the
next paragraph.
17
10 COMMON PART OF THE FLUID MECHANICS EQUATIONS
11 Cylindrical coordinate equations
The kinetic momentum theorem applied to the projections on the fluid trickle tangent writes
for a volume element:
−
∂p
d ( IΩ ) d ( Σiω )
r∂θ × ds × r − Mt ( Friction forces) =
+
r∂θ
dt
dt
Where I is the moment of inertia of the volume element related to the axis of the cylindrical
coordinate perpendicular to the flow plane. is the local angular speed of the flow.
i is the moment of inertia of the fluid components related to an axis perpendicular to the
flow plane and including their gravity centre and ω the angular speed of those elements
around this axis. The product of the moment of inertia and of the angular velocities shall be
summed up within the volume element.
Within the former dynamic approach of friction, the moments of the friction forces are
related both to the flow axis and to each of the volume elements' axis. The terms of the
equation related to friction in the former approach are easy to be retrieved and they write:
 d (τ × dz × rdθ )

Mt ( f . ft ) = (τ × dz × rdθ ) × r + 
× dr  × r − (τ × dz × rdθ ) × dr
dr


2
 ∂ V ∂V V 
Mt ( f . ft ) = µ  2θ + θ − θ2  × r
r∂r r 
 ∂r
The first term of the second member of this equation is the moment of the variation of τ
related to the distance, the second is the couple resulting from τ on each face involved of the
volume element and the third term is the deduction of the sold body rotation not involving
any relative movement.
The main moment of inertia may be written in a more usual way:
d ( ρdvr 2
d ( IΩ )
=
dt
dt
with :
r
dθ
)
dt
dθ
= Vθ
dt
18
Then it may be written
∂p
d ( ρdvrVθ ) d (Σiω )
r∂θ × ds × r − Mt ( Fft ) =
+
r∂θ
dt
dt
dV  d (Σiω )
∂p
 dr
−
r∂θ × ds × r − Mt ( Fft ) = ρdv Vθ + r θ  +
r∂θ
dt 
dt
 dt
dV  d (Σiω )
∂p

r∂θ × ds × r − Mt ( Fft ) = ρdvVrVθ + r θ  +
−
r∂θ
dt 
dt

1 ∂p Mt ( Fft ) VrVθ
∂V
∂V
1 d ( Σ iω )
−
−
=
+ Vr θ + Vθ θ +
r
r∂θ ρrdv dt
ρ r∂θ
ρrdv
∂r
−
12 Analysis of the complementary term
The equation includes a complementary term not found when it is directly derives from the
Cartesian coordinates equation. However this term is always negligible against the others as
far as the angular speed of the fluid components, should they be Helmoltz tubes or spinning
particles made of a limited amount of molecules, is only a result of a general rotating motion
of the fluid around the origin of coordinates. Nevertheless, when this motion is very slow,
and of course when there is no motion, this term related to the own moment of inertia of the
fluid particles may have major effects. In addition it may take non negligible values when
the fluid volume involved is great enough. This is the case for large Helmoltz tube. In this
case the motion of such tubes may be determined within a rotating fluid. This term is not
new. It is equivalent to what is called the CORIOLIS acceleration for the motion of a
massive point in the pure mechanical point of view. The result in fluid is a term equivalent
to a pressure, which could be named CORIOLIS pressure.
However, this term has no impact on the results of this report. Of course high moments of
inertia may be found within the two flows studied by this report. They are located where the
radius of curvature is very small, mainly near the edge of the hole for the whirl-well flow
and at the leading point for the flow around a cylinder. Nevertheless, the radius of curvature
has no effect on particles when their dimension is greater than this radius. It is not consistent
to reduce the integration step below this dimension aiming at an increase of the moment of
inertia. Even though this would be done, the effect of such a moment of inertia will not
allow for explaining by the same way both non conformances subject of this report, mainly
because the areas where such an event could occur are not in line between both flows, and
not in line within each flow with the areas where the non conformances occur.
19
13 Moments of friction forces
In most case, the inertia moment of fluid particles may be neglected. However, as soon as it
is considered that those particles may turn around themselves, there is friction between them.
The rotation of the fluid particles around themselves creates friction not existing otherwise.
Moreover if such a rotation is produced within flows deriving from a potential as it is the
case for all flows within this report, kinetic friction appears although the old approach of
friction did not allow for any friction.
This report is based upon this assumption that a gradient of speed within a flow makes the
fluid particles rotating around themselves. Thus the friction terms of the viscous fluid
mechanics equations are dramatically changed.
It shall be noted that quantitative results are only given in this report where the phenomena
are independent from the dimension of the fluid particles. The notion of "fluid particle"
remains undefined as previously. There are experimental results that may need the
knowledge of this dimension in order to enable explaining what happens, but this is the
purpose of a future report.
The rationale is to determine the kinetic friction parameter values for the whirl-well flow
both with vortex and without vortex, then to validate the new approach by implementing
these values of the parameters to the flow around a cylinder. The validation appears not to
be possible only for very small Reynolds numbers (below 10, i.e. without any industrial
impact) by lack of experimental data.
As impressive as is the compliance with the well-known experimental results, a coincidence
cannot be eliminated. Thus the new approach has been applied to the motion of superfluid
helium in a rotating bucket. Again most of the well-known experimental results are
explained, nevertheless only the approach is validated in this case for the value of the kinetic
friction parameters are not the same as those in water.
Note related to the kinetic moment of the fluid particles.
It could be thought that such a moment is internal and should not be taken into account and
the kinetic friction as well should not be considered to study the overall motion of fluids.
This is of course fully wrong. This would not be in line with the kinetic moment conservation
principle. The easiest way to be convinced is to consider the rotation of a gyroscope. The
overall kinetic moment depends mainly on the kinetic moment of the fly wheel inside. The
same applies to the fluid particles.
20
14. New expression of friction
The kinetic friction acts through their moments. The tangential friction, parallel to the fluid
flow, maintains the rotation of the fluid particles around themselves. This rotation generates
a radial friction between fluid particles as shown by the following sketch.
V fluide
 →
dτ
τft+ dr ft af
af
τfr
τ fr
τft
The index f is related to the friction within the fluid. Indices t and r are related to the
tangential and radial direction with regard to the local speed of flow although friction
between fluid particles is always tangential. Along the walls, f is replaced by p (as parietal).
There is no reason to believe that the friction parameters have the same value between
particles themselves and between particles and walls. This is by the way a mean to take into
account the wall roughness. Nevertheless within this report, it is always considered that the
roughness is very small so that the walls are covered with a fine layer of motionless fluid,
like painted, so that friction occurs every where between fluid particles only.
The kinetic friction may be related to either Helmoltz tubes or fluid particles relative
rotation. This friction is proportional to the surface of these tubes or particles in contact.
These surfaces are either part of cylinders or part of spheres; their radius is designated as af.
The kinetic friction coefficient is designated as the kinetic viscosity kf (dimension ML-1T-1).
The definition of this coefficient is similar to the definition of the dynamic viscosity. It is
assumed that the speed of rotation of either the tubes or the particles, is proportional to the
difference between their speed within the fluid. Thus the friction moments are balanced
within permanent flows independently from the speed of the flows.
The ratio of the angular speed to the speed gradient is designated as kkf. This is an
adimensional slipping coefficient of tubes or particles between them.
21
The friction strengths are proportional to the relative peripheral speeds of tubes and particles
and to the inverse of the contact surface da:
τ = kf ×
∆u
da
Moreover, the solid body rotation does not imply any relative motion so that the related
speed difference shall be corrected accordingly.
It shall be noted that fluid nooks are left by the Helmoltz tubes and the fluid particles
gathered within layers or sheets. These fluid nooks are subjected to friction and shear
stresses. The moments related to these effects have not been taken into account. Thus they
are implicitly introduced into the kinetic coefficient values obtained from the experimental
results related to the whirl-well flows. As far as the speed gradients have the same sign and
same order of magnitude in both the whirl-well flow and the flow around a cylinder, it may
be considered that the effect of nooks is similar for both flows. Any way, the results
obtained for the flow around a cylinder are complying with experiments when these values
of the kinetic parameters are used. This is no more the case for very high Reynolds numbers,
although the critical values are not changed. In such high speed conditions, those fluid nooks
have a major consequence. They allow for explaining the most remarkable phenomenon of
the fluid mechanics: the transition between laminar and turbulent flows.
22
The solid body rotation
15. Dynamic viscosity, kinetic viscosity and slipping
The kinetic friction
Along an indefinite plane, the kinetic viscosity may be calculated from the dynamic
viscosity and the slipping coefficient. The shear stress is the same in both cases as the
curvature radius is infinite. The only resulting shear stress is tangential. The shear stress
writes in each case:
23
∆u ft = u2 − ω2 × a f − (u1 + ω1 × a f )
du
× 2a f − (ω2 + ω1 ) × a f
dy
du
dω
∆u ft =
× 2a f − (2ω +
× 2a f ) × a f
dy
dy
du
with ω = kk f ×
dy
∆u ft =
 du
d 2u 
∆u ft = 2a f ×  × (1 − kk f ) − kk f × a f × 2 
dy 
 dy
The shear stresses of the dynamic and kinetic friction approaches are respectively:
du
ldx
dy
dx
dx
Τc = τ ft lda
= k f ∆u ft l
2a f
2a f
Τd = τldx = µ
avec :
Τd = Τc
µ
du k f ∆u ft
=
dy
2a f

d 2u 

dy 2 

µ = k f 1 − kk f − kk f × a f ×
du 


dy 

When the order of magnitude of the last term is small against kkf, it may be written:
kf =
µ
1 − kk f
In the other flows, there is always a curvature. Thus it is necessary to write the full equations.
Nevertheless it happens that the slipping coefficient is always linked to the radius of the
tubes or fluid particles for both the whirl-well flow and the flow around a cylinder. The
related terms may be neglected. The opposite situation occurs in the motion of superfluid
helium in a rotating bucket, and this is the reason why only qualitative results are drawn in
that case.
In order to make equation easier to be written, the slipping coefficient is always positive and
takes values between 1 (no slipping) and 0 (no rotation). It should have been more consistent
to define an algebraic coefficient (-1,+1), but the need to write the equations in each case
when using a positive coefficient only was considered as a good way to check several times
24
the correctness of the equations. At the end, an algebraic coefficient was used for helium
motion.
All the equations and most of the calculation codes have been written taking into account all
terms including the slipping coefficient. Its effect has been checked and does not affect the
results provided the radius af remains small enough (say below 1µm) excepted of course for
the motion of helium.
25
20 IMPLEMENTATION TO THE WHIRL-WELL FLOW
The conventional cylindrical coordinates are used together with the notation defined above
for friction. The friction on the bottom is calculated in § 25. Only the friction within the
fluid is first considered. It shall be reminded that the whirl-well flow is a typical irrotational
flow deriving from a potential, so that there is no friction in the old approach. It is not the
same within the new approach as shown hereafter.
dτ
τft+ dr ft af
af
τfr
τ fr
τft
21.1 Tangential relative speed:
 r − 2a f
∆u ft = Vθ 2 − ω2 × a f − (Vθ 1 + ω1 × a f ) − Vθ 2 × 1 −
r

 2a f 
∂V

∆u ft = θ × 2a f − (ω2 + ω1 ) × a f − Vθ 2 × 
∂r
 r 
 2a f 
∂V
∂ω

∆u ft = θ × 2a f − (2ω +
× 2a f ) × a f − Vθ × 
∂r
∂r
r


 ∂V V 
with ω = kk f ×  θ − θ 
r 
 ∂r
 ∂V V
∆u ft = 2a f ×  θ − θ
r
 ∂r



 ∂ 2Vθ ∂Vθ Vθ 

 2 −
(
)
×
1
−
kk
−
kk
×
a
×
+ 

f
f
f
r∂r r 2 

 ∂r
26
∆u ft = −
2 ka f
r2
af
( 2 − 2 kk f + 4 kk f
r
)
21.2 Radial relative speed:
∆u fr = 2 × a f × ω
with
 ∂Vθ Vθ 
− 
r 
 ∂r
ω = kk f × 
∆u fr = −
2ka f
r2
Vθ =
and
k
r
( whirl
2kk f
a) Moments:
1. tangential volume friction + τ ft × 2 a f × ds +
2. radial volume friction
− τ fr × 2 a f × ds
∂τ ft
× 2 a f × r × ds
∂r
b) Summation within the fluid volume element dx dy dz
∂τ


dx
dy
 + τ ft + ft × r − τ fr  × 2a f × da × dz ×
×
2 a f 2a f
∂r


c) Division by ρ r dx dy dz and replacement of strength by their value:
kf
∂∆u ft


 + ∆u ft +
× r − ∆u fr  ×
∂r

 rρ 2 a f
d) Replacement of the speed variations by their value:
2k f k  (2 − 2kk f ) 4a f kk f 2r (2 − 2kk f ) 12ra f kk f ) 2kk f 
−
−
+
+
+ 2 
ρr 
r2
r3
r3
r4
r 
The equation of §11 becomes, in the frame of the whirl-well flow:
2k f k 
a 
dV
dV
VrVθ
1 + 4kk f f  = 0
+ Vr θ + Vz θ +
3 
r
dr
dz
ρr 
r 
27
speed )
23.1 Whirl-well flow with vortex
The only results available are those related to the experiments performed by professor
OSTUBO, as reported by the "Traité d'Hydraulique Générale" of professor ESCANDE. The
case where the load is Zinfinite=0.207m with a 0.02m opening, has been chosen. The flow
rate has been calculated taking into account a flow rate coefficient of 1 applied to the surface
included between the edge of the hole and the vertical cylindrical surface of the flow in the
hole. This coefficient had been chosen in line with the bell-shaped downstream flow after
the hole. The most probable radial speed has been calculated accordingly. The tangential
speed has been calculated so that the theoretical surface coincides with the calculated water
surface, when the kinetic friction is not taken into account. At a distance designated as Rinf
= 0.1m where calculations begin, the tangential and radial speeds are respectively:
V0=0.004866m/s and U0=-V0/9.0524.
It shall be noticed that the mathematical model used takes into account the radial speed and
the surface lowering speed to calculate the level of the water surface. Thus the lowering of
the water free surface is beyond the value obtained in the basic whirl-well flow as presented
in the literature. As a result the swelling of the free surface shall be larger than expected
when those speeds are not taken into account. The free surface lowering excess results
mainly from the radial speed.
23.2 Whirl-well flow without vortex
Here again a curve given by the "Traité d'Hydraulique Générale" has been used. It is a curve
giving the tangential speed with regard to the distance to the axis of the hole. The speeds
have been calculated so that the theoretical curve coincides with the calculated curve when
the kinetic friction is not taken into account. These speeds are as follows: V0=0.0014m/s and
U0=-V0/9.0524 the other boundary conditions being the same as for the vortex case.
The theoretical flow mathematical model is stable by itself, provided the boundary
conditions are precisely entered as far as possible from the hole. Corrections have been
developed first to improve the secant method results. They were found very efficient for
polynomial solutions with powers included in the range -5 to +5. Although they were
installed in the codes, they are inhibited to reduce the calculation duration, already very long
even with computer processor 80486, 33Mhz. The gap with theoretical results remains
within the width of the curve line for the lower speeds.
23.3 Hole
Improvement of the results has been tested by fixing objectives downstream conditions after
the hole. Although many iterations were performed, it had not been possible to obtain
acceptable results. During these tests it had been noticed that the free surface was not
affected by the downstream conditions.
28
Both experimental curves of the "Traité d'Hydraulique Générale" have been scanerized and
printed thereunder.
Vitesse tangentielle d’un tourbillon sans vortex
(vitesse en mm/s, distance en mm)
29
free surface meridian (load 20.7cm)
surface lowering mm
50
40
30
20
10
0
0
0.01
0.02
0.03
0.04
0.05
0.06
1/(r*r) (r distance to the axis mm)
Kf = 100
Mu
Kf = 0
théorique
Kf = 150
Mu
Kf = 200
Mu
Whirl-well flow without vortex (Kf1=O.5mu)
0.03
speed m/s
0.02
0.01
0
0
0.005
0.01
0.015
0.02
0.025
distance m
fluide parfait
frottement cinétique
18
0.03
24 Results
It is not feasible to retrieve the tangential speed curve of the whirl-well flow without vortex
when only one value of the kinetic viscosity is chosen. The hypothesis is made that the
kinetic viscosity is very small for speed gradients lower than 2s-1. The ratio of kinetic
viscosity to the water mass
per volume unit enabling a good compliance of both the experimental curve and the
calculate one is kf/ρ= 0.5*10-6 m2s-1 up to a 2s-1 gradient. Beyond this value the fall is very
sharp whatever is the kinetic viscosity above kf/ρ=10-5m2s-1. It shall be noticed that the true
critical gradient is 4s-1 when the solid body rotation is taken into account to evaluate the
speed variation inducing the rotation of the fluid particles or tubes.
This first critical gradient was used to determine the value of kf/ρ allowing for the best
compliance between the experimental results obtained by professor OSTUBO and the whirlwell flow water surface calculated when the kinetic friction is taken into account. The most
suitable value is kf/ρ=1.5*10-4 m2s-1.
This value allows for compensating not only the discrepancy between the theoretical basic
flow and the experimental result, but also the additional fall of the surface due to the radial
speed being verticalized by the hole. It seems that the tangential speed is not verticalized as
it causes the bell-shaped flow of the downstream flow.
It could be noticed that a swelling of the free surface complying with the experimental value
at the hole level, is achievable with a kinetic viscosity 15 times less than the above value,
but without critical gradient. Nevertheless, the coincidence is only obtained at the top part of
the curve which deviate from the experimental curve for higher distance. In fact this value of
the kinetic viscosity leads to a straight line with a slope smaller than the curve. Such a result
cannot be considered as acceptable so that the hypothesis of the existence of two values of
the kinetic viscosity can be considered as consistent with the experimental results.
It could be thought that first HELMOLTZ tubes are generated. When the speed gradient
increases, they are cut to form fluid particles made of a large number of molecules.
The curves obtained from the calculation have been printed close to the experimental curves.
25 Friction effect on the bottom
It could be thought that as the friction causes a swelling of the free surface, the same applies
to the friction on the bottom. This is wrong, for the signs of the related moments are not the
same. The kinetic friction moments act in the same way as the fluid motion which is
accelerated towards the axis although the bottom friction moment acts in the opposite way.
Notwithstanding this fact, the effect of the friction on the bottom has been calculated. The
value obtained is so tiny that it is included in the curve line width.
19
The calculation was done according to the old friction approach with a Reynolds number as
great as 4200. Then one obtains:
0.644  V
τds = ρ
4200  2
2

ds


with
τ × rdθ × dr = ρgdz × z × rdθ
dr
ρgz
dr
dz ≈ 5V 2
ρgz
dz = τ
26 Lowering of the air pressure as a result of the sweeping by the vortex
Conversely, the effect of the air sweeping by the flow in the vortex of the whirl-well flow
has the same direction as the kinetic friction. Thus it induces also a swelling of the free
surface.
The most extreme conditions have been taken to favour the air effect. All the air includes in
the vortex is considered as swept by the water and with the same speed. The vertical water
speed responsible for the air sweeping is taken as large as the full radial speed thus
considered as fully verticalized. In addition the sweeping is calculated on the full height of
the water between the surface level far away and the bottom.
Vθvortex = 0. 45m / s
and
Vθvortex
9 . 0523
then
Vzvortex =
Vzvortex ≈ 0. 05m / s
ρ eau gh = ρ air ×
V2
2
h = 1. 3 × 10 −5 mm
water .
Definitely this is not enough. The value obtained from this extreme hypothesis remains far
below 4 orders of magnitude compared with the experimental results.
20
30 IMPLEMENTATION TO THE FLOW AROUND A CYLINDER
31.1 Tangential relative speed
As far as the curvature is the main topic of this new friction approach, conventional
notations for cylindrical coordinates are also used in this part. Nevertheless, the study of the
boundary layer is made using the Cartesian coordinates, as usual in that case, while the
curvature will not be neglected. This lack of consistency presents the advantage of allowing
for an easy verification of the equations obtained in the related part of this chapter against
the large amount of literature related to the boundary layer.
The equations are the same as those obtained in §§ 21.1 and 21.2, as the tangential speed
signs are the same.
 ∂Vθ Vθ 
 ∂ 2Vθ ∂Vθ Vθ 
∆u ft = 2a f × 
−  × (1 − kk f ) − kk f × a f ×  2 −
+ 
r 
r∂r r 2 
 ∂r
 ∂r
31.2 Radial relative speed
V 
 ∂V
∆u fr = −2kk f × a f ×  θ − θ 
r 
 ∂r
k f  ∂V V
a f  Vθ
∂ 3Vθ ∂Vθ
τ
= − × θ − θ −
×  − 2 + r
+
ρ
ρr  ∂r
r
2  r
r∂r
∂r 3




or by approximation
k  ∂V V 
τ
= − f × θ − θ 
ρ
ρr  ∂r
r
The calculation uses the tangential speed of the theoretical irrotational flow around a
cylinder for integration stability purpose (see chapter 50).
 a2 
Vθ = V0 × sin(θ ) × 1 + 2 
 r 
21
The equation of the motion within the fluid writes:
1
ρ
×
k f V0 a 2

∂V
∂V
r2 
∂p Vr Vθ
+
+ Vr θ + Vθ θ +
× 4 sin(θ ) ×  3 + 2  = 0
r∂θ
r
r∂θ ρ
∂r
r
a 

The code CYLINDRE.BAS allows for calculating the tangential speed as modified when
taking into account the kinetic friction (reservations are explained in chapter 50 CODING)
It shall be noticed that only the tangential speed gradient and only the moment with regard
to the axis of the cylinder have been considered although there are other speed gradients and
thus other kinetic strength moments in this kind of flow. This is mainly the case at the
leading point. However the calculation is in that case rather intricate although a code was
developed to calculate the effect of the moments of inertia, and delivers both the radius and
the curvature centre position evolving along the flow. Some improvements remain to be
done.
The calculations have been performed for a 0.04m cylinder radius in water. The boundary
conditions are those of the theoretical flow at a distance of 100 times the step. In most cases
the step is as small as 0.001m, but in some unstable cases it has been decreased to 0.0001m
and sometimes even below.
No assumption was made about the wall speed in this part. This is not the case within the
part hereafter devoted to the study of the boundary layer, leading to the precise separation
point calculation. No friction is first assumed on the wall.
The calculations deliver the tangential and radial speeds as well as the pressure for each step.
34 Results: the flow configurations
The flow around a cylinder with a given radius is characterised by a sequence of stable
states fully differentiated obtained when the speed is increased. These states are reminded in
the table thereafter. They refer to the Reynolds number as usual.
22
Reynolds
number
1
5
de 50 à 5000
from 5000 to 2/300 000
(separation azimuth 82°)
above 2 to 300 000
(separation azimuth 120°)
23
The main characteristic of the equation of the § 32 is that the remote speed
module is implicitly present in each of its terms. The kinetic friction effect
has thus a relative effect decreasing when the speed is increasing. Thus the
flow should reach a separation point before an azimuth of 90° for very low
Reynolds numbers (1 to 10) if the first kinetic viscosity kf1=0.5m2s-1
obtained from the whirl-well flow was acting from the lowest speed. As it is
not the case, this first lower kinetic viscosity shall be also subject to a first
critical speed gradient. That is to say that the rotation of fluid tubes or
particles is not possible when the tangential speed gradient remains below a
minimum value. This value cannot be determined by the whirl-well flow
without vortex because the friction is too low to be seen on the
experimental curve. This first critical gradient should be below 1.5*10-2s-1;
this is the value used for the calculation.
In these conditions, flows around a cylinder with Reynolds numbers below
50 do not involve kinetic friction with the exception of the boundary layer
where the gradient is higher as explained in the related part of this report.
Friction shall be considered in the former dynamic point of view.
From a speed of 0.0006m/s, thus for a Reynolds number of 50, the
tangential speed gradient reaches the first critical value 1.5*10-2s-1. Above
that speed a sharp fall of the tangential speed occurs for an azimuth close to
85°.
m/s
1.30E-03
tangential speed
Tangential speed for Reynolds=50 a=0.04m Kf=0.5mu
1.20E-03
1.10E-03
1.00E-03
60
70
80
90
100
110
120
azimuth degree
For Re=50 the theoretical and calculated curves are identical., but for Re=
2000, the fall is fully characterised.
Vitesse tangentielle Reynolds=2000 a=0.04m Kf=0.5mu
vitesse tangentielle en m/s
5.20E-02
5.10E-02
5.00E-02
4.90E-02
4.80E-02
4.70E-02
4.60E-02
4.50E-02
60
70
80
90
100
110
120
azimut en degré
vitesse calculée
vitesse théorique
The second critical gradient (2s-1as determined by the whirl-well flow with
vortex) is reached only for a tangential speed of 0.626m/s with Re=5000.
The kinetic viscosity is then kf/ρ=1.5*10-4 m2s-1 where the gradient is above
the critical value. The fall occurs at an azimuth of about 80°.
vitesse tangentielle en m/s
Vitesse tangentielle Reynolds=5000 a=0.04m Kf1=0.5mu
Kf2=150mu
1.00E-01
5.00E-02
0.00E+00
0
10
20
30
40
50
azimut en degré
25
60
70
80
90
Reynolds number = 50 000 V0=0.625m/s R0=0.04m
Kf2=150mu
1.40E-04
Tangential speed m/s
1.20E-04
1.00E-04
8.00E-05
6.00E-05
4.00E-05
2.00E-05
0.00E+00
0
15
30
45
60
75
90 105 120 135 150 165 180
azimuth degree
For Reynolds numbers in the range of 5000 to about 300 000 the lower
viscosity effect disappears so that the fall occurs near 82°. The relative
effect of the second value of the kinetic friction decrease slowly and finally
disappears completely above Re=6 to 800 000.
Reynolds number = 200 000 V0=2.5m/s R0=0.04m
Kf2=150mu
Tangential speed m/s
5.00E+00
4.00E+00
3.00E+00
2.00E+00
1.00E+00
0.00E+00
0
15
30
45
60
75
90 105 120 135 150 165 180
azimuth degree
26
Reynolds number = 800 000 V0=10m/s R0=0.04m Kf1=0.01mu
2.50E+01
Tangential speed
m/s
2.00E+01
1.50E+01
1.00E+01
5.00E+00
0.00E+00
0
15
30
45
60
75
90
105 120 135 150 165 180
azimuth degree
In the mean time, the kinetic friction is responsible for another phenomenon
above Re=200 000. The peripheral tangential speed of the rotating water
particles belonging to two adjacent sheets resulting from the slipping,
increases in the meantime and the gradient related to their distance reaches
the second critical gradient.
This gradient is approximately 4*V0*(1-kkf)/r. The critical gradient is 4 s-1
when the solid body rotation is taken into account. With (1-kkf) equal to the
value obtained from §21 when kf=150µ the related Reynolds number is
then 240 000, provided the approximation made are acceptable.
For such a gradient value, new rotating particles should take form, and the
most probable is that they take form from the water nooks between the
existing rotating particles. As many bolts as nooks take form, locking the
sheeted structure of the laminar flow. However as the flow shall continue
under the effect of the friction of the remote sheets, the structure is
destroyed although the gradients still exist and maintain the rotation of the
fluid particles now distributed at random.
It shall be added there that the Reynolds number set values for the transition
between the flow configurations have been determined for a given radius of
the cylinder. As the gradient does not vary as the Reynolds number, it is
evident that the set values are change when the radius is changed. This
27
could explain a part of the large discrepancies between the experimental
results.
Conclusion
Although the separation point can only be determined by the detail
study of the boundary layer, it appears that the configurations of the
flow around a cylinder are mainly determined by the kinetic friction
which modify deeply the flow itself.
Moreover the kinetic friction allows for an interpretation of the
transition between laminar and turbulent flows.
28
configuration
kf1
and
fluid
Reynolds number
1 à 10
no
kf2
kf1
kf2
fluid boundar boundar
y
y
layer
layer
no
no
no
separation azimuth, flow within the fluid,
speed profile within the boundary layer,
method and remarks
variable,
perfect
fluid,
separation non calculated,
boundary layer!
Polhausen,
very thick
no
no
yes
no
≈90°, perfect fluid, Polhausen modified, all
layers affected by the lower kinetic friction
viscosity, conforming calculated separation.
no
no
yes
yes
≈ 90°, perfect fluid, Polhausen modified,
wall layer affected by the higher kinetic
friction viscosity, conforming calculated
separation
"Karman street" yes
no
yes
yes
≈90°, Equation within the fluid modified by
the lower kinetic friction viscosity,
Polhausen modified, all layers affected by
the higher kinetic friction viscosity, wake
vortices explained, conforming calculated
separation.
5000
yes
yes
yes
yes
82°, Equation within the fluid modified by
the higher kinetic friction viscosity,
Polhausen modified, all layers affected by
the higher kinetic friction viscosity,
conforming calculated separation.
yes
yes
yes
yes
120°, higher kinetic friction viscosity
without influence on the speed curve,
destruction of sheeted laminar structuree,
speed profild to the n<1, higher kinetic
friction viscosity, without influence except
in the boundary layer parietal layer,
calculated separation depending on n,
conforming for n=0.2
10
50
"laminar"
200 000
"turbulent"
29
35 Boundary layer equation
35.1 relative speeds within the boundary layer and within the fluid tubes or
particles wall layer
35.1.1 Within the fluid of the boundary layer
a) Tangential relative speed:
The effect of the solid body rotation is neglected
∆u ft = u2 − ω2 × a f − (u1 + ω1 × a f )
du
× 2a f − (ω2 + ω1 ) × a f
dy
du
dω
∆u ft =
× 2a f − (2ω +
× 2a f ) × a f
dy
dy
du
with ω = kk f ×
dy
∆u ft =
 du
d 2u 
∆u ft = 2a f ×  × (1 − kk f ) − kk f × a f × 2 
dy 
 dy
b) Radial relative speed
∆u fr = 2 × ω × a f
with
ω = kk f ×
∆u fr = 2 a f × (
du
dy
du
× kk f )
dy
35.1.2 Wall/wall layer hypothesis
The speed gradient within the boundary layer is large enough to allow in all
cases a wall layer of tubes or fluid particles to be formed. The equations are
the same in both cases. Only the fluid element's radius is changed. The
radius of those elements is written ap. These radii ap and af have no reason
to be constant. Certainly they are changing along the wall; they are
probably increasing with the wall gradient. Nevertheless taking into account
such an evolution leads to rather intricate calculations. Even more, the
solving of the differential equations is fully unstable when such evolution is
taken into account when it has been neglected during the derivation process.
It has been checked in some stable enough cases that the separation azimuth
seems not to be increased by an evolution of these radii.
30
a) Tangential relative speed:
ω = kk p × u p / a p
∆u pt = u p × (1 − kk p )
b) Radial relative speed
∆u pr = 2 × a p × ω p
ω = kk p × u p / a p
with
∆u pr = 2 × kk p × u p
35.1.3 Fluid/wall layer hypothesis


 du  
∆udt = −(u p + ω p × a p ) +  u p +    × a p + ω f × a f 


 dy  p 
with
ω p = kk p × u p / a p
 du 
∆udt = −u p × kk p +   × a p + ω f × a f
 dy  p
35.2 Wall layer equation
35.2.1 General equation
The inertia forces being negligible close to the wall, the friction force
moments are approximately balancing the pressure force moment on the
volume element dx dy dz. The situation is exactly identical to the Prandtl's
hypothesis.
dp
× dy × dz × R0 =
dx
or :
∑ M (τ
nn
× ds )
du
dp
= − ρ × uδ × δ
dx
dx
therefore :
ρ × uδ ×
duδ
dx
× dy × dz × R =
∑ M (τ
nn
× ds )
35.2.2 Friction moments' calculation (first term of §11 equation):
a) Friction force moments:
1. tangential friction
− τ dt × a p × ds − τ dt × R0 × ds
2. radial friction
2 × τ pr × a p × ds
31
3. Wall/fluid particles friction
− τ pt × a p × ds + τ pt × R0 × ds
b) Summation in space dx *2ap:
1.
(−τ dt × a p × ds − τ dt × R0 × ds ) ×
2.
2 × τ pr × a p × ds ×
3.
( − τ pt × a p × ds + τ pt × R0 × ds) ×
dx
2a p
dx
2a p
dx
2a p
c) Value of τ strengths:
Generally speaking, τ strengths are proportional to the relative speed and to
the inverse of the contact surface da. The friction coefficient is the kinetic
viscosity kp .
τ = k × ∆u / da
d) Replacing τ by their value:
1.
2.
3.
− ∆udt ×
kf
( a p + R0 ) ×
dx
× ds
2a p
× ( R0 − a p ) ×
dx
× ds
2a p
da
kf
dx
2 × ∆u pr ×
× a p × ds ×
da
2a p
∆u pt ×
kp
da
e) Replacing ds by dzda:
1.
2.
3.
− ∆udt ×
kf
( a p + R0 ) ×
dx
× dz × da
2a p
da
kf
dx
2 × ∆u pr ×
× a p × dz × da ×
da
2a p
∆u pt ×
f) Division by 2apρR0dxdz:
32
kp
da
( R0 − a p ) ×
dx
× dz × da
2a p
1.
− ∆u dt ×
2.
∆u pr ×
∆u pt ×
kf
4a 2p ρR0
( a p + R0 )
kf
2 a pρR0
kp
( R0 − a p )
2
4 a p ρR0
g) ap and af are very small against R0; the §22.1 expression becomes:
3.
− uδ
duδ
1
= 2
dx 4a p ρ
2a p


× k f × ∆u pr + ∆u pt × k p 
− ∆udt × k f +
R0


h) Replacing ∆u par their value:
− uδ
+


k 
a  2a
 du 
duδ
= 2f + u p × kk p −   a p × 1 − kk f f  − p × 2kk p × u p 

dx 4a p ρ 
a p  R0
 dy  p


kp
× u p (1 − kk p )
4a pρ
i) ap/R0 terms may be neglected:
− uδ
kf
duδ
= 2
dx
4a p ρ



kp
k f  du   a f
  1 −
(1 − kk p ) × u p −
kk f 
kk p +

kf
4a p ρ  dy  p  a p



.
33
(35.2.2 i)
35.3 Global boundary layer equation
a) Moments:
1. tangential volume friction − τ ft × 2 a f × ds −
∂τ ft
× 2 a f × R0 × ds
∂y
τ fr × 2 a f × ds
2. radial volume friction
3. Wall/fluid particles radial and tangential friction
τ cr × 2 a p × ds + τ pt × ( R0 − a p ) × ds
3. Wall/fluid particles radial and tangential friction
b) Summation in space dx dy and integration from 0 up to δ :
1.
δ
 δ
 dx
∂τ ft
dy
−
×
2
a
×
ds
−
× 2a f × R0 × ds  ×
×
τ
 ∫ ft
f
∫
∂y
0
 0
 2a f 2 a f
δ
2.
∫τ
fr
× 2a f × ds ×
0
3.
dx
dy
×
2 a f 2a f
τ cr × 2a p × ds + τ pt × ( R0 − a p ) × ds ×
dx
2a p
c) Replacing τ by their value:
1.
2.
3.
 δ
k f dadz δ ∂∆u ft
k dadz 
−
∆
u
×
−∫
× R0 × f
 ∫ ft
 × dx × dy
da 2a f 0 ∂y
da 2a f 
 0
δ
kf
∫0 ∆u fr × 2a f da × dz × da × dx × dy
kf
kp

 dx
∆ucr × 2a p × da dadz + ∆u pt × da × ( R0 − a p ) × dadz  × 2a


p
d) Simplifications
1.
2.
3.
δ
δ

∂∆u ft

−
×  ∫ ∆u ft × dy − ∫
× R0 × dy  × dx × dz
2a f  0
∂y
0

δ
kf
∫0 ∆u fr × 2a f × dz × dx × dy
kf
∆ucr × 2 a p × k f + ∆u pt × k p × ( R0 − a p ) × dz ×
34
dx
2a p
e) Replacing ∆u par their value:
δ
 du

d 2u
− k f × ∫  × (1 − kk f ) − 2 × kk f × a f  × dy × dx × dz
dy
dy

0
1.
δ
2
3
d u

d u
− k f × ∫  2 × (1 − kk f ) − 3 × kk f × a f  × dy × R0 × dx × dz
dy
dy

0
δ
du
× dz × dx × dy
dy
0
2.
k f × kk f × ∫
3.
u p × ( 2 kk p × k f + k p × (1 − kk p ) ×
( R0 − a p )
ap
) × dx × dz
f) Integration processing and division by ρR0dxdz:
δ


du
−
× u × (1 − kk f ) −
× kk f × a f 
dy
ρR0 
0
kf
1.
δ
 du

d 2u
− ×  × (1 − kk f ) − 2 × kk f × a f 
dy
ρ  dy
0
k f × kk f
δ
× u0
ρR0
kf
2.
3.
up
ρR0
× ( 2 kk p × k f + k p × (1 − kk p ) ×
( R0 − a p )
ap
)
g) Replacement by boundary values


kf
 du 
−
× (u δ − u p ) × (1 − kk f ) −   × kk f × a f 
ρR0 

 dy  p
1.

k f  du 
 d 2u 
−
×   × (1 − kk f ) −  2  × kk f × a f 
ρ  dy  p

 dy  p
2.
k f × kk f
× (uδ − u p )
ρR0
3.

( R0 − a p ) 
× 2kk p × k f + k p × (1 − kk p ) ×

ρR0 
ap

up
35
h) Gathering u and subsequent derivative terms:
k p kk f 
 kf
(uδ − u p ) × −
× (1 − kk f ) +
ρR0 
 ρ R0

 du   k f kk f × a f k p
+   × 
−
× (1 − kk f )
ρ R0
ρ
 dy  p 

+
k f kk f × a f
ρ
 k p ( R0 − a p )
2k f kk p a f 
 d 2u 
×  2  + u p × −
× (1 − kk p ) +

ρ R0 a p
ρR0 
 dy  p

i) ap/R0 terms may be neglected:
k kk 
 k
(uδ − u p ) × − f × (1 − kk f ) + p f 
ρR0 
 ρR0

 du   k
+   × − p × (1 − kk f )
 dy  p  ρ

+
k f kk f × a f
ρ
 k

 d 2u 
×  2  + u p × − p × (1 − kk p )
 ρa p

 dy  p
j) Gathering uδ and up terms:
τp
 k kk × a f
 du   k
u
= δ × [k p kk f − k f × (1 − kk f ) + ] +   × − p × (1 − kk f ) + f f
ρ ρR0
ρ
 dy  p  ρ

+
 d 2u 
×  2 
 dy  p
k

×  p × R0 × (1 − kk p ) + k f × (1 − kk f ) − k p kk f 
ρR0  a p

up
k) Deleting terms very small against those including R0/ap:
τp

u
 du   k p
= δ × [k p kk f − k f × (1 − kk f )] −   ×  × (1 − kk f )
ρ ρR0
 dy  p  ρ

+
k f kk f × a f
ρ
up kp
 d 2u 
×  2  +
×
× (1 − kk p )
 dy  p ρ a p
36
35.3.k
35.4 Boundary layer speed profile:
As it has been assumed that the tubes or fluid particles are rotating, the
speed up of the wall layer is not null. The POLHAUSEN polynomial form
becomes:
2
u
y
 y
 y
= A + B + C   + D 
uδ
δ
δ 
δ 
3

du
y
y2 

= uδ ×  B + 2C 2 + 3D 3 
dy
δ
δ 

d 2u
y
 C
= uδ ×  2 2 + 6 D 3 
2
dy
δ 
 δ
when
 du 
  = 0
 dy δ
A+B+C+D = 1
B+2C+3D = 0
2C+6D = 0
yδ = δ
then:
u = uδ
 d 2u 
 2  = 0
 dy δ
thus : B=-C, D=-C/3 and A=1+C/3.
Replacing up, du/dy and d2u/dy2 by they value, the expression 35.2.2.i
becomes:
− uδ
−
kf
duδ
= 2
dx
4a p ρ


kp
kf B
(1 − kk p ) × A × uδ −
× uδ
kk p +
kf
4a p ρ δ


 δ2
kf 
kp
kf B δ2
du δ δ 2
×
= 2 kk p +
(1 − kk p ) ×
A−
dx ν
kf
4a p ρ 
4a p ρ δ ν
 ν
while writing
successively :
duδ δ 2
dx ν
 δ2
kf 
kp
ε = 2 kk p + (1 − kk p ) ×
kf
4a p ρ 
 ν
Λ=
γ =
af
δ
× (1 −
kk p )
4a p ρ ν
ap
kf
×
then one obtains:
37
Λ+ε
ε / 3+ γ
35.5 Boundary layer thicknesses and subsequent derivatives
C=−
35.5.1 Thicknesses
It is then possible to evaluate δ1 (displacement thickness) and δ2
(momentum thickness), as function of Λ, ε and γ:
u
δ1 δ
= ∫ (1 − )dy
uδ
δ 0
B C D
δ1
=1− (A + + + )
δ
2 3 4
δ2 δ u
u
=∫
(1 − )dy
uδ
δ
0 uδ
δ2
B C D 1
= A+ + + −
2 3 4 δ
δ
2
δ

By Cy 2 Dy 3 

+
A
∫0  δ + δ 2 + δ 3  × dy
δ2
B C D  2 B2 C 2 D2 

= A + + + −  A +
+
+
δ
2 3 4 
3
5
7 
AC
AD
BC BD
CD 
 AB
− 2
+2
+2
+2
2
+2

3
4
4
5
6 
 2
35.5.2 Thicknesses' derivatives
The Karman relation involves the derivatives of δ and δ2 :
δ2
δ dδ
dδ 2
=δ δ + 2
dx
dx
δ dx
d
d 2uδ
dδ
δ dΛ δ dx 2
=
−
dx 2Λ dx 2 duδ
dx
Bringing the second relation within the first one obtains:
38

d 2 uδ 
 δ dΛ δ

dx 2 

35.5.2
−
 2Λ dx 2 duδ 

dx 
From now the expression δ2/δ is designated as FF.
δ2
δ
dδ 2
=δ δ + 2
δ
dx
dx
d
35.5.3 dFF/dx calculation from § 35.5.1:
dFF dA 
2 B 2C 2 D 
=
1− 2A −
−
−

dx
dx 
2
3
4 
+
dB  1 2 A 2 B 2C 2 D 
−
−
−
−
dx  2 2
3
4
5 
+
dC  1 2 A 2 B 2C 2 D 
−
−
−
−
dx  3 3
4
5
6 
+
dD  1 2 A 2 B 2C 2 D 
−
−
−
−
dx  4 4
5
6
7 
what may write :
dFF dC  1 A B C
D 
=
− −
−
−

dx
dx 12 6 30 90 210 
that can be written:
dFF dC
=
× LL
dx
dx
35.5.4 dC/dx calculation from 35.5.1:
dC
1
dΛ dδ 
2ε
Λ + ε  2ε γ 
=−
+
+
 + 
− (
dx
ε / 3 + γ dx dx  δ ε / 3 + γ ) (ε / 3 + γ )2  δ δ 
dC
1
dΛ dδ
=−
+
JJ
dx
ε / 3 + γ dx dx
39
35.5.4
35.6 Karman relation:
τ p  dδ 2 duδ 2δ 2 + δ 1  2
=
+
×
 × uδ
dx
uδ 
ρ  dx
Carrying the second term in the §35.3k expression it comes:
 dδ 2 duδ 2δ 2 + δ1  2 uδ
+
×
× k p kk f − k f × (1 − kk f )

 × uδ =
ρR0
dx
uδ 
 dx
[
]
u
k
 k kk × a f  d 2uδ 
 du   k
−   ×  p × (1 − kk f ) + f f
×  2  + p × p × (1 − kk p )
ρ
 dy  p  ρ

 dy  p ρ a p
dδ 2 duδ (2δ 2 + δ1 )
1
+
×
=
× k p kk f − k f × (1 − kk f )
ρR0uδ
dx
dx
uδ
[
 d 2uδ
 du 
 2
 
 k f kk f × a f  dy
 dy  p  k p
−
×  × (1 − kk f ) +
×
uδ2
uδ2
ρ
ρ

]


p
+
up
uδ ρ
2
×
kp
ap
× (1 − kk p )
as the second term does not included dL/dx, one can write:
dδ 2 duδ (2δ 2 + δ 1 )
EE
+
×
=
dx
dx
uδ
δ × uδ
replacing the thicknesses and their successive derivatives by their value
within 35.5.2:

δ2
d 2uδ
dδ 2
δ  δ dΛ δ dx 2
=δ δ + 2 
−
dx
dx
δ  2Λ dx 2 duδ
d
reminding 35.5.2:

δ2
δ
δ× δ + 2
δ
dx
d
40

d 2 uδ
 δ dΛ δ
dx 2

−
 2Λ dx 2 du δ

dx





dx 

 du
(2δ 2 + δ 1 ) EE
+ δ ×
=
uδ
δ × uδ
 dx


d 2 uδ
2
du
(2δ 2 + δ 1 ) EE
dFF FF dΛ FF
+
×
−
× dx + δ ×
=
duδ
2Λ dx
2
dx
δ × uδ
δ × uδ
dx
dx
d 2uδ
dC FF dΛ FF dx 2
du (2δ 2 + δ1 )
EE
LL
+
×
=
×
− δ ×
+
duδ
2
dx 2Λ dx
dx
δ × uδ
δ × uδ
dx
finally, replacing dC/dx by its value, one obtains:
d 2uδ

dΛ  FF
1
LL × δ  dx 2
du (2δ 2 + δ1 ) EE
δ   FF
×
+ LL ×  −
+ JJ ×
 = 
+ JJ ×
×
− δ ×
+

dx  2Λ
2Λ    2
2  duδ
dx
δ × uδ
uδ δ
 ε /3+ γ
dx
d 2uδ
dΛ  FF
LL × δ  dx 2
GG
  FF
×
+ LL × MM  = 
+ JJ ×
×
−

du
dx  2Λ
2 
δ uδ
δ
  2
dx
dΛ
II
=
dx HH
41
35.6.
35.6 Reynolds numbers higher than 200 000 (turbulent flows):
The POLHAUSEN polynomial profile is no more suitable. Moreover the
existence of a wall layer made of rotating particles allows for not taking
into account the former assumption of a null fluid speed derivative on the
wall surface to obtain the separation point. The separation condition is more
simply now the collapse of the wall layer speed. Thus the KARMANPOLHAUSEN method may be used in turbulent flows when the polynomial
profile is replaced by the following one:
u
 y
= A + B 
uδ
δ 
n
du
y n −1
= uδ × n × B × n
dy
δ
d 2u
yn−2
=
u
×
n
×
(
n
−
1
)
×
B
×
δ
δn
dy 2
with n = 0.1 to 0.2.
This kind of profile is tangent to the theoretical profile only for very thick
boundary layer. Thus the tangent boundary condition is replaced by making
equal the upper part boundary layer speed to the theoretical speed.
A+ B =1
B= −
Λ
2
et
A = 1+
Λ
2
The separation occurs after Λ is null that is to say after 90°.
δ1
Λn
=−
2(n + 1)
δ
δ2
Λn
Λ2 n 2
=−
−
δ
2(n + 1) (2n + 1)(n + 1)

dFF
dΛ  n
2 Λn 2

=−
× 
+
dx
dx  2(n + 1) (2n + 1)(n + 1) 
When only the wall layer is considered, one obtains:
τ p up kp
= × × (1 − kk p )
ρ ρ ap
42
36 Results: the separation points
Note: the synthesis of the results is given page 28 together with the
configurations of the flow around a cylinder.
36.1 Very low Reynolds numbers (1 <ℜe < 10)
The tangential speed gradient is below the first critical gradient value
everywhere in the boundary layer as well as within the main flow. This
should be the typical case where the PRANDTL-KARMAN-POLHAUSEN
should apply. Unfortunately the boundary layer is very thick compared to
the cylinder radius. The calculated profile may be over the main theoretical
flow speed by more than 50% at the boundary layer thickness distance.One
of the many curves obtained is presented thereunder. It is possible to reduce
the maximum speed by a similarity transformation. However, the
connection is no more complying with the null value of the speed
derivatives so that a part of the momentum remains outside the boundary
layer, and this is not in line with the hypothesis used to calculate the
boundary layer thickness. For θ=115.8°, ℜ e=5, V0=0.0000625m/s one
obtains Λ=2.39 and δ=0.0072m.
Udelta for Reynolds=5 teta=115.8°
R0=0.04m
0,0005
Speed m/s
0,0004
0,0003
0,0002
0,0001
0
0
0,002
0,004
0,006
0,008
distance from the wall en mm
uth
u
udelta
But the worth is that so thick is the boundary layer that the mass
conservation principle is no more coped with. As the wall speed is null and
as it increases very slowly, an overspeed occurs before the boundary layer
thickness is reached. It may be thought that the separation occurs by excess
of centrifugal acceleration when the boundary speed exceeds the theoretical
speed always balancing exactly the pressure drop.
43
It may be reminded that the existence of a kinetic friction for so low
Reynolds numbers would make the separation occurring at 80°. After all
this situation has no engineering interest.
36.2 Low Reynolds numbers (10 <ℜe < 50)
The only change against the previous case, is that the speed gradient is now
above the first critical value within the boundary layer. Although this case
has no more engineering use than the previous one, the separation point was
calculated assuming that all the speed gradient is covered by the wall layer
made of HELMOLTZ' tubes; the remaining part of the boundary layer is
under dynamic friction condition. The code LAM1NAP.BAS allows for
calculating the separation point some fraction of degree after the maximum
speed of the main flow reached for θ=90°.
36.3 Karman street (50 <ℜe < 5000)
The first critical gradient is reached within the main flow as soon as ℜe=50
close to a 90° azimuth. This value is reached very fast upstream as the
speed varies with the azimuth sinus. The speed law may be considered as
the sinusoid of a lower angle (ratio among 82 and 90) with a lower module
(see curve §34).
The KARMAN-POLHAUSEN method applies. However, the speed
gradient within the boundary layer reaches the upper critical gradient value
at least within the wall layer. The code LAMNNAP.BAS using two values
of the kinetic viscosity allow for calculating the separation point in that case.
The separation occurs immediately after the speed maximum at 82°.
The wall layer should be made of rotating particles while the upper layers
are made HELMOLTZ's tubes. It may be thought that these tubes are
growing along the cylinder so that they are visible in the wake.
Another problem cannot be hidden.
From the shore, one may hear the skipper shouting "de l'eau" (that skipper
only speaks French) in order to obtain his right to pass first the buoy, just
before he plunges in the cockpit of his Star, pushing the tiller, while the sail
boom sweeps the deck.
How the upper vortex will inform the lower one that it is now its own turn
to go? And what will do this other one while waiting?
In fact, the vortices are generated from the upstream flow before the leading
point. There, within the main flow, the first critical gradient is reached, on
each side of the axis, but the sign of the gradient is the opposite on each
44
side and in addition it is respectively the opposite of the speed gradient sign
along the cylinder wall. So that the lower vortex will roll up on the cylinder
upper wall, while the lower vortex will roll down over the lower vortex then
along the lower wall of the cylinder.
36.4 Laminar flow (5 <ℜe < 2 to 300 000)
When ℜ e=5000, the speed gradient within the main flow reaches the
second critical value near the wall for a 90° azimuth. There also the critical
value is very fast reached upstream for the same reason. The maximum
speed is reached at 82°. The separation calculated with the code
LAMNNAP.BAS with the upper kinetic viscosity everywhere, occurs also
some fractions of degree after the maximum speed at 82°.
36.5 Turbulent flow (ℜe > 2 to 300 000)
The boundary layer thickness is very small. The use of a polynomial speed
profile will lead to an immediate separation just after the maximum near
90°. The reason why the profile using a speed ratio to the n remains
unexplained. Nevertheless this profile gives a separation at 120° for n=0.2,
while it is assumed that the wall layer is made of very tiny particles ( not far
from molecules) with a slipping coefficient much lower than calculated in §
15.
The Code used is CYLTU.BAS.
36.6 Turbulent transition of the flow along a plane
The upper critical gradient is reaches within the boundary layer of the flow
along a plane for:
∆u ft
2a f
=
∂u
∂ 2u
× (1 − kk f ) − a f × kk f
> 2 s −1
2
∂y
∂y
It is no more possible to neglect the term including the particles radius
because the second derivative is very large. The upper limit of the radius of
particles so that the speed gradient becomes positive and then reaches the
upper critical value, is δ/300 that is to say of the order of magnitude of 1µm.
For this value the flow becomes turbulent by destruction of the laminar
layer structure as a result of the generation of additional rotating particles
within the fluid nooks.
The transition from the laminar flow to the turbulent flow within the flow
along a plane can only involve the boundary layer because there is no
gradient at all outside. Thus it appears again that it is necessary to
determine the particle radius to obtain quantitative results.
45
40 IMPLEMENTATION TO THE MOTION OF SUPERFLUID
HELIUM IN A ROTATING CYLINDER
41.1 Within the wall layer
a) tangential relative speed
(the solid body rotation is deducted)
2a 

∆u pt = rΩ c − rΩ f − kk p ×  rΩ c − rΩ f − Vθp p 
r 

b) radial relative speed
 ∂V 
V  
∆u pr = 2a p kk p  θ  −  θ  
 ∂r  p  r  p 
41.2 Within the fluid
Values were calculated in §31.
a) tangential relative speed
 ∂V V
∆u ft = 2a f ×  θ − θ
r
 ∂r
b) radial relative speed
 ∂ 2Vθ ∂Vθ Vθ 

 2 −
×
(
1
−
kk
)
−
kk
×
a
×
+ 2 

f
f
f
∂
r
r
∂
r
r 


 ∂V V 
∆u fr = 2 × a f × kk f ×  θ − θ 
r 
 ∂r
42 Calculation of the moments
42.1 Within the wall layer
a) tangential friction on the wall side:
− τ pt × a p × ds − τ pt × rp × ds
− τ ft × a p × ds + τ ft × rp × ds
b) tangential friction on the fluid side
:
c) radial friction
2 × τ fr × a p × ds
42.2 Within the fluid
− τ ft × 2 a f × ds −
a) tangential friction
46
∂τ ft
× r × 2 a f × ds
∂r
2 × τ fr × a f × ds
b) radial friction
43 Overall moments
43.1 Overall equation
a) Within the wall layer
kp
4 ρrp a 2p
− ∆u ft a p − ∆u pt rp + ∆u ft rp − ∆u ft a p + 2 ∆u fr a p
b) Within the fluid
∂∆u ft


− ∆u ft −
r + ∆u fr 

2 a f ρr 
∂r

43.2 Replacing ∆u par their value
kf
a) Within the wall layer

kp 
ap
(
)
(
)
(
)
1
1
1
2
(
1
)
−
Ω
r
−
kk
+
V
−
kk
−
+
kk
−
kk
V
−
r
Ω
−
kk

θ
θ
c p
p
p
p
f
p
c
p 
rp
4 ρa 2p 

kp 
∂V 
2a p (1 − kk f + kk p ) θ 
2 
∂r 
4 ρa p 
b) Within the fluid (while neglecting terms including ap):
+
k f  Vθ
∂Vθ
∂ 2Vθ
(1 − kk f )
−
kk
+
kk
−
r
f
f

2
ρr  r
∂r
∂ r

43.3 Bringing friction within the general equation:
In this case, all the derivatives related to the azimuth are null and so is Vr.
a) Within the wall layer (while neglecting terms including ap):
rp Ωc − Vθ
=
2a p
×
(kk
f
+ kk p )
rp Ω c
rp 1 − kk p − 2kk f
This expression gives the angular speed of the wall layer as a function of
the angular speed of the bucket as far as both the slipping coefficient and
the fluid rotating particles are known.
b) Within the fluid):
∂Vθ k f  Vθ
∂Vθ
∂ 2Vθ
(1 − kk f )
=
−
kk
+
kk
−
r
f
f

2
∂t
ρr  r
∂r
∂ r

47
When the flow is permanent, the solid body rotation is of course a particular
solution of this equation. The general solution is obtained after the
following variable subsequent changes:
Vθ = y
y' −
1 − kk f
y
− ry "
=0
r
kk f
(type
homogène)
y = ez
z' 1
−
− cz " − cz '2 = 0 ( type sans ordre 0 )
r r2
dz
=u
dx
u 1
−
− cu ' − cu 2 = 0 ( type Riccati )
r x2
v = u − u1
where u1 is the particular solution related to y=ar (solid body rotation)
u1 =
1
r
2
v 1 1
1
1

+ 2 − 2 − c (v ' − 2 ) − c  v +  = O
r r
r
r
r

v
2cv
− cv ' − cv 2 −
=O
r
r
w=
(type Bernoulli )
1
v
dw
dv 1
=−
dr
dr v 2
w
(1 − 2c ) + cw ' − c = O
r
(type linéaire)
The solution is well-known, nevertheless the computerised solution of the
Bernoulli type is precise enough for the purpose of this report (Code
HELIUM.BAS).
48
43.4 General case
The second and third derivatives are preponderant when r is small. The
general equation writes as follows:
ra f kk f
a

∂ 3Vθ
∂ 2Vθ
(
)
−
r
(
1
−
kk
)
+
a
kk
+ kk f 1 + 2 f
f
f
f
3
2
r
∂r
∂r

a
 ∂Vθ


− kk f 1 + 2 f
r
 ∂r

 Vθ
 = 0
 r
This equation may be integrated by a threefold fourth order Runge-Kutta.
Nevertheless it is necessary to know before both the slipping coefficient
and the fluid particles' radius otherwise there is always a couple of their
values allowing to calculate a surface fall where intended.
44 Results
44.1 Reminding the experimental results
When the bucket containing superfluid helium is set in rotation, helium first
remains at rest. The measured surface curve shows that all the helium mass
contained in the bucket rotates like a solid body.
At a critical value Ωc1 = 3x10-3 rad/s with a bucket radius of 0.01m, a single
vortex appears in the bucket axis. Then for larger angular speed, an array of
vortices is formed. The photographs taken of the bucket show a vortices'
line, roughly spiral at the beginning, evolving towards disorder although the
related report (see chapter 60 References) does not state whether is natural
or the result of the vibration induced by the camera.
This situation persists up to a second critical value Ωc2 = 1012 rad/s for the
same bucket radius when the cores start to overlap.
44.2 Construing these results by kinetic friction.
No reference is made there to any assumption on the nature of liquid helium.
Only one single fluid is assumed to exist, but this fluid involves kinetic
friction although the relative critical gradients and both the upper and lower
kinetic viscosities should be very small.
The lower critical gradient appears at very low bucket angular speed. As the
gradient is the result of a wall effect, there is always a wall layer made of
rotating particles or molecules (atoms for helium).
49
The same occurs in capillary tubes. If the slipping coefficient is small
enough, the wall layer may absorb the whole speed gradient. This is
achieved when the kinetic viscosity is very low. Thus the viscosity has no
more influence inside the fluid outside the wall layer. The fluid appears not
to have any viscosity, mainly when the particles are atoms.
Let us come back to the bucket. As small as may be the kinetic viscosity,
the wall layer always has a tangential speed on the inner side, (in the
capillary tube this speed is the flow speed). Thus the inner fluid is always
set into rotation, and according to the equation there under this is the solid
body angular speed.
For higher angular speed, the tangential speed gradient reaches the upper
critical value. The kinetic viscosity is increased accordingly and the
slipping coefficient is lower than before. The simplified equation (Bernoulli
type) leads to a free surface curve identical to the vortex' one for kkf=0.3.
Nevertheless it shall be noticed than the surface falls sharply already for
higher value of kkf.
50
Hélium II superfluide dans un cylindre en rotation, solution simplifiée
0.00004
0.000035
vitesse en m/s
0.00003
0.000025
0.00002
0.000015
0.00001
0.000005
0
0.0000
0.0020
0.0040
0.0060
0.0080
0.0100
distance à l'axe en m
KKf=0.99
KKf=0.9
KKf=0.8
KKf=0.5
KKf=0.3
KKf négatif
As af is not known, nothing more can be added, and mainly the general
equation would not give more information. Nevertheless it could be thought that
the upper critical gradient value is reached within the fluid progressively at
smaller distance from the axis. Thus it may happen that the helium motion
comes to the KARMAN street configuration. If a Reynolds number of 1000,
fully within that configuration, then the kinetic viscosity (the upper one) may be
as small as ten to the minus 3 lower than the water kinetic viscosity. Such a
value may be found to high according to some indications mentioned by the
references, but it shall be reminded that this is not the value of the dynamic
viscosity but the kinetic viscosity, and their ratio is deeply depending upon the
slipping coefficient value.
For much larger bucket angular speed, the transition between laminar and
turbulent should appear by local particles peripheral tangential relative
speed exceeding the upper critical gradient value. However the way this
leads the vortices to overlap remains unexplained.
51
50 CODING OF THE IMPLEMENTATIONS
Note: All the codes have been developed in QUICKBASIC, including a
good editor, now improved by VISUALBASIC, and a compiler as fast as
TURBOPASCAL, although BASIC language is very poor as it does not
include any variable internal validation and is not intended for engineering
use.
51 Equations of the whirl-well flow
(CODES PTAVEVO, PTSANVO and PTFRORA.BAS)
The codes PTAVEVO and PTSANVO are identical but the boundary
conditions are dedicated to each case.
Initialisation shall be done within SUB ENTREE procedure. The options
are details in the header ( they are in French language) Two of these options
are intended for testing the trends of results according to parameters' value
when they have no impact beyond one of the two distances from the axis
where the calculation begins. The full run of the code save all the value at
both these predetermined distances so that they are automatically loaded
when the code is ran at one of those distances.
The method used is based on finite elements on a double mesh (2dR,2dθ).
The values of the variables are taken at the middle point 2 or 3 according to
the integration direction chosen once only for each code. The step is
nevertheless (dR,dθ). The value at points sup3 or sup4 in line with the
integration direction, are calculated by the secant methods in the case of the
whirl-well flows both with and without vortex. (This is not the case for the
cylinder as explained hereafter.).
'Mesh schematic
'
'
'
[sup2]
'
:
'
^
'
:
' [2I]--<---[2=4I]-----------[4]
'
:
:
' itert
I
V en cours ^ dT*R:
' :
:
:
' : [1I]------[1=3I]-----<------[3]--->---[sup3]
' :
=2K
: -dR
' :
: K
v
' :
:
:
' :
[1K]-------[3K]
'rang------------------iterr
52
(en cours = being processed)
These codes do not include any stabilisation feature. They are stable by
themselves provided the boundary conditions are defined with a sufficient
accuracy far enough from the axis. The optimum distance allowing a good
stability was obtained by testing. Any change in the boundary conditions is
a very fastidious operation. At least, the boundary pressure P3# shall be set
to 0# after any change as described in the procedure included in the SUB
ENTREE.
The code may be run out without any kinetic viscosity ( there are no mean
to include dynamic viscosity of course) nevertheless the Bernoulli method
to calculate the free surface level shall be chosen: in that case the vertical
step correction disappears as the vertical speed is set to 0#. In the other case
Bernoulli is not applicable; it is a global method and there is no reason to
think that the upper part of each step coincides with the same stream line
along the horizontal iteration. The vectorial method applied to the
equipression surfaces was installed but it does not work correctly probably
because the steps are not constant along the horizontal iterations.
During the first tests devoted to the effect of the kinetic moments alone, the
accuracy has been improved by correcting and advancing the values given
by he secant method. As the solution is not far from a polynom, this is very
efficient, but it was not necessary for the calculation related to the kinetic
friction and this feature is not operated.
As for all codes involved by this report, the results are saved on a text
format on the A: floppy disk drive at the end of the run. The related files
may be read by NOTEPAD of WINDOWS or any by other text processor
including a DOS text format importing driver. A paragraph mark or any
other separator readable by the spreadsheet used shall be added after each
value. This fastidious work may be accelerated by a macro when available,
only when the figure format and length are identical all over the file. (the
curves within this report were obtained by EXCEL4, series' name could not
be translated in English; the comments of some curves improved with
DESIGNER are translated, the same apply to drawings).
52 Equations of the flow around a cylinder
(CODE CYLINDRE.BAS, CYCLTU1.BAS)
The equations are fully unstable whatever is the accuracy of the boundary
conditions. The secant method cannot be improved as the derivatives' signs
are changing. It has been tried to improve the situation by changing the
correction sign in line with the second derivative. It was not accurate
enough so that the first derivative is used to calculated the values at point
sup3 or sup4 according to the case. The results are impaired by this feature
53
as soon as the derivative of the calculated flow differs too much from
theoretical solution. Nevertheless, only the point where the derivative sign
change is searched. It is the point where speed begins to fall.
Initialisations are made within SUB ENTREE. The step shall be taken small
enough: no more than 0.00001#m and sometimes less, for instance for ℜ
e=5000 where both kinetic viscosities have a significant impact respectively
in the main flow and in the boundary layer. The saving features are
identical to the codes thereunder. There is no choice of the distance, but a
choice in the step and the distance is changed accordingly.
53 Equations of the superfluid helium
(CODE HELIUM.BAS)
This code is based on a double implementation of a forth order RungeKutta method applied to a Bernoulli type differential equation. There are no
stability problems but some singular points. The solid body rotation is not a
solution of the Bernoulli type equation because it disappears when changing
the variable just before. Of course it remains a solution of the motion
equation.
54 Polhausen and modified Polhausen methods
(CODES POLHA50,LAMNNAP and LAM1NAP.BAS)
The code POLHA50 used the well-known Karman-Polhausen equation by a
single forth order Runge-Kutta calculating the derivative of Λ#.
The codes LAMNNAP and LAM1NAP.BAS are similar to POLHA50 but
they are based upon the equation giving the derivative of Λ# with a wall
layer speed different from 0#. The first only consider one single wall layer
below a classical viscous fluid ( dynamic viscosity) although the second
one involved a fully layered boundary layer with two possible kinetic
viscosities according to the configuration calculated and entered within
SUB ENTRY. These two codes require very small angle step ( no more
than 0.0001# degree and even less near the stating point always taken at the
maximum tangential speed azimuth where Λ=0#). Several steps are
predetermined within SUB ENTRY, but they shall be referred to within the
WHILE loop within the main program in order to be used.
54
60 REFERENCES
Auteur
titre
éditeur
L.Escande
Hydraulique générale
Privat
1947
Oudart
La couche limite
1948
Brun
Introduction à l'étude de la couche limite
Sdit
Gauthiers
Villars
IG Maillard
Cours de Mécanique des fluides
F.Halbwachs
Les fluides à spin
Le Fur
Brun,Martinot
alt
Couche limite laminaire
Mécanique des fluides
A technique for photog.vortex in rot
A.Gary and alt superfluid
Ensta
Gauthiers
Villars
année
1955
1957
1960
1962
Dunod
Jo of low
temp
1980
1980
M.Ichiyinagi
A microscopic theory of vortices in superfluid Physica
1981
A.L.Fetter
Vorices in rotating superfluid He3
Rotation of a tangle of quantized vortex line
HeII
Phys rev
Phys.
Rev.Letters
1983
Spins in deforming continuum
Mart Nijhoff
1984
C.Swanson
Guo Zhong
Hen
M.Solomaa
E. Sonin
G.Volovik
Classification of axisymmetric vortices in He3 Hels Un Tech
Rev of Mod
Vortex oscillations of rotating superfluids
phys
Rev of Mod
Quantized vortices in He3
Phys
1983
1987
1987
1987
Ph.Petitjean alt Instabilité des couches limites parois concaves 92PA066580
Résultats récents sur les fluides parfaits
P.Gérard alt
incompr
Astérisque
Jo of statis
A.J.Chorin
Vortex model with superfluid and turb
Phys
1992
Bebentec
6th order polynom for laminar boundary layer Meca appli
Ann Rev Flu
Quantized vortices and turbulence in HeII
mech
Europhys
Slow modes and pining in a vortex array He3 letters
1992
1993
1993
P.S.Bernard
Single scale surface perturbation on a cylinder Exp Fluids
Jof applied
Stability flow around a cylinder
math
Vortex dynamics and production of Reynolds Jof Fluid
st.
mech
C.Airiau alt
Stabilité linéaire des couches limites
1993
R.J.Donnelly
E.Sonin
J. Nebres
S.R.Otto
55
La reche
1992
1992
1993
1993
1993
aérosp
P.A. Durbin
H.Abarbanel
alt
Eddy viscosity transport model turbulent
flows
Vortex filament stability and boundary layer
High Re
Geropp alt
Brun
Phys of fluids
1994
1994
2D body flow with wake and wall effect
Phys rev
Zeits ange
M&M
Couche limite
CSEM
56
1994
sd
2
Physical modelling
of volume waves within
free surface liquids
June 2008
57
1 Introduction
This report is related to the calculation of the propagation speed of
volume waves within free surface liquids. Volume waves are resulting
of a change in the fluid free surface level. They are not linked to the
compressibility of the fluid.
2 Modelling of volume waves within free surface liquids
The basic model chosen by the associated engineers is made of vertical
plates. Distance between plates is the same when at rest. They are linked
at top and bottom by springs and remain parallel.
The plates, allowed an unit mass m, are moving according to the
dynamics fundamental equation :
d2 X p
= - ℜ (Xp - Xp-1 ) + ℜ (Xp+1 - Xp)
m
dt2
where Xp is the displacement of the plate number p related to its position
at
rest
xp0 = p l0
The propagation speed of waves within this model is equal to : l0 √ℜ /m ,
where l0 is the initial distance between plates, m the mass of each plate
and ℜ the stiffness of each spring.
The fluid model is obtained by replacing the springs and the mass of the
plates by an identical mass of liquid within each of the cells between the
plates. The force of the springs is replaced by the variation of the liquid
height within the cells when moving.
58
p(p-1) pp
H(p-1)
x(p-1) = (p-1)l0
displacement X(p-1)
position
pp
p(p+1)
p(p+1)
Hp
xp = p l0
Xp
H(p+1)
x(p+1) = (p+1)l0
X(p+1)
Continuity equation writes :
H0 l0 = H(p-1) (l0 + Xp - X(p-1)) = Hp (l0 + X(p+1) - Xp)
Horizontal forces acting on the plates of cell number p write:
ρ g (l0 + Xp - X(p-1)) (Hp - H(p-1))
H(p-1)
and
ρ g (l0 + X(p+1) - Xp) (H(p+1) - H p)
Hp
The equation of the fluid model with vertical plates is :
( 2 Xp - Xp-1 - Xp+1 )
1
d2 X p
= - Hp-1
2
g
dt
(l0+Xp+1-Xp) (l0+Xp-Xp+1)
This equation gives, by analogy with the spring model :
V = √gH
3 Limit of validity of the parallel plates model.
The calculation performed within the previous paragraph includes a
simplification limiting its validity. A pressure difference of pressure on
each side of the plates is resulting from the level difference between each
side of a plate. This pressure difference is assumed to be the same from
59
the top to the bottom of the plate. First, the vertical acceleration of the
liquid within cells has not been taken into account. Moreover, it has been
assumed that the pressure difference is established instantaneously over
all the height of the plates. This is only acceptable when the time needed
to establish that pressure variation could be neglected with respect to the
free level variation time scale.
In other words, The volume wave period shall be much higher than the
time needed to establish the pressure variation over the height of the
plate :
T >> H/a
where a
is the speed of sound within the
liquid.
λ >> (H/a) √gH
The associated engineers fail to understand the justification of the
relation proposed by many books where the second term is limited to H.
They cannot understand how the time needed to establish the pressure
might not play a part.
There is no condition on the liquid height. The propagation speed of
waves within free surface liquids is valid for both the ocean volume
waves and for shallowness canals.
The model is valid for ocean deepness of 5000m when the wavelength is
over 150m. This is typical of Tsunamis.
It is valid as well for channel 10m deep when the wavelength is over
10cm.
The propagation speed of volume waves is related to a critical value of
the bodies moving within fluids. This is similar to sound speed. But the
shock waves, which are compression waves, are replaced by volume
waves associated to a variation of the free surface level.
The free surface fluid flow is very similar to a wind-tunnel flow where
the body is motionless.
60
For a given water height within a channel, the volume wave propagation
speed is characterising the transition between torrential and quiet flows..
The calculation of the canal critical speed is not performed presently on
the basis of an undulatory phenomenon. The calculation is rather
performed by considering a level raising motion. Thus, there are no
vertical accelerations to be taken into account. There is only horizontal
accelerations exactly as for the calculation of the speed of volume waves.
The calculation conditions are very similar so that it is not surprising to
found the same result.
Whatever is the deepness, waves not complying with the wave length
condition are surface waves. This is the case of swell. In that case, the
calculation shall take into account both the vertical and the horizontal
accelerations.
It would be interesting to calculate the wave speed for the model
proposed by the associated engineers in the April 2008 report. The
global method proposed presently in all books could be used for this
model and will of course give the same result. But the calculation of
accelerations within each cell, within the model of the associated
engineers, is not that simple. They abandon the attempt.
4 Belarras, breaking solitary waves
As a result of the calculations hereunder, Belarras and breaking solitary
waves are volume waves, breaking when a deepness change occurs. This
is similar to a Cherenkof effect. The swell wavelength shall be over the
limit calculated by the associated engineers in relation with the deepness.
This condition involves a high wavelength. This occurs within ocean
tempest when an opposite current reinforces the swell.
As far as Basque’s Belarras are concerned, The swell is resulting from
high sea tempest in the Bay of Biscay. The deepness needed is only
reached during equinox high tides.
5 Conclusion
61
The associated engineers succeed in making the link between volume
waves in deep sea and the shallowness canal waves. They explained
why the Froude number is common to both cases and also to the
propagation of a level raising within a canal.
Additionally, they determined the validity condition of the volume
waves speed calculation. This is the transition condition between swell
and volume waves.
They determined in the meantime, the process and the condition for
swell to be a volume wave instead of a surface wave. This was the main
objective to explain the Belarras.
Nevertheless, the associated engineers give up for the last objective. The
calculation of the friction within each cell is rather intricate. Moreover it
appears that it will not be possible to compare the friction calculation
resulting from the de Gennes flows friction concept and the present
theory relying on friction between solid. This cannot give significant
results because of the wave dispersion during propagation.
62
3
Modélisation
physique
de la houle
Swell
physical
modelling
April 2008
63
1 Introduction
La surface marine, agitée en tous sens, n’est guère propice aux visions
simplifiées que permettent les outils mathématiques. La conception des
navires et des ports se contente de quelques données accumulées par
l’expérience. Si l’on excepte les Belarras, aucun besoin nouveau n’est
exprimé. La thèse des ingénieurs associés, exposée dans la présente note,
n’a donc, là encore, qu’un caractère pédagogique.
L’origine de ces réflexions vient de l’évocation de trajectoires des
particules fluides dans le cadre des théories existantes. Des spécialistes
de mécanique des fluides ne peuvent pas ne pas éprouver une forme
d’étonnement au spectacle des croisements des trajectoires des particules
fluides. Cette note résulte aussi de la nécessité de disposer d’un modèle
reposant sur la mécanique des fluides pour l’analyse du phénomène des
Belarras.
Avant de proposer une solution, les ingénieurs associés ont établi une
liste des phénomènes ondulatoires connus dans les fluides.
Cette note est la première des trois dédiées au problème de la houle. La
seconde note sera relative à la vitesse de propagation des houles dans le
cadre des simulations proposées par les ingénieurs associés. Enfin, la
troisième portera sur l’amortissement des houles. Les frottements seront
évalués à partir des équations de Navier-Stokes, puis de l’extension de la
thèse du professeur de Gennes à l’ensemble des écoulements laminaires,
conformément à leur note de 1994. Cette approche, comme on sait,
consiste à considérer que les frottements dans les fluides ne résultent pas
des vitesses différentielles entre éléments parallélépipédiques, comme
entre deux solides. Il s’agit en fait des vitesses différentielles angulaires
entre particules fluides, mises en rotation par les frottements. Les
écoulements de Gennes se limitent à la mise en rotation pariétale.
L’objectif est de tenter une nouvelle validation de l’approche
« cinétique » des frottements par opposition à l’approche dynamique
classique.
2 Les ondes dans les fluides
Il existe, curieusement, davantage de types d’ondes dans les fluides que
dans les solides, bien que les fluides n’aient pas d’équivalent des ondes
64
transversales. Une certaine striction peut se produire dans les fluides
très visqueux, mais la viscosité s’oppose à la propagation qui est, quand
même, la caractéristique commune aux phénomènes ondulatoires.
2.1 les ondes sonores
Les ondes sonores sont des ondes de pression. La propagation de ces
ondes est étudiée sur la base de la théorie cinétique des gaz. Elles
résultent de la compressibilité du fluide. Leur utilisation pratique est
particulièrement vaste. Les coups de béliers dans les conduites forcées
font évidemment partie de tous les traités d’Hydraulique. Leurs
conséquences sont essentielles dans les domaines de l’aérodynamique et
de la conception des réacteurs et des tuyères.
Elles sont modélisées dans tous les ouvrages par la propagation d’une
onde de compression dans un ressort.
Leur vitesse de propagation dans l’eau de mer s’élève à 1500m/s environ
(5400km/h). Il n’y a pas de direction de propagation privilégiée.
2.2 les ondes de gravité de surface, les houles marines
La seconde force de rappel qui permet la transmission d’ondes est la
gravité. L’oscillation qui se propage est donc verticale, mais elle se
propage horizontalement à la surface libre des fluides.
Elles n’ont pas de modélisation physique, mais des présentations
mathématiques diverses.
Elles se propagent, très approximativement, à une vitesse égale à la
racine carrée du produit de g par leur longueur d’onde.
2.3 les ondes internes en milieux stratifiés
La force de rappel conjugue une variation de densité du fluide et la
poussée d’Archimède liée à la gravité. La principale cause de
stratification est d’origine thermique. La stratification n’est pas
nécessairement verticale, mais l’oscillation est verticale. Ce type d’onde
revêt une importance toute particulière dans les installations nucléaires,
par exemple, caractérisées par des températures élevées. Les dispositifs
65
hydrauliques de sûreté ne sont pas utilisés en fonctionnement normal.
L’eau stagnante présente des gradients thermiques élevés.
2.4 les ondes capillaires
La capillarité est la quatrième et dernière force de rappel susceptible de
produire des ondes dans les fluides. Ce sont des ondes de surface. La
célérité de ces ondes diminue quand la longueur d’onde augmente.
2.5 les ondes de canal, dites de faibles profondeurs
Il ne s’agit pas à proprement parler d’un phénomène ondulatoire. Les
ondes dites de faibles profondeurs sont la propagation d’une
surélévation du niveau de l’eau en amont de nombreux phénomènes.
Les exemples vont du mascaret, lié aux marées, à la fermeture des
vannes dans les canaux.
Les ondes de canal se propagent à une vitesse égale à la racine carrée du
produit de g par la hauteur initiale de l’eau dans le canal. Cette
approximation n’est valable que si la surélévation est faible devant la
hauteur d’eau. Il s’agit d’une application directe des équations de la
mécanique des fluides incompressibles.
2.6 les ondes volumiques, les Tsunamis
Ces ondes sont considérées comme des ondes de canal et sont appelées
« ondes de faible profondeur » dans la littérature. Leur vitesse de
propagation est celle des ondes de canal. La profondeur des océans
conduit cependant les ingénieurs associés à s’interroger sur la validité
d’un concept développé pour les faibles profondeurs.
La propagation des marées dans le plateau continental est considérée
par les ingénieurs associés comme une telle onde.
2.7 les Belarras, les déferlantes solitaires
66
Ces vagues se forment dans des conditions très mal connues. Elles ne
peuvent se confondre avec les barres d’embouchure des fleuves. Le
déferlement des houles résulte, dans ce cas, de la présente de barres de
sables constituant des hauts-fonds qui provoquent le déferlement. Les
Belarras se produisent en eau profonde. La position actuelle des
ingénieurs associés est que les Belarras sont des effets Cherenkof, c’est-àdire provoquées par une modification brusque de la vitesse de
propagation des ondes volumiques. L’analogie des équations des
écoulements transsoniques dans l’air et des écoulements transcritiques
dans les canaux, et donc entre les rôles respectifs du nombre de Mach et
du nombre de Froude, constituent la motivation essentielle des
ingénieurs associés. Toutefois, les équations dans l’eau sont établies pour
de faibles profondeurs. C’est dans ce cadre que s’explique le passage des
écoulements tranquilles aux écoulements torrentiels. En sorte que la
signification du nombre de Froude pour les grandes profondeurs reste à
expliciter.
3 Les théories de la houle
De nombreux mathématiciens se sont intéressés à la houle. Ils se sont
efforcés de montrer que leurs solutions sont conformes aux équations de
la mécanique des fluides.
Le modèle le plus connu est la houle rotationnelle trochoïdale de
Gerstner. Les particules d’eau se déplacent sur des cercles fixes avec une
vitesse angulaire constante. Leur vitesse tangentielle réelle au point haut
coïncide avec la vitesse apparente de la propagation des crêtes de la
houle. Le rayon des trajectoires décroît de façon exponentielle avec la
profondeur. Enfin, à un niveau donné, les particules se déplacent sur
leur cercle avec un décalage proportionnel à la distance entre les centres
des cercles.
Ces conditions sont conformes aux équations de la mécanique des
fluides et la surface libre est bien équipotentielle.
La houle irrotationnelle de Stokes est définie par un potentiel de vitesses
conforme aux équations de la mécanique des fluides. La surface est
également équipotentielle. Les particules d’eau décrivent des cercles
exactement comme dans la solution de Gerstner. La surface libre est
67
également trochoïdale. Mais le caractère irrotationnel de la houle de
Stokes exclut la présence d’une distribution de tourbillons qui
caractérise la houle de Gerstner.
Les autres solutions sont des variantes de ces deux types de houles
mathématiques.
Pour comprendre que la notion de trajectoires des particules d’eau,
fondamentale dans ces solutions, ne peut avoir aucun rapport avec la
mécanique des fluides, il convient de prendre un exemple.
Le liquide contenu dans une bouteille pleine n’est l’objet d’aucun
mouvement pourvu qu’elle ne soit soumise qu’à des accélérations
négligeables devant les frottements. Les particules du liquide contenu
dans cette bouteille ont néanmoins une trajectoire, approximativement
décomposable en sinusoïdes, lorsqu’elle se trouve dans la poche d’une
personne qui marche. On peut lui attribuer un tangage et un roulis, plus
ou moins accentués, ce qui n’est pas nécessairement incompatible avec le
fait que la bouteille soit pleine.
Le liquide contenu dans la bouteille respecte toutes les conditions
vérifiées par les houles mathématiques. Pourtant, il est sans aucun
intérêt de penser que le comportement de ce liquide est entièrement
conforme aux équations de la mécanique des fluides. Les trajectoires des
particules du liquide contenu dans la bouteille n’ont aucun rapport avec
la mécanique des fluides.
Il en va de même pour la houle. La notion de base de la mécanique des
fluides n’est pas la trajectoire, mais le filet fluide. Les particules ne sont
pas considérées isolément, mais comme appartenant à des filets fluides.
Les équations de la mécanique des fluides ont été établies pour des filets
fluides, mais en aucun cas pour des trajectoires de particules. Il est, en
particulier, totalement exclu que des filets fluides se croisent, ce qui est
parfaitement possible pour des trajectoires. Les trajectoires relèvent de la
mécanique newtonienne, mais en aucun cas de la mécanique des fluides.
Il serait absurde de vouloir appliquer les équations de Navier-Stokes aux
mouvements relatifs des particules d’eau sur leur trajectoire respective.
Les frottements se produisent entre filets fluides et en aucune manière
entre trajectoires des particules fluides.
68
Bien sûr, la théorie cinétique des gaz utilise la notion de trajectoire. Elle
ne peut être complètement dissociée de la mécanique des fluides. Sans la
théorie cinétique des gaz, la notion de pression ne serait qu’une fiction.
Une erreur très fréquente est de penser qu’un fluide parfait n’exerce
aucune action sur un corps en translation dans le fluide. C’est le
paradoxe de d’Alembert. C’est une profonde erreur. Il n’a pas été
possible d’écrire, à ce jour, des équations de la mécanique des fluides qui
tiennent compte de la nature même de la pression. Le paradoxe de
d’Alembert résulte de cette lacune. Il faut donc s’appuyer simultanément
sur les deux approches pour comprendre que le paradoxe n’est que
mathématique. Physiquement, la vitesse de déplacement du corps dans
le fluide augmente d’autant la vitesse quadratique moyenne des
particules du fluide à l’amont et la diminue à l’aval du corps. Cette
vitesse étant la cause de la pression, celle-ci se trouve donc augmentée à
l’amont et diminuée à l’aval. Il y a donc un effet des fluides sur les corps
en mouvement dans les fluides, même en l’absence de traînée de
frottements et a fortiori de traînée d’ondes.
La conformité d’une solution, même irrotationnelle, aux équations de la
mécanique des fluides, n’est ainsi nullement une preuve de sa validité.
69
4 Les modélisations de la houle
Les phénomènes ondulatoires longitudinaux sont modélisés,
traditionnellement, par des ressorts. C’est le cas essentiellement des
ondes sonores. Cette simulation n’apporte rien en ce qui concerne la
houle. Les ingénieurs associés proposent un modèle physique utilisant
également des ressorts, comme dans la théorie de l’élasticité. Les ressorts
peuvent être associés à des amortisseurs pour l’étude des frottements, de
manière semblable aux modèles de la plasticité.
Le modèle des ingénieurs associés est constitué de plaques fines
verticales montées sur rotule en pied et reliés par des ressorts en tête.
Les plaques sont planes et indéformables. Les intervalles entre plaques
sont remplis d’eau. Dans un premier temps, le poids de l’eau est négligé.
La simulation est réalisée par le tableur Excel avec une centaine de
plaques. L’ensemble est animé par une action sinusoïdale entretenue.
L’écartement des plaques étant variable, le niveau d’eau dans chaque
cellule est donc variable.
Schéma des plaques à ressorts sur rotules
Schéma des plaques à ressorts
(hauteur d'eau entre plaques numérotées de 1 à 11)
9
8
7
6
5
4
3
2
1
0
0
2
4
6
8
10
12
14
Les « trajectoires » des particules d’eau contenues dans chaque cellule
sont bien des cercles. Vues de l’extérieur, les particules d’eau décrivent
des cercles; comme les particules contenues dans une bouteille d’eau qui
décrirait un cercle, mais les particules n’ont, au sein de leur cellule, du
point de vue de la mécanique des fluides, que des mouvements
verticaux liés à la variation du niveau dans la cellule, et en conséquence,
également, de légers mouvements transversaux. Les surfaces libres de
70
l’eau, dans chaque cellule, se répartissent, bien évidemment, sur une
trochoïde.
Houle modélisée par plaques à ressorts sur rotules
12
10
8
6
4
2
0
0
2
4
6
8
10
12
Le problème de ce modèle est la discontinuité créée par les rotules. Si les
cellules sont ouvertes en bas, il y a nécessairement des échanges avec
l’eau située en dessous. Aussi les ingénieurs associés ont essayé un
modèle semblable dans lequel les plaques de séparation sont encastrées
en pied et déformables. L’épaisseur des plaques peut-être variable du
côté de l’encastrement. Le moment transversal d’inertie peut être
modifié. La déformation de la plaque peut donc prendre des formes
variées. La simulation sur Excel permet de considérer des déformations
des plaques souples allant de la solution plane précédente aux formes de
courbes de toutes les puissances souhaitées. Les ingénieurs associés ont
essayé la forme parabolique et le profil en puissance 3. La liaison avec
l’eau sous-jacente peut ainsi être renvoyée à de très grandes profondeurs,
en tous cas sans discontinuité différentielle.
Houle modélisée par plaques à ressorts sur rotules
12
10
8
6
4
2
0
0
2
4
6
71
8
10
12
Schématisation par plaques souples à ressorts encastrées en pied (profil en y3)
10
9
8
7
6
5
4
3
2
1
0
0
20
40
60
80
100
La courbe de la surface libre est, bien entendu, toujours tracée en
fonction de la position de la surface libre de la cellule et non de sa base.
La simulation réalisée avec le tableur Excel conduit systématiquement à
une courbe de la surface libre dissymétrique. La fonction sinusoïdale
choisit met la cambrure la plus forte vers la droite. La cambrure est
d’autant plus faible que la puissance utilisée pour définir la déformation
des plaques souples est élevée.
Ce résultat provient seulement du fait que les calculs des surfaces
conduisent, indirectement, à faire la somme de déplacements fonctions
de la hauteur d’eau à la puissance choisie. Cette somme diffère
légèrement de la puissance de la somme des déplacements. La forme des
plaques n’est donc pas exactement fonction de la puissance choisie de la
hauteur. Le calcul exact n’est pas trop complexe, mais les choses ont été
laissées en l’état car on obtient des courbes équivalentes à celles qui
résulteraient d’une variation de la hauteur de l’eau au repos dans les
cellules. On peut obtenir des houles jusqu’au point critique de
déferlement par cette sorte de simulation de la remontée du fond.
4 Conclusion
Cette première note a permis aux ingénieurs associés de vérifier qu’il est
possible de modéliser physiquement la houle. Les résultats
correspondent à ceux des modèles mathématiques qui ne sont pas
72
acceptables du point de vue de la mécanique des fluides, même s’ils
suffisent parfaitement dans la pratique pour calculer les structures à la
mer et vérifier la stabilité des carènes.
L’approche proposée permet cependant le calcul de l’amortissement des
houles en appliquant les méthodes relevant strictement de la mécanique
des fluides.
Mais l’objectif prioritaire des ingénieurs associés est de déterminer la
vitesse de propagation des houles marines dans le cadre des simulations
proposées par la présente note. L’objectif évident est de rattacher les
ondes volumiques et les Tsunamis aux houles et non aux ondes de
canaux dites de faibles profondeurs.
73
1 Introduction
Although there are no need for improved development of sea swell
theories, the associated engineers propose a physical modelling.
Fluids mechanics specialists can hardly accept that trajectories of fluid
particles cross one another if it means crossing of fluid threads. On
another hand, the associated engineers need a physical model for
analysing the breaking high waves such as Belarras.
Before detailing the proposed solutions, the associated engineers issued
a list of all known wave-motions within fluids.
This report is the first of the three reports dedicated to swell. The second
one will deal with the swell propagation celerity within the frame of the
swell modelling proposed by this first report. The last one will be a
tentative validation of the angular kinetic approach of friction within
fluids as described by the 1994 report of the associated engineers. This
approach is an extension of the de Gennes model of heavy oil flow
within small pipes. The rotation of the parietal layer of the fluid within
de Gennes’ model has been extended by the associated engineers to the
full extent of laminar flows. The result will be compared to the dynamic
present approach of frictions. The dynamic approach is based upon a
concept of friction used for friction between solid bodies.
2 Waves within fluids
There are several kinds of waves within fluids:
2.1 Sound waves
Sound waves are pressure waves. Their properties are quite well
described by the kinetic theory of gases. They are a result of the fluid
compressibility. They are used in many areas from liquid speed
measuring to aeroplanes design.
The model used for such pressure waves is made of springs.
The celerity of sound waves within sea water is about 1500m/s
(5400km/h). There is no preferred propagation direction within fluids.
74
2.2 surface gravity waves: sea swell
The second back action enabling wave propagation is gravity. The
oscillation is vertical but it propagates horizontally at the free surface of
the fluids.
Such waves have presently only mathematics description but no
physical modelling.
Their celerity is about the square root of the product of g by the wave
length.
2.3 Internal waves within stratified media
The back action is including both the fluid density gradients together
with gravity. Thermal gradients are the main causes of stratification.
Stratification may not be vertical but the oscillation occurs always
vertically. Safety devices of nuclear power plants with high temperature
gradients within motionless water are one of the main areas where such
waves occur.
2.4 capillary waves
Capillarity is the forth and last back action enabling generation of waves
within fluids. They are surface waves. Their celerity is decreasing for
increasing wave lengths.
2.5 canal waves also known as shallowness waves
Those waves are not properly undulatory. They involve propagation of
heightening of the water level in canals. They occur when closing water
gates as well as tide water level changes in many estuaries.
Their celerity is the square root of the product of g by the depth. The
ratio of the actual speed in canals and that celerity is the Froude’s
number. The Froude number plays the same part for water canals as
Mach number for nozzles.
2.6 volume waves, Tsunamis
75
Those waves are considered as canal waves. Their speed is
approximatively the same as shallowness or canal waves. However as
they occur within Oceans deeper than 5000m as well as in the
continental shelf about 200m deep, the associated engineers are
questioning the validity of such an interpretation.
The associated engineers report dated March 2008 states that the tidal
waves within the continental shelf are such volume waves when tide
within ocean deeps are an overall upheaval of both the soil and the sea
occurring in the meantime in every point of meridians.
2.7 Belarras and breaking solitary waves
Generation conditions of such waves are not clearly identified. Belarras
occur only during Spring high equinox tides, about four nautical milles
offshore the French Basque coast. They are not linked to any estuary
tidal bore.
The position of the associated engineers is presently that such waves are
Cherenkof effects occurring when the celerity of waves is sharply
changed by sea depth upheaval.
3 Swell theories
There are several mathematical solutions for swell. They comply with
the fluid mechanics equations.
The most used is the Gerstner’s rotational trochoïdal swell. Water
particles described motionless circles with a constant angular speed. The
tangential speed at the upper part of the circle is the apparent swell
speed. Circles radius is decreasing exponentially with depth. For a given
level, particles are moving along their circle out of phase with an angle
proportional to the distance between the centres of circles.
Those conditions are complying with the fluid mechanics equations and
the water free surface is equipotential.
The Stokes’ irrotationnal swell is defined by a speed potential complying
also with the fluid mechanics equations. The water free surface is also
equipotential. Water particles moves along circle as for the Gerstner’s
76
swell and the free water surface is trochoïdal as well. But there is no
vortex distribution within water as for Gerstner’s swell.
Other solutions are said to be variant of both those solutions (in fact the
associated engineers have no knowledge about those variants).
In order to explain that the water particle trajectories have not the
slightest relation with fluid mechanics, the best is to give an example.
The liquid contained within a bottle, fully filled in, has no motion
involving the implementation of fluid mechanics equations as long as
the bottle is handled carefully so that acceleration effects remain very
small against frictions inside the liquid. Nevertheless, the liquid particles
do move along trajectories when the bottle is slipped into the pocket of
somebody walking along. This fellow may have both rolling and
pitching motions. The trajectories of the liquid particles in the bottle are
the same as the bottle trajectory. Their motion may be split in sinusoids.
All conditions enabling for implementing the fluid mechanics equations
are fully coped with. The free surface of the liquid in the bottle is
equipotential. However those motions have not the slightest relation
with fluid mechanics.
The situation is exactly the same for fluid particles within swell. The
basic concept of fluid mechanics is not the trajectories but the fluid
threads. Particles are not considered as material isolated point as for
Newtonian mechanics but as belonging to these fluid threads. Fluid
mechanics equations have been issued for such fluid threads. Mainly, it
would be nonsense to implement the Navier-Stokes equations to
calculate the friction within swell using the speed of fluid particles along
their trajectories. These equations are definitely only valid for fluid
threads. Frictions occur between fluid threads and not between
trajectories.
Of course, trajectories play a main part within the kinetic theory of gases.
This theory cannot be dissociated from fluids mechanics. Fluid pressure
would be a pure fiction if not considered through the concepts of the
kinetic theory of gases. Such a distinction leads to one of the most
frequent errors when studying fluids. The perfect fluid equations give a
null effect of the fluid on a body moving within the fluid. This is known
77
as the d’Alembert paradox. But it is only a mathematical paradox. In fact,
the effect of a perfect fluid on a moving body shall be obtained through
the implementation of the kinetic theory of gases. The mean quadratic
speed of the fluid molecules is increased by the speed of the body so that
the upstream pressure on the body is, in turn, increased, and conversely
decreased downstream. So that a fluid has always an effect on moving
bodies even when not taking into account the friction drag or the wave
drag.
Finally, a mathematical solution, should it be irrotational, coping with
fluid mechanics equations, is not necessarily a fluid mechanics solution.
4 Swell modelling (for figures and curves, please refer to the French text)
Longitudinal Waves are traditionally modelled by springs. This is
mainly the case for sound. This is not applicable to swell. Swell is
characterised by vertical motions of water.
The associated engineers propose a physical model based upon thin
vertical plates linked by perpendicular springs at their upper part and
mounted on axis at their lower part. Plates are flat and indeformable.
They initially form a series of identical cells filled in with water. Within
this report the weight of water is neglected. It will be taken into account
by the next report, in place of springs, to calculate the speed of the swell.
Dashpots may be installed, parallel to the springs, to simulated friction
losses in a similar way to plasticity modelling.
The cells are actuated by a sinusoidal action. The distance between
plates is not constant so that the level of the water in the cells is changing.
The trajectories of water particles are of course circles. They seem to
describe circle exactly as would particles inside a bottle when the bottle
is describing a circle, but within each cell, they have only a small vertical
motion and a related horizontal motion linked to the motion of the level
of the water in the cells. The surfaces of the cells altogether form, of
course, a trochoïd.
This model, as the next ones, has been simulated with Excel. Calculation
sheets, including formulas, are attached to this report.
78
This model cannot prevent water motion at the bottom of cells when the
bottoms are opened. There is a discontinuity at the axis level.
This is the reason why the associated engineers propose a better model.
The axes of the plates are replaced by an embedded part. The inertia
module of the plate is variable so that any kind of curve may be obtained
by changing the law of deformation of the plate under the action of the
springs. The associated engineers tested this model with parabolic and
power 3 curves.
The free surface of the water within the cells altogether described an
unsymmetrical curve. This is a result of the calculation method.
Distances between plates have been added to determine the top width of
each cell. As the displacement is a power of the height of the water in the
cells, the total calculated displacement exceeds the exact displacement.
Although this would not have been too difficult to be changed, it has
been left as it is because it gives a very simple way to consider the effect
of a raising of the bottom part of the cells. This occurs when the swell
reaches the seashore. It has been possible to obtain curves up to the
critical point where the swell breaks.
4 Conclusion
This first report enables the associated engineers to check that it is
possible to model the swell with physical devices. The results are similar
to those obtained with purely mathematics models.
This physical approach allows for calculating the damping out of the
swell by friction using methods fully complying with fluid mechanics
principles.
But the main objective of the associated engineers is to determine the
celerity of swell using the models they propose within this report. Their
objective is to show that volume waves as well as Tsunamis are relevant
to swell and in no case of canal waves called in the literature
shallowness waves. The Froude’s number has no meaning within deep
sea, because torrential flows are irrelevant.
79
4
Ocean
Tides
March 2008
This report is public. Its use as a whole or the use of any part thereof cannot involve any subsequent right.
Ce rapport n'est couvert d'aucune protection. Son utilisation, en tout ou partie, ne peut générer aucun droit.
80
1 Introduction
Several theories of ocean tides may be found in Internet. As it seems to
be a problem involving Fluid Mechanics, the associated engineers
decided to look at it.
Several theories are based upon the idea that tides are very large waves
generated in the equator zone.
In fact, it appears that it is not the case. The main part of the tide
phenomenon is a result of the Newtonian mechanics principles. There is
an overall swelling of the sea level from the Equator up to 50°N and S
latitudes. Tides are waves only within the continental shelves where
deepness is sharply smaller.
2. Generating accelerations
There are two generating accelerations: the Moon gravitational effect
and the Sun gravitational effect. The overall effect of planets is 10 000
times weaker than the Sun effect in the most favourable conditions.
The inertia of the Earth generates centrifugal accelerations as a reaction
to the generating accelerations.
As a consequence of the Newtonian principles, the centre of the Earth is
in weightlessness condition within its rotations around the gravity
centres of the Sun and Earth and of the Earth and Moon yokes.
The rotation of the Earth centre around these centres of gravity does not
imply any angular momentum. There are no links between the Earth
centre motion and the Earth angular momentum.
The Moon is an exception. The Moon does have an angular momentum
linked to its revolution around the Earth. But this is, in no case, a result
of the Newtonian principles. This situation may be a result of the
solidification of a gravitational buckling. This is an indirect consequence
of gravitation.
Thus, the centrifugal acceleration balancing the Moon gravitation at the
centre of the Earth is the same everywhere on the Earth.
81
The same applies to the Sun. There is not any mechanical reason for the
Earth to have an angular momentum, bound to its rotation around the
Sun.
Ocean tides are a result of the vectorial combination of the Sun and
Moon gravitation at the surface of the Earth with the related centrifugal
accelerations of the centre of the Earth within its revolutions around the
gravity centre of both yokes made of the Sun and the Earth of the Earth
and the Moon.
Calculation shows that the centrifugal accelerations shall not be
calculated from the distance of the surface of the Earth to the gravity
centres of the yokes. Such a distance will multiply by 100 the effect of the
Moon. The effect of the Sun would then be negligible. The change will
not affect mainly the time of high tides but their height. In the case of
Brest (France), taken as reference within this paper, there will be only
one maximum per month instead of two as it is the case for the months
used in the calculation (January, March and June 2008).
The main result is, of course, well known. There are two cycles per day.
Gravitation of the Sun or of the Moon is larger on these asters side than
in the centre of the Earth, and lower the other side of the Earth. But the
very same centrifugal accelerations are subtracted, in each case, from the
gravitation acceleration. The resulting acceleration is positive on the
attracting aster side and negative on the other side. The positive
acceleration is toward that aster, the negative is in the opposite direction.
Thus there is a bulging each side of the Earth.
The soil of the Earth is submitted to the same accelerations.
3. The tide mechanism
The time schedule of tides is mainly a result of the position of the Moon
and of its crossing of the mean meridian of the Atlantic Ocean.
The first hypothesis involved a swelling of the sea in the equatorial zone.
The highest swelling would have occurred at latitude corresponding to
the Moon declination. It would have been modified by the Sun effect
82
according to its own position. The swelling would have generated a
large wave propagating within the Atlantic Ocean. This wave should
propagate with a speed of about 800km/h up to the continental shelves.
Such a scenario is fully incompatible with the time schedule of high tides
in Brest (France). The volume wave Froude’s speed would be either over
1500 km/h, or below 300 km/h. These speed are the volume waves speed
for sea deepness of 20 000m and 800m respectively. But, the mean
deepness of the Atlantic Ocean is 5000m. This is not consistent. These
mean speeds do not take into account the continental shelf, included in a
separated calculation.
In addition, the declination variations of both the Moon and the Sun
generate changes in the distance to be covered by the tide waves. They
account for one third of the total distance from the Equator to Brest. The
result is fully inconsistent with the Brest time schedule of high tides.
Moon on Pacific side
Moon on Atlantic side
BREST
HEURE
calcul K=0
20,00
20,00
calcul K=1,04
15,00
15,00
10,00
10,00
5,00
5,00
0,00
0,00
0
10
20
30
0
10
20
Within the graphs hereunder (January 2008), K is linked to the propagation speed of volume wave in the sea
(Froude’s speed). K=1,04 is related to a speed of 800 km/h. For K= 0, the tide phenomenon is characterised by
an overall swelling from the equator zone up to the North of the Atlantic Ocean.
83
30
Damping had been tested. The damping includes both frictions and
diffusion of the wave. It has been determined from the Tsunamis
damping as measured for Alaska and Chile earthquakes. This had not
improved the curves.
As a consequence, oceanic tides are not waves. They are rather an
overall bulging or swelling of the ocean up to latitude as high as 50° N
and S. The speed of the swelling is about 1800km/h at the Equator. This
is the speed of the Moon with regard to the surface of the Earth. The
swelling is the same at all point for a given latitude when the moon is
passing by. But the cohesion of soil is greater than water. For low depth
(<250m), the additional swelling is only some centimetres when it is over
one meter for depth about 5000m as it is the case for oceans compared to
about 50 cm for the sea bed in both cases. This gap generates a volume
wave at the cliff edge of the continental shelf. This volume wave
propagates at the Froude’s speed through the continental shelf. The
associated engineers are in process of determining the critical conditions
for a change of sea level, such as surface waves, to generate a volume
wave, within their analysis of Belarras.
This is the only available explanation to the delay between the time
swelling is produced and the time the high tide occurs in Brest. This
delay is about an hour and a half. This is precisely the duration needed
for a Froudian wave to propagate through the continental shelf when the
sea is about 150m deep. A more precise calculation should involve the
speed of the swelling displacement.
4. Calculations
The implementation of the Newton law is the only calculation done. As
centrifugal accelerations always keep their values at the centre of the
Earth, they were not calculated. Rather they were replaced by the related
values of the gravitation at this centre.
The orders of magnitude of the resulting accelerations at the Earth
surface on the side of the attracting bodies and at the opposite side of the
Earth are well known:
resulting acceleration of the moon
84
:
1.2161E-06 m/s²
resulting acceleration of the Sun
resulting acceleration of Planets
:
:
5.0575E-07 m/s²
8.2542E-11 m/s²
(total of all planets assumed to be aligned at their shortest distance to the Earth)
Moon and Sun data were found in Internet. Tables used are valid for
Paris meridian. Linear corrections were implemented to obtain the
needed values for the mean meridian of the Atlantic Ocean (30°W). Tests
were performed to check that slightly modified data would not impair
the validity of the results within the objective of this paper.
The tables used give the values for each day. Interpolations were
performed to obtain the increment for half a day. The step of the
calculation is ten minutes. As no better data smoothing was performed,
there are small flaws in the curves when high tides occurs near 18H TU,
the time of data change in the calculation. Those calculations are not
iterative.
MS Excel worksheet was used for all calculations. They are split among
several sheets in order to easily validate each step. The time base is from
6H TU to 6H TU, so that there are no risk of error for the Sun.
Nevertheless the trigonometric calculation were performed with a
parallel base from 0 à 360°. High tides schedule obtained by calculation
is thus compensated by addition of 6H (plus 10/60 related to the starting
point of the calculation). An additional 2.155 hours is also included in
the compensation in order to take into account the angle between the
Paris Meridian and the mean Atlantic Ocean meridian. Moreover, 1.5
hour is finally added to take into account the time needed for the tide
wave to reach Brest through the continental shelf. (And last, of course, it
is necessary to subtract 24h from time to time and to jump to the next
day… and not to forget the legal time changes because the time table
found in Internet are given in legal time). These compensations are
always the same for all the high tide time schedule curves.
The heights of high tides makes use of typical French system: the tide
coefficient. Tables give this coefficient in place of height in meter as
usual. It is quite interesting because this coefficient is generally unique
for each tide all over the country.
The tide coefficients in Brest were directly calculated from the value of
the resulting accelerations through a linear transformation. This is a
rather rough approximation. Accelerations depend upon the inverse of
the square of trigonometric functions. And those functions are not the
85
same for the Sun as for the Moon. Slightly different linear
transformations were implemented for the high tides in Brest related to a
Moon position on the Atlantic side and on the Pacific side.
The calculations were performed for the months of January, March and
June 2008.
4. The curves
January 2008 data was used to validate the approach. The K parameter
indicated on the curves is related to the hypothesis of a tide wave
propagation from the Equator zone to Brest. This parameter is intended
for including various propagation speeds, that is to say to test that
solution for various deepness. This solution has been eliminated, so that
K=0 for all the curves.
The curves of January 2008 include also an « angle » option. This was a
test of the impact of the attracting aster declinations. Although the
variation of declinations are trigonometric, a linear variation was
implemented using a device intended first for the impact of the
declination on the distance to be covered by the tide waves. This does
not produce any improvement of the curves. Nevertheless the change
was maintained for all curves.
At last, the angular momentum of the Earth within its revolution around
the Sun was removed. This is the curve « sans rotation Terre » appearing
in the January 2008 tide curves. This is consistent with the same
provision made systematically for the Moon so that it has been
maintained for all curves.
The compensations detailed within the previous paragraph have been
implemented at the end of the calculations. Please note that the high tide
times and the related coefficients have been obtained « manually »
looking at the calculation results. As explained, these compensations
include both the impacts of the units and origin chosen for the
calculation and the duration of the wave propagation through the
continental shelf (1.5-hour). As also explained, there are two slightly
different linear transformations of the resulting accelerations to obtain
the tide coefficient, one for each case: the first when the tide is produced
by the Moon located on the Atlantic side, the other when the Moon is
located the other side round, about half a day later.
86
Conclusion
This paper shows that it is possible to present the ocean tide problematic
in a very simple way. This is fully convenient for pedagogic approaches.
It is an opportunity to deepen the concept of the Newtonian mechanics
while allowing for a first approach of the angular momentum
problematic which is rather more difficult to perceive.
The Ocean tides are overall swelling of the sea level from 0 to + 50° and
– 50° latitudes. This swelling is the result of both a deformation of the
Earth and of the Sea. These swellings have been calculated and
confirmed by satellite levelling. The values are about 50cm for the soil
and one meter for the Open Sea level. The tide amplitude within the
continental shelf would be negligible without any other action. This
amplitude is a result of the propagation of a volume wave generated by
the swelling on the cliff of the continental shelf. Thus, the amplitude is
independent from the dimension of the shelf. A general additional
swelling of the Open Sea is produced by the reflection of the forming
volume wave on the cliff itself. This is typically the case both in Alaska
and between Madagascar and Africa.
The tide amplitude within the continental shelf and the coast amplitude
are higher than the swelling for hydraulic reasons which could be
compared to Tsunamis. The continuity equation would give a ratio of 25.
This is an additional proof that there are no volume waves in the Open
Sea. The volume wave is generated on the cliff. The deepness involved
should be about 500m.
Then this swelling propagates as waves through the continental shelves.
The speed of these volume waves is given by Froude’s law i.e.
proportional to the square root of sea deepness. According to the mean
deepness of the continental shelf, the high tide should occur in SaintMalo (France) 2 hours later than in Brest, 10 minutes before in La
Rochelle and 30 minutes before in Saint Jean de Luz, near the Spanish
border.
In fact, this is not that easy.
Ocean tide is an intricate problem. But the replacement of the wave
propagation hypothesis by an overall swelling opens a way to explain
the rather more intricate problem of tides in the Pacific Ocean, although
87
that Ocean is divided in three vertical zones so that it could be analysed
in three steps.
Appendices (Attached to the French text):
January, March and June 2008 tide curves for Brest (France).
The first two curves of each month give the high tide for each day of the month.
The two other ones give the related French coefficients. Colour codes are
indicated on the graphs.
88
5
L’application
du principe
de Hamilton
en mécanique des fluides
Implementation
of the Hamilton’s principle
in fluid mechanics
May 2008
89
1 Introduction
Plusieurs sites Internet exposent le problème de la rotation des cyclones de
manière enfin conforme à la réalité. L’accélération de Coriolis, dont le caractère
mystérieux a le don de plaire, selon le célèbre jugement d’Erasme, ne joue, au
mieux, qu’un rôle marginal dans la rotation des cyclones, et ce d’autant plus
qu’à proximité de l’équateur son effet est nul.
La rotation des écoulements de condensation n’est pas le résultat du hasard ou
d’une quelconque « loi de la Nature ». C’est un problème d’énergie minimale
comme dans la théorie de l’élasticité, ou, si l’on préfère les visions plus
mathématiques de la physique, une conséquence directe du principe de
Hamilton.
2 Cas des fluides parfaits et visqueux (3 dimensions).
2.1 Applications du principe de Hamilton
NOTA : Dans ces équations, v est la vitesse du fluide en un point donné, p la
pression, g la gravité, ρ la masse volumique. Dans la suite µ est la viscosité.
2.1.1 Continuité
div v = 0
(1)
2.1.2 Equation d’Euler (fluide parfait) :
(v ∇) v = - 1 grad p + g
r
2.1.3 Bernoulli (le long d’un filet fluide)
1 v2 + g z + p = Cte
2
r
2.1.4 Conservation de l’énergie (Equation de Lagrange)
90
(2)
L’équation est identique à l’équation de Bernoulli, mais elle est valable dans
toute la masse du fluide.
2.1.5 Application au puits-tourbillon.
De (1) on tire :
vr = - k
r2
et de (2) :
1 ∂p
=
2 k 2 + v θ2
r5
ρ ∂r
1 ∂p
r
k ( ∂vθ + vθ )
=
r2
ρ r∂θ
(3)
∂r
(4)
r
Si l’écoulement spatial des puits est à symétrie sphérique, il n’en va pas de
même de l’écoulement tourbillon qui est axial. L’axe permet de définir le plan
principal ou équatorial de l’ensemble des deux écoulements. Le théorème de
Poincaré entraîne la formation d’un tourbillon concentré dans la zone principale
ou équatoriale ainsi définie. L’équation de Bernoulli donne :
1 ∂ p = 1 ∂vθ2 + 2 k2
ρ ∂r
2 ∂r
(5)
r5
Les équations (3) et (4) donnent :
vθ = - k’
r
L’écoulement puits-tourbillon est irrotationnel. Ce genre d’écoulement
dérive d’un potentiel.
91
f(z) = A Log(z) + i B Log (z)
où a et b sont des constantes. On prendra B = - b² pour définir un sens de
rotation
et
A = - a² pour définir le sens d’écoulement du puits par opposition à la source.
Le Hamiltonien doit être un extremum. Le principe de Hamilton s’écrit :
t2
∂H = 0
∂c
avec
H=
t
∫ dt Σmv²
1
dans l’expression de l’action, c représente le paramètre d’une famille de
solutions possibles. Dans le cas du puits tourbillon, c’est le rapport des
constantes des deux écoulements. La généralité de la démonstration
n’est pas affectée en choisissant pour intervalle de temps celui qui
correspond à une seule rotation des particules fluides dans le tourbillon,
donc la variation de θ de 0 à 2π. On obtient :
H=2π
a4+b4
b²
Le hamiltonien est représenté par la courbe ci-après. On remarque que la
solution puits intégral, sans tourbilon, est théoriquement possible, mais
hautement instable. Les équations des écoulements puits tourbillon étant
identiques pour les fluides parfaits et les fluides réels, il apparaît que la
théorie n’est nullement conforme à la réalité puisque, d’une part,
l’écoulement puits pur peut se maintenir en eau très calme et que,
d’autre part, le rapport entre les constantes du puits et du tourbillon
n’est pas de un. Ces deux faits viennent confirmer qualitativement
l’extension de la conception même des écoulements de Gennes, posant la
mise en rotation de la couche pariétale des huiles lourdes dans les
canalisations de faible diamètre, à tous les écoulements laminaires,
extension objet de la note des ingénieurs associés de 1994. L’application
de cette conception était d’ailleurs basée sur les écoulements puits
tourbillons. L’extension ne pourrait être justifiée mathématiquement ici
que par l’application du principe de Hamilton à cette conception des
92
écoulements laminaires. C’est ce que les ingénieurs associés n’ont pas
réussi à ce jour.
HAMILT ONIEN DE L'ECOULEMENT PUIT S-T OURBILLON
H = a2/b2+b2
10
9
8
7
6
5a
4
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
coefficient de rotation b/a
La pression s’annule pour la distance rc telle que :
p0 = r ( k’ - k )
r2c r4c
Dans un domaine très proche de la mécanique des fluides sur le plan
purement mathématique, l’élasticité, l’habitude est plutôt d’appliquer le
principe de l’énergie minimum, conséquence du Principe de Hamilton,
mais beaucoup plus intuitif. On obtient, bien évidemment le même
résultat.
2.2 Applications du principe de l’énergie minimum
Entre l’instant - ∞ relatif à l’écoulement à l’infini et l’instant tc où la pression
s’annule, la variation d’énergie cinétique, représentant le travail des forces
résultant de la pression, est représentée pour une particule fluide dm par :
93
δ Ec
2
2
= 1 δm ( vr + vθ )
2
2
2
δ Ec dt = 1 δm ( k +k’ ) dt
4
2
2
rc rc
Sur un cercle de rayon r :
dt = rc dθ
k’
δ Ec t =
δm 2 π 1
rc
(k
2
2
+k’ )
2
k’
rc
k est le paramètre du débit, et sa valeur ne peut donc être une condition de
l’énergie minimum. Il doit y avoir au moins une solution pour chaque valeur de
k. La dérivation doit donc être faite par rapport à k’ :
∂(δ Ec t ) =
∂k’
2
δm 2 π 1
rc
( k’
2
k’
Le minimum est obtenu pour k’ = + k .
Rc
94
2
- k )
2
rc
3 CAS DES FLUIDES A MOMENTS CINETIQUES
Les équations de la mécanique des fluides ont été établies sur la base de
plusieurs hypothèses fondamentales. Il a été principalement supposé
que les molécules, ou atomes, n’ont pas de moment cinétique. Ce n’est
pas le cas le plus général.
Il convient d’abord de modifier les lois des chocs élastiques de la théorie
cinétique des gaz, en ajoutant la transmission des moments cinétiques.
Cette transmission ne peut se faire que dans le cadre de la théorie de
l’élasticité. Les déformations élastiques doivent comporter une
dissymétrie induite par les moments cinétiques des corps lors du choc.
La surface de rencontre est gauche au lieu d’être plane. C’est la condition
de la transmission d’un moment cinétique transversal : le gauchiment
contribue, avec l’aplatissement, à empêcher le glissement. La restitution
de l’énergie de déformation suit le processus inverse. Les particules d’un
fluide, douées d’un moment cinétique, ont, dès lors, six degrés de liberté.
Dans le cadre de la théorie cinétique des gaz, les particules ne rentrent
pas dans le concept utilisé dans la mécanique des fluides, mais
s’applique aux atomes ou molécules du fluides. Le principe
d’équipartition de l’énergie permet une modification radicale des
équations fondamentales de la mécanique des fluides, qui conduit
principalement à une solution puits-tourbillon à vitesse tangentielle en
1/SQRT(R), au lieu de la forme bien connue en 1/R. Si, malheureusement,
l’écoulement n’est plus irrotationnel, il peut trouver des applications
intéressantes. Le caractère rotationnel n’est justement pas sans
remarquables conséquences comme on le verra dans le dernier
paragraphe.
Chaque particule du fluide est donc supposée avoir 6 degrés de liberté
3.1 Continuité
div v = 0
(1)
qui donne comme dans les fluides parfaits sans moments cinétiques :
95
vr = - k
2
r
Cette relation est la loi des flux fluides. Elle exprime qu’en l’absence de
condensations et d’évaporations internes autres qu’au centre du puits, le flux
massique se conserve. Tant que le fluide peut-être considéré comme
incompressible, le flux volumique se conserve également.
3.2 Théorème du moment cinétique.
Le théorème du moment cinétique pour un élément de volume dv = rdθ ds avec
ds = dr dz, s’écrit :
−
∂p
d ( IΩ) d (Σiω )
r∂θ × ds × r − Mt ( Forces de frottement ) =
+
r∂θ
dt
dt
I étant le moment d’inertie de l’élément de volume par rapport à l’axe
perpendiculaire au plan de l’écoulement passant par l’origine des coordonnées
cylindriques, Ω la vitesse angulaire locale de l’écoulement plan.
i est le moment d’inertie des composants du fluide par rapport à l’axe
perpendiculaire au plan de l’écoulement passant par leur centre de gravité et ω
la vitesse de rotation des composants de l’élément de volume autour de cet axe,
le produit de ces deux quantités étant sommés dans l’élément de volume.
Le moment cinétique principal peut se mettre sous une forme plus habituelle :
d ( IΩ)
=
dt
avec :
r
d ( ρdvr 2
dθ
)
dt
dt
dθ
= Vθ
dt
96
On peut donc écrire :
∂p
r∂θ × ds × r − Mt ( Fft ) =
+
r∂θ
dt
dt
dV  d (Σiω )
∂p
 dr
−
dt 
dt
r∂θ
 dt
∂p
dV  d (Σiω )

−
r∂θ
dt 
dt

∂V
∂V
1 d ( Σ iω )
−
−
=
+ Vr θ + Vθ θ +
r
r∂θ ρrdv dt
ρ r∂θ
ρrdv
∂r
−
Les frottements seront négligés dans la suite. Dans le cas du puits-tourbillon
cette dernière équation devient :
0=
VrVθ
∂V
1 d (Σiω )
+ Vr θ +
r
∂r ρrdv dt
3.3 Principe d’équipartition.
L’équipartition de l’énergie dans le fluide a pour conséquence que la moyenne
des moments cinétiques des particules de fluide n’est pas nulle. Dans les fluides
à 6 degrés de liberté, l’énergie cinétique de rotation qui apparaît en raison de la
rotation du fluide doit être compensée par un apport d’énergie qui ne peut
provenir que des particules elles-mêmes. En l’absence de toute force extérieure,
l’équipartition de l’énergie se ramène à l’égalité des variations instantanées des
moments cinétiques :
rρdv
∂Vθ dr d (Σiω )
=
∂r dt
dt
soit :
Vr
∂Vθ
1 d (Σiω )
=
∂r
ρrdv dt
L’équation du § 3.2 devient :
97
0=
VrVθ
∂V
+ 2Vr θ
r
∂r
et donc :
Vθ = - k’
toujours avec Vr = - k
√r
r2
La pression s’annule pour la distance rc telle que :
p0 = ρ ( k’ - k )
4
rc rc
Il faut noter que si l’écoulement puits lui-même est toujours irrotationnel,
l’écoulement tangentiel, le tourbillon, ne l’est pas. Les frottements
devraient donc être pris en compte dans les équations. La solution
exposée n’est donc valable que pour les vitesses tangentielles
suffisamment faibles par rapport à la vitesse quadratique moyenne
d’agitation des composants du fluide, c’est-à-dire par rapport à la
célérité des ondes dans le fluide.
Les fluides à moments cinétiques sont composés de particules dotées d’un
moment cinétique brownien dont l’énergie de rotation correspondante est égale
à l’énergie cinétique d’agitation. Dans ces fluides, la notion de température est
exclue, et, puisque les frottements doivent être considérés comme négligeables
aux vitesses extrêmement faibles envisagées, les échanges d’énergie que
nécessitent les déplacements forcés se produisent par pompage ou apport
d’énergie cinétique de rotation à l’énergie cinétique de rotation brownienne.
3.4 Application du principe de l’énergie minimum
Entre l’instant - ∞ relatif à l’écoulement à l’infini et l’instant t, la variation
d’énergie cinétique, représentant le travail des forces résultant de la pression, est
représentée pour une particule fluide dm par :
98
δ Ec
2
δm ( Vr
=
2
+ Vθ )
2
2
δ Ec dt = δm ( k + k’ ) dt
4
r’ c
δ Ec t =
avec dt = rc √ rc dθ
k’
r’c
δm 2 π √ r’ ( k2 + k’2 )
c
3
k’
r’ c
k est le paramètre du débit, et sa valeur ne peut donc être une condition de
l’énergie minimum. Il doit y avoir au moins une solution pour chaque valeur de
k. La dérivation doit donc être faite par rapport à k’ :
∂(δ Ec t ) =
δm
π √r’c
2
∂k’
2
( 2 k’
2
- k )
3
k’
r’ c
Le minimum est obtenu pour
k’
k
= _______
r’c √r’c
Il n’existe donc qu’un seul écoulement puits-tourbillon conforme au principe de
l’énergie minimale, mais il faut noter que cet écoulement dépend de rc , et donc
de la manière dont s’établit le débit du puits, représenté par k. En particulier,
l’écoulement puits seul est exclu n’étant pas stable.
3.5 Transfert d’énergie
Il faut noter qu’une particule fluide située à une distance r du puits, dans un
fluide parfait sans moments cinétiques, est animée de deux mouvements de
99
rotation opposés : le premier dû à la rotation d’ensemble dans le tourbillon, le
second dû à la variation de la vitesse tangentielle du tourbillon entre les
extrémités de la particule fluide la plus proche et la plus éloignée du puits.
Or, dans le puits-tourbillon en fluide parfait, les vitesses angulaires de ces deux
rotations sont égales et opposées :
ω1 = Vθ
=
k’
ω’1 = 1∂Vθ dr /dr
2
r
2 ∂r
r
2
= - k’
2
r
Il n’y a donc aucune rotation des particules du fluide. Toutefois, il en résulte une
rotation différentielle de particules du fluide qui ne correspond pas au modèle
classique puisque le Laplacien est nul et qu’en conséquence il n’y a pas de
frottements au sens classique. Cet écoulement est dit irrotationnel. Par contre, il
existe des frottements qualifiés de cinétiques, car provenant de rotations
relatives qui permettent d’ailleurs d’expliquer l’anomalie liée au soulèvement de
la surface libre près de la bonde par rapport à la surface théorique obtenue en ne
considérant pas les frottements cinétiques, donc dans ce cas, en l’absence de
tout frottement. On peut noter au passage, que cette approche « cinétique » des
frottements dans les fluides permet également d’expliquer le comportement des
fluides dans leur écoulement autour d’un cylindre et, beaucoup plus
fondamentalement, le comportement de l’hélium superfluide dans un cylindre
en rotation sans faire appel, en aucune manière, à des considérations
probabilistes, entièrement étrangères à la mécanique des fluides, et de résoudre
les paradoxes du bi-fluide de Landau. Ce qui résulte, aussi bien, du concept de
mise en rotation pariétale des écoulements de Gennes, étendu à tous les
écoulements laminaires.
Par contre, dans l’écoulement puits-tourbillon en fluide à moments cinétiques, la
vitesse angulaire de la rotation d’ensemble est l’opposé du double de celle de la
rotation due à la variation de vitesse tangentielle :
ω2 = Vθ
r
=
k’
r√r
2 ∂r
100
2
=- 1
k’
2 r √r
Il y a donc bien transfert d’énergie. Il en a été tenu compte dans les équations.
Dans le cas des fluides à moments cinétiques, on peut avoir l’ordre de grandeur
de l’énergie pompée dans l’énergie cinétique de rotation brownienne du fluide,
en intégrant l’énergie cinétique de rotation de l’écart entre les deux vitesses de
rotation ci-dessus depuis l’infini jusqu’à la distance R du puits :
R
R
2
2 3
∫ Σ ½ i ω2 /4 = ∫ ½ (2 π r dr dz λ) i (1/4) k’ /r
∞
2
= (π/4) λ dz i k’ /R
∞
λ étant le nombre de particules du fluide par unité de volume, i le moment
d’inertie de ces particules et dz la hauteur de la zone considérée
4 Conclusion
L’approche des ingénieurs associés pour les fluides réels, par opposition
aux fluides parfaits, ne serait pleinement validée que par la
détermination du Hamiltonien des écoulements à frottements cinétiques
c’est-à-dire par rotations différentielles des particules du fluide selon le
concept des écoulements de Gennes et non par déplacements
différentiels selon le modèle actuel emprunté à la mécanique des solides.
Quant aux fluides à moments cinétiques internes, c’est un tout autre
problème qui n’a été ajouté à la présente note que pour être formalisé.
101
1 Introduction
Condensation or well flows rotation are not occurring as a result of some “law
of Nature” or by chance. It is a direct consequence of Hamilton’s principle. It
can be considered as well as an implementation of the minimum energy as used
within Elasticity theory.
2 Perfect and viscous fluids (3D).
2.1 Hamilton’s principle implementation
Note : within the following equations, v is the speed of the fluid at a given point,
p the pressure, g gravity, ρ volume mass. µ is the fluid viscosity.
2.1.1 Continuity
div v = 0
(1)
2.1.2 Euler’s equation (perfect fluid):
(v ∇) v = - 1 grad p + g
(2)
r
2.1.3 Bernoulli (along a fluid thread)
2
1 v + g z + p = Cte
2
r
2.1.4 Energy Conservation (Lagrange’s Equation)
This equation is the same as Bernoulli, but it is valid within the whole fluid.
2.1.5 Implementation to the whirl well flow.
From (1) it comes:
vr = - k
2
r
102
and from (2):
1 ∂p
2
2 k + vθ
=
5
ρ ∂r
1 ∂p
r
(3)
r
k ( ∂vθ + vθ )
=
2
ρ r∂θ
2
r
∂r
(4)
r
The 3D well flows have a spherical symmetry although the whirl flows are axial.
The axis defines the main or equatorial plane of the combined flow. The
Poincare’s theorem entails the whirl to concentrate in the main plan of the
combined flow. Bernoulli’s equation gives:
2
1 ∂ p = 1 ∂vθ + 2 k
ρ ∂r
2 ∂r
2
(5)
5
r
Equations (3) et (4) give:
vθ = - k’
r
The whirl-well flow is irrotationnal. As the Laplacian is null, viscous
fluids are the same as for perfect fluids. The flow is deriving from a
potential.
f(z) = A Log(z) + i B Log (z)
where A and B are constants. It may be assumed that B = - b² in order to define
the rotation direction and A = - a² for the condensation flow.
The Hamiltonian shall be an extremum. Hamilton’s principle writes:
103
WHIRL-WELL FLOWS HAMILTONIAN
H = a2/b2+b2
10
9
8
7
6
5a
4
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
rotation coefficient b/a
t2
∂H = 0
avec
∂c
H=
t
∫ dt Σmv²
1
Within the “action”, c is the parameter of the possible flows family. For
Whirl-well combine flows, it is the ratio of the constants. The
demonstration remains fully general when assuming the time interval to
be limited to one turn i.e. for θ increasing from 0 to 2 π. Then it comes:
H=2π
a4+b4
b²
The Hamiltonian variation is given by the curve hereafter. It appears
that the sole well flow is highly unstable. Viscous and perfect flows are
identical in that case. Thus the theoretical approach is not complying
with reality, for a pure well may occur within initially motionless water.
Moreover, the theoretical ratio is not complying with observation. This is
qualitatively confirming the extension of de Gennes’ flows involving a
rotation of the parietal layer of heavy oil flows within small pipes to all
laminar flows as proposed by the associated engineers report dated 1994.
This can be confirmed fully only by implementing the Hamilton’s
principle to the associated engineers’ approach of kinetic angular friction
within laminar flows. The associated engineers failed to obtain such a
proof up to now.
104
The pressure is null for the value of rc giving:
p0 = r ( k’ - k )
2
4
rc rc
Instead of implementing the Hamilton’s principle, the intuitive principle
of minimum energy, well known in Elasticity theory, may be used as
well.
2.2 Minimum energy principle implementation
Between t = - ∞ at an infinite distance and t = tc where the pressure is null,
the kinetic energy variation is equal to the work of pressure forces for a fluid
particle dm:
δ Ec
2
2
= 1 δm ( vr + vθ )
2
2
2
δ Ec dt = 1 δm ( k +k’ ) dt
4
2
2
rc rc
Along a circle with r radius:
dt = rc dθ
k’
δ Ec t =
δm 2 π 1
rc
(k
2
2
+k’ )
2
k’
rc
k is the flow parameter and cannot be a condition of the minimum energy.
There shall be only one very single value for each value of K. The derivation
shall be perform on k’:
∂(δ Ec t ) =
∂k’
2
δm 2 π 1
rc
( k’
2
- k )
2
2
k’
rc
The minimum is obtained forr k’ = + k .
105
Rc
3 FLUIDS WITH ANGULAR MOMENTUM OF PARTICLES
Fluids mechanics equations have been issued with some fundamental
hypotheses. Mainly, it is assumed that the atoms or molecules of the
fluid don’t own any angular momentum. The general case shall include
such an angular momentum although this is definitely not the case
neither for perfect fluids nor for viscous fluids nor for superfluids.
The first point is to change the elastic impact laws. The angular
momentum shall be added. The angular momentum exchange is only
possible within the elasticity theory. Elastic buckles are not symmetrical.
This is a result of the angular momentum. The impact zone is then
buckled. The buckling combine with the flattening is preventing the
fluid particles to slip on one another. Then the fluid particles have six
degrees of freedom. The equipartition principle applies. The fluid
mechanics equations are dramatically changes accordingly. The
tangential speed law is 1/SQRT(R), instead of 1/R. Unfortunately the
flow is no more irrotationnal. This is involving in fact some interesting
consequences as underlined within the last paragraph.
3.1 Continuity
div v = 0
(1)
the relation gives the same result as for perfect fluids:
vr = - k
2
r
This is, of course, the fluxes law.
3.2 Angular momentum theorem
The angular momentum theorem writes for a fluid volume element dv = rdθ ds
with ds = dr dz:
106
−
∂p
d ( IΩ) d (Σiω )
+
r∂θ × ds × r − Mt ( friction forces) =
r∂θ
dt
dt
I is the inertia angular momentum of the fluid volume element related to the
axis perpendicular to the flow and including the cylindrical co-ordinates origin;
Ω is local angular speed of the plane flow; i is the inertia angular momentum
of the fluid particles related to the axis perpendicular to the flow plane and
including the gravity centre of the particles and ω the revolution speed of the
fluid particles around this axis. The product of those two quantities is summed
up within the fluid volume element.
The main angular momentum may writes:
d ( ρdvr 2
d ( IΩ)
=
dt
with :
dθ
r
= Vθ
dt
dθ
)
dt
dt
So that it may be written now:
∂p
r∂θ × ds × r − Mt ( Fft ) =
+
r∂θ
dt
dt
dV  d (Σiω )
∂p
 dr
−
r∂θ
dt 
dt
 dt
∂p
dV  d (Σiω )

−
r∂θ
dt 
dt

∂V
∂V
1 d ( Σ iω )
−
−
=
+ Vr θ + Vθ θ +
r
r∂θ ρrdv dt
ρ r∂θ
ρrdv
∂r
−
Friction are nor taken into account hereafter. In this case, the whirl well flow
equation hereunder becomes:
0=
∂V
VrVθ
1 d (Σiω )
+ Vr θ +
r
∂r ρrdv dt
3.3 Equipartition principle.
107
As a consequence of energy equipartition within the fluid, the average value of
angular momentum of the fluid particles is not null. Within fluids with 6
degrees of freedom, the kinetic angular energy involved by the fluid rotation
shall be balanced by an energy contribution. The only possible contributors are
the particles own angular momentum. The energy equipartition is equivalent to
the equality:
rρdv
∂Vθ dr d (Σiω )
=
∂r dt
dt
then:
Vr
∂Vθ
1 d (Σiω )
=
∂r
ρrdv dt
3.2 paragraph equation becomes :
0=
VrVθ
∂V
+ 2Vr θ
r
∂r
then:
Vθ = - k’
together with Vr = - k
r2
√r
The pressure is null for rc complying with :
p0 = ρ ( k’ - k )
rc
4
rc
Although the well flow remains irrotationnal, the tangential flow, i.e. the
whirl is not irrotational. Friction should be taken into account. The
solution hereunder is only valid for very slow flows. The fluid speed
shall remain very small compared to the quadratic mean Brownian
speed.
3.4 Minimum energy principle implementation
Between t = - ∞ and t, the kinetic energy variation is equal to the pressure force
work for a fluid particle dm:
108
δ Ec
2
δm ( Vr
=
2
+ Vθ )
2
2
δ Ec dt = δm ( k + k’ ) dt
4
k’
r’c
r’ c
δ Ec t =
with dt = rc √ rc dθ
δm 2 π √ r’ ( k2 + k’2 )
c
3
k’
r’ c
k is the flow parameter; its value cannot be a condition of the minimum energy.
It shall exist at least one solution for each value of k. Derivation shall be
performed on k’:
∂(δ Ec t ) =
∂k’
δm
π √r’c
2
( 2 k’
2
- k )
2
3
k’
r’ c
The minimum is obtained for
k
k’ = _______
r’c √r’c
There is only one whirl-well flow complying with the minimum energy
principle. This flow is depending upon rc, thus on how the well flow is
established.
3.5 Energy transfert
Within perfect fluid, a fluid particle located at the distance r from the centre of
the well has to opposite motions: the first one is linked to the overall rotation of
the fluid within the whirl, the second one is a result of the tangential speed
variation within the whirl. In perfect fluid, these two angular speed are equal.
ω1 = Vθ
= k’
109
= - k’
2
r
2 ∂r
r
2
2
r
So that fluid particles have no rotation at all. This is why the flow is irrotational.
The discrepancy between the theoretical surface curve and the experimental
curve shows that the conventional friction approach is not correct as it is not
complying with experiment. This was the reason for proposing a new approach
taking into account the angular motion of fluid particles. The very same
approach deriving from the de Genness flows of heavy oil in small pipes, allows
not only to explain the anomaly of laminar flows around cylinder but also the
behaviour of superfluid within rotating bucket without any reference to any kind
of probabilistic approach fully contradictory with fluid mechanics theory and to
explain the paradoxes resulting from the Landau’s bi-fluid. This would results
as well directly from the de Gennes’ flows approach.
This rationale is only applicable when there are no angular momentum within
the fluid. In the case there are such angular momentum, the angular speed of the
overall flow is the double of the rotation involved by the tangential speed
rotation within whirl-well flows, and I the opposite direction:
ω2 = Vθ
=
r
k’
r√r
2 ∂r
2
=- 1
k’
2 r √r
So that there is an energy transfer needed. This is taken into account within the
equations.
The energy pumped out from the Brownian angular speed energy my be
calculated:
R
R
2
2 3
∫ Σ ½ i ω2 /4 = ∫ ½ (2 π r dr dz λ) i (1/4) k’ /r
∞
2
= (π/4) λ dz i k’ /R
∞
λ is the number of fluid particles in the volume element, i the inertia
momentum of those particles and dz the height of the zone involved.
110
4 Conclusion
Full validation of the associated engineers approach, extending the de
Gennes’ flows, involving fluid friction within laminar flows based upon
differential angular motion friction instead of longitudinal differential
motion as for solids, requires the calculation of the related Hamiltonian.
Implementation of Hamilton’s principle to fluids with internal angular
momentum has been added to this paper for information only.
111
6
La polarisation
de la couche
coronale K
du Soleil
dans le plan
de la Galaxie
Polarization
of the K coronal layer
of the Sun
within the Galaxy plane
February 1997
Le présent rapport n’est couvert d’aucune protection. Son utilisation en tout ou partie ne peut générer aucun droit.
112
Sommaire.
00 INTRODUCTION
01 Objectifs
02 Principes
10 DISPOSITIF EXPERIMENTAL
11 Le tube à diaphragmes instrumenté
12 Le viseur solaire
20 RESULTATS
21 Mesures
22 Exploitation
30 CONSEQUENCES
Contents
00 INTRODUCTION
01 Objectives
02 Principles
10 EXPERIMENTAL DEVICE
11 The intrumented diaphgram tube
12 The solar sight
20 RESULTATS
21 Measurements
22 Analyses
30 CONSEQUENCES
113
Sommaire:
On observe, dans de nombreuses circonstances, une polarisation de la lumière plus ou
moins complète. La polarisation de la lumière diffusée par l’atmosphère à 90° du
Soleil est fortement polarisée. On observe une légère polarisation de la lumière
réfléchie par la Lune. Ce phénomène fait intervenir un passage de la lumière dans des
milieux transparents, qu’il se produise une réfraction ou une réflexion totale.
Par contre, il n’en va pas de même pour la forte polarisation de la couche coronale K
du Soleil et pour la polarisation partielle des étoiles. La réflexion sur des électrons ne
peut en aucun cas être assimilée à une réflexion vitreuse, et la polarisation des étoiles
ne fait pas intervenir de réflexion.
Des considérations sur l’application du théorème du moment cinétique ont amené à
proposer une cause unique pour ces deux derniers types de polarisation et à penser
que la couche coronale K du Soleil doit présenter une seconde direction de
polarisation dans le plan de la Galaxie, en plus de la polarisation déjà connue et
globalement parallèle au plan équatorial du Soleil.
Cette seconde polarisation de la couche coronale K du Soleil a été mise en évidence le
14 décembre 1996, une semaine avant la date du passage de la direction du Soleil
dans un plan parallèle au plan de la Galaxie, à l’aide d’un tube à diaphragmes
instrumenté.
Abstract:
In many circumstances, polarization of light may be observed with a variable degree
of completeness. The polarization of the light diffused by the atmosphere within an
angle of 90° from the Sun is highly polarized; the light reflected by the Moon surface
is slightly polarized. In both cases, the polarization is linked to a reflection or a
refraction within transparent media and the related phenomena are well identified.
The situation is neither the same for the large polarization of the K coronal layer of
the Sun nor for the partial polarization of the Stars. Reflection by electrons cannot be
considered as a reflection on atoms, and the polarization of Stars does not involve any
reflection.
From the implementation of the angular momentum theorem, it has been deducted
only one explanation for these both kinds of polarization, and to think that the K
coronal layer of the Sun should also be polarized in the Galaxy plane, in addition to
the well known polarization approximately parallel to the equatorial plane of the Sun.
This second polarization of the K coronal layer of the Sun was found on Saturday
December 14, 1997, one week before the direction of the Sun is exactly parallel to the
Galaxy plane, using an instrumented diaphragm tube.
114
00 INTRODUCTION
01 Objectifs
Les circonstances répertoriées de polarisation de la lumière, résultent d’un passage de
la lumière dans des corps transparents. Les champs électromagnétiques sont
susceptibles de provoquer ou de modifier la polarisation de la lumière à la traversée de
certains milieux, mais il est essentiel de noter que les champs électromagnétiques
n’ont aucun effet direct sur la lumière. Leur action sur la matière en modifie la
structure et cette modification peut agir sur la lumière et en particulier sur son état de
polarisation.
Il y a toutefois des exceptions : la polarisation de la couche coronale K du Soleil; il
s’agit dans ce cas d’une réflexion de la lumière solaire sur les électrons de la couche
K. Il n’y a donc pas de traversée d'un milieu transparent. On considère que le champ
magnétique du Soleil agit directement sur les électrons pour provoquer la polarisation
de la lumière.
Un autre cas de polarisation a reçu une explication dédiée: la polarisation partielle de
la lumière émise par les étoiles. Dans ce cas en effet, la polarisation résulterait de
l’orientation des poussières métalliques sidérales par le champ magnétique dans les
bras de la Galaxie, ce qui permettrait d’expliquer pourquoi cette polarisation est
globalement parallèle au plan de la Galaxie. Il s’agirait bien du résultat d’une action
structurante d’un champ électromagnétique sur la matière, mais l’explication est
dédiée dans la mesure où il ne s’agit pas de corps transparents; la lumière ne peut
nullement traverser des poussières métalliques.
00 INTRODUCTION
01 Objectives
The known light polarization circumstances are linked to the behavior of light through
transparent bodies. Electromagnetic fields may initiate or modify the light polarization
status when going through some specific media, but it is essential to notice that
electromagnetic fields have no direct effect on light. Their action on matter modifies
its structure and the related change may act on the light polarization status.
Nevertheless there are exceptions. The polarization of the K coronal layer of the Sun
is understood as the result of the reflection of the solar light by the electrons of the K
layer. There is no transparent medium. It is considered that the electromagnetic field
of the Sun acts directly on the electrons to initiate the polarization of light.
An other polarization case received also a dedicated explanation: the polarization of
light coming from stars. In this case, polarization is considered as the result of an
orientation of the metallic dust by the electromagnetic field of the Galaxy arms. This
allows for explaining that the polarization direction is parallel to the Galaxy plane. It
may be seen as an action on an electromagnetic field on mater, but the explanation is
dedicated because metallic dust is not transparent.
115
On pense que cet ensemble structuré de particules métalliques constitue un milieu, luimême transparent, mais ce processus de polarisation n’a jamais pu être reproduit, pas
plus que le précédent d’ailleurs.
L’objet de cette note est de proposer une explication unique pour ces deux exceptions
et de présenter une expérience réalisée pour mettre en évidence un cas de polarisation
similaire, mais ne pouvant pas résulter de l’action d’un champ électromagnétique : la
polarisation de la couche coronale K du Soleil dans le plan de la Galaxie.
La généralisation de cette explication de la polarisation de la lumière, au delà de ces
trois cas, à l’ensemble des phénomènes de polarisation de la lumière, a des
conséquences théoriques et expérimentales exposées dans la dernière partie.
02 Principes
La vitesse tangentielle de rotation du Soleil dans la Galaxie est du même ordre de
grandeur que la vitesse képlerienne de rotation autour du Soleil à la distance de sa
surface égale à une fois et demi son rayon, c'est-à-dire d’environ 250km/s. Cette
coïncidence peut conduire à s’interroger sur les phénomènes qui se produisent dans
cette circonstance.
It could be thought that the metallic dusts could be a transparent medium, but this had
never been neither found elsewhere nor reproduced.
The subject of this report is to propose a common explanation for those two
exceptions and to present an experiment performed to show a similar case of
polarization which, nevertheless, can in no case be the result of the action of an
electromagnetic field. This is the polarization of the K coronal layer of the Sun in the
Galaxy plane.
The proposed explanation of the polarization of light may be extended to all the
polarization cases. The consequences, both theoretical and experimental are presented
in the last part of this report.
02 Principles
The tangential speed of the Sun within the Galaxy is within the same order of
magnitude as the keplerian speed around the Sun at a distance of one and an half
radius of the Sun to its surface They are 250km/s. This is a coincidence which may
induce a review of the phenomena occurring in such a circumstance.
116
Malheureusement aucun corps ne tourne aussi près de la surface du Soleil. Mais
l’existence même des couches coronales, montre qu’il a des électrons, des atomes, des
molécules et des poussières, qui, même si elles ne sont pas en orbite, se trouvent dans
une zone où à la fois la vitesse képlerienne de rotation autour du Soleil et la vitesse
tangentielle de rotation dans la Galaxie sont très voisines.
Or il se trouve que la lumière solaire réfléchie par les électrons de la couche coronale
K est fortement polarisée globalement dans un plan parallèle au plan équatorial du
Soleil. En est-il de même pour cette même lumière réfléchie par ces mêmes électrons
de la couche K, mais dans la direction du plan de la Galaxie?
Toutefois, il n’est possible pas de comprendre comment la vitesse peut faire
apparaître une polarisation, d’autant moins que les électrons sont bien animés de la
même vitesse galactique que le Soleil, mais ils ne gravitent pas autour du Soleil. Si la
vitesse était la cause de la polarisation, seule existerait la polarisation galactique. Or la
polarisation de la couche K, connue depuis plus d’un siècle, ne peut se rattacher à
aucune grandeur, telle que le champ magnétique, ni à aucune position qui serait liée à
la Galaxie. Le champ magnétique de la Galaxie, au niveau de la couche K, est
extrêmement faible par rapport à celui du Soleil.
L’égalité des vitesses est la manifestation d’un autre phénomène.
Unfortunately there are no bodies turning around the Sun at a so low distance. But
there are many electrons, atoms, molecules and dusts in the vicinity of the Sun within
the so called coronal layers. Although they are not turning around the Sun, they are in
a zone where the keplerian rotation speed is the same as the speed of the Sun around
the Galaxy center.
However, the solar light reflected by the electrons of the K coronal layer of the Sun is
highly polarized. This polarization is approximately parallel to the equatorial plane of
the Sun. Is it the same for that same light reflected by those electrons, but in the
direction of the Galaxy plane?
But how may the speed initiate polarization by its own? In addition, the electrons of
the K layer are turning around the Galaxy center with the Sun, but they are not at all
turning around the Sun. If the speed was the root cause of polarization, the galactic
polarization would only exist. And this is not possible for the polarization of the K
layer is known since many decades and it has never been found any link with the
Galaxy properties. The magnetic field of the Galaxy is extremely weak compared to
the Sun field.
The equality of speeds is the result of another phenomenon.
117
On peut définir une grandeur, pour l'instant toute théorique, mesurant le défaut
potentiel de moment cinétique local par la sommation de la variation du moment
cinétique d'un point matériel de masse unité, tournant autour d'un astre, qui se
rapprocherait de l'astre depuis l'infini jusqu'à une distance définie. Or il se trouve que
l’énergie cinétique de rotation correspondant à cette grandeur est du même ordre pour
un corps tournant autour du Soleil qui se rapprocherait jusqu'à une distance de sa
surface égale à une fois et demi son rayon, que pour un corps tournant autour du
centre de la Galaxie qui se rapprocherait de l’infini jusqu'à la distance où se trouve le
Soleil.
Le rapprochement du corps envisagé doit cependant avoir une cause physique et la
compensation des moments cinétiques est nécessairement liée à cette cause. Les
frottements avec les poussières et atomes de l'Espace ne peuvent être invoqués pour
compenser tout à la fois l'extension et l'effondrement des systèmes galactiques ou
planétaires. D'autre part, la compensation globale des moments cinétiques,
apparemment satisfaisante pour l'esprit, ne peut expliquer la compensation des
variations, sans qu'il soit nécessaire d'ajouter un transfert à distance à chaque instant,
question aussi irritante que celle de l'action à distance de la gravitation. La
conservation du moment cinétique n'est pas moins impérative que celle de la
conservation de la quantité de mouvement. C’est-à-dire que le système de la
gravitation doit pourvoir à la compensation des variations du moment cinétique à
chaque instant et en chaque point de l'Espace indistinctement.
One may define the Angular Momentum Lack as a theoretical value obtained by
integrating the angular momentum variation of a body, with a mass equal to one unit,
turning around a star, while it is pushed toward the star. The kinetic energy related to
this value is within the same order of magnitude both for the a body pushed from an
infinite distance down to a distance of 1.5 radius of the Sun surface and for the same
body pushed down to the Galaxy center down to the distance of the Sun.
Of course the motion of that body toward the Sun or toward the Galaxy center shall
have a cause and this cause shall provide for the angular momentum variation
clearing. Friction with sidereal matter and dust cannot be taken as a root cause both
for system collapsing or extending. On another hand, the global overall clearing of
angular momentum variation, although it seems to be consistent with principles,
cannot explain the clearing of local variations without adding a transfer to distant
points of space to be added to the distant action of gravity. The angular momentum
conservation is exactly as mandatory as the linear momentum conservation. That is to
say that the gravitation system shall provide simultaneously, and in every point of
space for both a clearing of the linear momentum variations and a clearing of the
angular momentum variations.
118
Dès cet instant, il faut reconnaître que c'est la totalité du moment cinétique des corps
des systèmes galactiques et planétaires qui est compensée localement par le système
de la gravitation quelle que soit la forme mathématique sous laquelle on l'exprime.
Maintenir une compensation globale et universelle, qui devrait, dès lors, se trouver
inscrite dans le processus de formation de l'ensemble des systèmes galactiques et
planétaires, conduirait à une redondance fatale. En effet lors de la formation des
systèmes galactiques et planétaires, les variations de moment cinétique s'intègrent
dans le temps sans que l'on puisse en arrêter la sommation jusqu’à des valeurs qui
atteindront enfin exactement l'amplitude du moment cinétique actuel, supposé
pourtant déjà compensé globalement.
La compensation est nécessairement locale. Elle est dès lors partout potentielle.
Chaque point d’un système, est affecté d’un défaut potentiel de moment cinétique par
rapport au centre du système, mesuré par la grandeur définie ci-dessus. Il s’agit
toujours d’un défaut de moment cinétique et la variation de ce défaut est positive ou
négative selon que le système est en extension ou en effondrement.
La gravitation a ainsi deux composantes indissociables et nécessaires. Un moyen de
compensation des forces pour la conservation de la quantité de mouvement et qui se
traduit par l'égalité de l'action et de la réaction et un moyen de compensation du
moment cinétique pour la conservation du moment cinétique.
At this point, it shall be recognized that the total amount of the angular momentum of
all the bodies belonging to a system shall be locally cleared by the gravitation system
whatever is the mathematical formalism use to describe it.
A global and universal clearing would have to be operated at the very beginning of
any motion of all the existing matter, and this will lead to a fatal redundancy. During
the formation of galactic and planetary systems, the angular momentum variations are
added to themselves without any possibility to stop the process, up to the actual value
of the angular momentum, although it was assumed that this angular momentum
would have been globally and universally cleared only at the very beginning.
There is no way to escape to a local and simultaneous clearing. Therefore the clearing
is everywhere as potential. Every point of a system is affected with a lack of angular
momentum with regard to the system center; and this lack is measured by the value
defined here under. This is always a lack and the variation of this lack is either
positive or negative when the system is either extending or collapsing.
Thus the gravitation has two components. One clearing mean for the angular
momentum variation and one clearing mean for the linear momentum variation. The
linear momentum conservation is represented by the equality of the action to the
reaction.
119
Il faut noter au passage que la conservation de la quantité de mouvement ne peut pas
davantage être globale que celle de la conservation du moment cinétique. Un
raisonnement semblable au précédent conduit à considérer la nécessité d'une égalité
locale de l'action et de la réaction et non pas seulement d'une égalité globale ou à
distance. Le système de la gravitation doit inclure une égalité locale de l'action et de la
réaction, et une compensation locale du moment cinétique, quel que soit le
formalisme mathématique utilisé pour l'exprimer.
La gravitation, sous ses deux composantes ainsi reconnues, est supposée agir
physiquement sur les corps de deux manières différentes. Si l’on fait l’hypothèse que
ces deux composantes sont deux manifestations d’une même forme de la Nature, alors
il faut également tenir compte du principe d’équipartition de l’énergie entre les degrés
de liberté de cette forme unique, au nombre de six. Rien n’exclut d’en donner
davantage, encore faut-il en trouver l’utilité et la justification.
En outre, cette forme ne peut disposer d’une énergie cinétique infinie. Si l’on prend
comme limite 1/2 mc2, le principe d’équipartition conduit à attribuer la même limite à
l’énergie cinétique de rotation, la limite totale étant naturellement mc2.
Sans qu’il soit nécessaire de faire aucune hypothèse sur la nature même de cette
forme, une constatation fondamentale s’impose pour les systèmes galactiques et
planétaires: l’énergie cinétique de rotation croît plus vite, en se rapprochant du centre
du système, que l’énergie cinétique. La première varie avec l’inverse du cube de la
distance au centre du système, la seconde avec l’inverse de cette distance seulement.
The linear momentum conservation cannot be obtained neither by a global and
preliminary clearing, nor by distant actions. It is also local. The same reasoning as for
the angular momentum conservation would show that a global and preliminary
clearing would lead also to a fatal redundancy. The action and the reaction shall be
locally equal. The gravitation system shall include both a local equality of the action
and the reaction and a local clearing of the angular momentum whatever is the
formalism used.
The gravitation, made of these two components, is assumed to act physically on
bodies by two different ways. If it is assumed in addition, that those two components
are evidence of one single form of the Nature, thus the equipartition principle shall
apply between the freedom degrees of this single form. And nothing can prevent to
take more or less than 6 degrees.
At last, this single form cannot have an infinite available energy. If the limit of energy
attached to the linear momentum component is assumed to be 1/2 mc2, then the total
amount available for both components is mc2.
Without any further assumption on the nature of that form itself, there is an immediate
consequence for the galactic and planetary systems. The angular momentum grows far
faster than the linear momentum while approaching the system center, when the
motions are keplerian. The first is growing as the distance to the center to the minus 3,
although the second is growing to the minus 1.
120
C’est-à-dire que l’énergie cinétique de rotation disponible dans la forme de la
gravitation peut s’annuler bien avant l’énergie cinétique disponible. Les distances
d’annulation respective ne peuvent se faire sans hypothèses sur la nature même de
cette forme.
Cependant, les conditions de rotation des astres des systèmes doivent être modifiées
bien avant que ces limites soient atteintes. Il semble que ce soit le cas dans le système
solaire où une anomalie mesurable de la rotation de Mercure a été constatée. Mais
c’est assurément le cas des galaxies, puisque la rotation de deux galaxies voisines
n’est pas képlerienne. A fortiori la rotation des astres des galaxies n’est pas non plus
képlerienne.
La diminution de l’énergie de la composante relative au moment cinétique, de la
forme de la gravitation, provoque un pompage des moments cinétiques des corps
placés au point considéré.
En conséquence, si le moment cinétique propre des électrons qui composent la couche
K, est entièrement pompé par le défaut local lié au Soleil, il sera, de la même manière,
entièrement pompé par le défaut galactique en résultat de la remarque liminaire.
The energy related to the angular momentum may be fully swallowed far before the
energy relates to the linear momentum. The distances where those energies are
annealed cannot be calculated without further assumptions on the gravitation form.
However, far before the energy related to the angular momentum is annealed, the
rotation conditions of bodies within systems are modified. It seems that this is the
case for Mercury in the Solar system. But of course it is undoubtedly the case for
galaxies, as the relative rotation of two galaxies close to one another is not at all
keplerian. Of course it is also the case within the galaxies themselves.
As the available energy related to the angular momentum is decreasing, the angular
momentum of bodies is pumped accordingly.
As a consequence, if the angular momentum of electrons of the K layer of the Sun,
perpendicular to the Solar equator, is fully pumped out by the angular momentum
local lack resulting from the Sun, the angular momentum of electrons of the K layer
of the Sun, perpendicular to the Galaxy plane will be also fully pumped out by the
angular momentum local lack resulting from the Galaxy as a result of the preliminary
remark.
121
Or la lumière solaire réfléchie par les électrons de la couche K est fortement polarisée.
De plus cette polarisation est parallèle globalement au plan équatorial du Soleil, alors
que le défaut de moment cinétique lui est perpendiculaire. Si cette polarisation est liée
à ce défaut de moment cinétique des électrons, alors la même lumière solaire réfléchie
par ces électrons doit aussi être polarisée dans un plan parallèle au plan de la Galaxie
puisque le défaut galactique de moment cinétique est lui-même perpendiculaire à ce
plan.
Il se trouve que le plan équatorial du Soleil et le plan de la Galaxie font un angle
voisin de 70°. Circonstance remarquable en cela qu'elle conduit à deux critères
caractéristiques. On doit observer d'une part 4 extinctions par tour de l'analyseur,
d'autre part les maxima doivent être inégaux, mais inférieurs au tiers environ de
l’amplitude observée pour la seule polarisation liée au Soleil.
L’observation de la couche coronale K lors du passage du Soleil dans une direction
parallèle au plan de la Galaxie, circonstance qui se produit actuellement 39 minutes
avant le solstice d’hiver, a mis en évidence cette polarisation supplémentaire le 14
décembre 1996, quelques jours avant le solstice d'hiver où la couverture nuageuse n'a
pas permis d'observation. L’observation effectuée en février 1997, plus de 60 jours
après le solstice d’hiver, montre que seule subsiste la polarisation due au Soleil, l’effet
galactique ayant disparu.
Les résultats de ces observations sont exposés au chapitre 20 après une présentation
du dispositif expérimental utilisé.
But the solar light reflected by those electrons is highly polarized. Furthermore, this
polarization is globally parallel to the Sun equator, the lack of angular momentum of
electrons being perpendicular to the Sun equator. If the lack of angular momentum is
the cause of the polarization, then that same Solar light reflected by those electrons
shall be also polarized in a direction parallel to the Galaxy plane, the related lack of
angular momentum being perpendicular to the Galaxy plane.
The angle between the Sun equator and the Galaxy plane is about 70°. This leads to
two criteria. First, 4 extinctions shall be observed for each revolution of the analyzer,
second, the maxima are not equal but both lower than a third of the normal amplitude
observed for the solar polarization alone.
The measurements of the light reflected by the Sun by the coronal K layer, when the
Sun is seen from the Earth in the Galaxy plane, show this supplementary polarization
of the K layer. This event occurs twice a year, presently 39 minutes before the winter
solstice. The measurements had been performed on the 14th December 1996, some
days before the winter solstice. On the 21st, the heavy cloud cover didn’t allow for
any observation. The measurements were performed again 60 days after the winter
solstice. The galactic effect had disappears.
The results are detailed in chapter 20 after a presentation of the device used.
122
10 DISPOSITIF EXPERIMENTAL
Le dispositif utilisé se compose essentiellement d’un tube à diaphragmes, équipé d’un
photomultiplicateur, fixé sur une monture équatoriale.
11 Le tube à diaphragmes instrumenté
111 Partie mécanique et optique
Une boîte à diaphragmes dont le plus petit a une ouverture de 0.3 mm est fixée à une
extrémité d’un tube principal en cuivre de 40 mm de diamètre, de 1 mm d’épaisseur et
de 1m de long, dans l’axe du tube. Un obturateur tournant est fixé à l’extérieur.
Le tube principal est revêtu intérieurement d'une couche de peinture antiréfléchissante et comporte dans sa partie médiane une série d'une dizaine de bourres
de 1 cm de long en mousse de polyuréthane noire percées d'un orifice tubulaire de
2mm de diamètre dans l'axe du tube principal.
Un tube secondaire en aluminium, de 5 mm de diamètre, monté dans deux roulements
à billes, est inséré, dans l’axe, à l’autre extrémité du tube principal. L’extrémité
intérieure du tube secondaire porte un diaphragme de 0.3 mm de diamètre également.
L’extrémité extérieure porte un analyseur. Un tube de 10 mm est fixé à cette extrémité
du tube secondaire pour former une chicane lors de l’insertion du tube de 8 mm fixé à
la face interne de la bride support du photomultiplicateur sur le tube principal et dans
son axe. La face externe de la bride support du photomultiplicateur permet la fixation
d’un carter de protection en plastique.
10 EXPERIMENTAL DEVICE
The device used is essentially made of a tube with diaphragms fitted with a
photomultiplier, attached to an equatorial mount.
11 The instrumented diaphragm tube
111 Mechanical and optical part
A diaphragm box with the smallest diaphragm diameter not greater than 0.3 mm, is
fitted to one end of a copper main tube 1 mm thick, 1 m long and 40 cm diameter, in
the axis of the tube. A rotating shutter is attached outside the tube.
The main tube is internally coated with an antireflection painting and includes, near its
middle, about ten bored black wads 1cm long made of polyurethan. The bores are 2
mm diameter in the axis of the main tube.
A secondary aluminium tube 5 mm diameter mounted on rolling bears, is fitted inside
the main tube. A diaphragm, also 0.3 mm diameter, is fixed to the internal end of the
secundary tube. The external end bears an analyzer. A 10 mm copper tube is welded
at this end of the tube in order to form a chicane whith the 8 mm diameter tube
welded to the internal face of the photomultiplier support flange. The external face of
that flange allows for fitting a plastic protection tube.
123
Le tube secondaire est entraîné en rotation par une courroie reliée à un moteur par
l’intermédiaire d’un volant d’inertie. Le volant et le moteur sont fixés à l’extérieur du
tube principal. Les deux traversées de la courroie sont protégées en sorte que l’entrée
de lumière soit limitée au minimum. La chicane décrite ci-dessus complète la
protection.
L'ouverture du tube à diaphragmes est de 2.5'.
Un viseur optique réticulé, équipé d’un filtre solaire est fixé au tube principal.
Schéma du tube à diaphragmes instrumenté
moteur / motor
analyseur/analyzer
Photomultiplicateur / Photomultiplier
Principle of the instrumented diaphragm tube
The secondary tube is actuated by a motor fitted outside the main tube. A fly wheel is
used both for rotation smoothing and speed reduction. The crossings of the strap
through the main tube are properly protected to reduce the path for light to a
minimum. Any remaining light which would enter the tiny remaining gaps is stopped
by the chicane described here above.
The opening of the diaphragm tube is no more than 2.5'.
An optical sight with a reticle is attached ouside the main tube.
124
112 Partie électrique.
Un boîtier de connexion et de contrôle est fixé au tube principal. Il comporte un
connecteur SUBD9M relié aux câbles d’alimentation et de contrôle du
photomultiplicateur, une prise BNC relié au câble de sortie du photomultiplicateur, un
potentiomètre et un multimètre digital pour le réglage de la tension de contrôle. Le
boîtier est relié aux alimentations ± 12 V et à la masse.
Ce boîtier comporte également un potentiomètre de réglage de la vitesse du moteur et
un connecteur SUBD9M relié aux cables d’alimentations du moteur lui-même et du
compte-tours en 9 V et 1.5 V respectivement. Les moto-réducteurs utilisés pour les
mesures tournent entre 6 et 24 tours par minute ce qui correspond, compte tenu du
volant d’inertie qui joue également le rôle de réducteur par son mode d’entraînement,
à une vitesse du tube secondaire porte analyseur de 0.75 tour par minute à 3 tours par
minute.
Le compte-tours consiste en deux plaques souples isolées en cuivre fixées à l’intérieur
du tube principal. Le courant est établi par le passage des plaques au contact d’un fil
soudé longitudinalement au tube de la chicane fixé au tube secondaire. Un bruiteur
électronique fixé au tube principal est actionné par le compte tours.
Les alimentations ± 12 V sont régulées et stabilisées avec une ondulation crête à crête
inférieure à 3 mV.
Les autres alimentations sont des piles du commerce.
112 Electrical part
A connecting and control box is fixed to the main tube. It includes a SUBD9M
connector linked to the power supply and control unit of the photomultiplier and a
BNC connector linked to the output of the photomultiplier. It includes also a
potentiometer and a digital voltmeter for the setting of the control voltage. The box is
connected to the ground and to the bipolar power supply.
The box includes also a potentiometer for the setting of the motor speed and a
connector SUBD9M also for connection to the power supplies of the motor, the
multimeter and the revolution counter (9 V DC, 9 V DC, 3 V DC). The electrical DC
motors include reducers. Their speeds are within the range 6 to 24 rpm. The rotation
speed of the analyzer, taking into account the flywheel reduction factor, is within the
range of 0.75 to 3 rpm.
The revolution counter is made of two thin copper strips bolted inside the main tube
by insulated bolts. The circuit is closed by a copper transversal wire welded to the
external tube of the chicane. An electronic buzzer is actuated by the revolution
counter.
The ± 12 V power supplies are regulated and stabilized so that the peak to peak ripple
is below 3 mV.
The other power supplies are batteries.
125
113 Partie photométrique
Un photomultiplicateur est fixé à la bride mentionnée ci-dessus. Les caractéristiques
essentielles du photomultiplicateur sont données dans le tableau ci-après.
114 Partie acquisition
La prise BNC du boîtier de connexion et de contrôle et la sortie du compte-tours
peuvent être reliées à un module d’acquisition par un connecteur SUBD25M. Le
module d’acquisition est lui-même relié à la sortie parallèle d’un ordinateur portable.
Un adaptateur d’impédance à amplificateur opérationnel à transistors à effet de champ
peut être inséré entre le module d’acquisition et le connecteur SUBD25M.
Le logiciel d’acquisition, fourni avec le module, permet de reporter les valeurs
obtenues sur EXCEL.
Un connecteur SUBD25F permet de remplacer le dispositif d’acquisition par des
voltmètres d’une sensibilité inférieure au mV.
12 Le viseur solaire
Le viseur solaire est conçu pour déclencher une alarme de proximité de la surface du
Soleil. Le signal d'alarme peut être envoyé au dispositif d'acquisition dont le logiciel
peut actionner des voyants d'écran ou le haut-parleur de l'ordinateur.
113 Photometric part
A photomultiplier is fixed by four 2mm screws to the flange mentioned here above.
The main characteristics of the photomultiplier are indicated by the table here after.
114 Acquisition part
The BNC connector of the connector and control box and the output of the revolution
counter may be connected to an acquisition module through a SUBD25M connector.
The acquisition module is plugged in a parallel port of a laptop computer. An
impedance adapter with field effect transistors may be placed between the acquisition
module and the SUBD25M connector. The acquisition software, supplied with the
module, allows for transferring data to EXCEL.
A SUBD25F connector linked to multimeters may be plugged instead of the module.
The sensitivity of the multimeters is lower than 1 mV.
12 The solar sight
The solar sight is intended for preventing the photomultiplier to be placed toward the
Sun disk. The alarm may be sent to the acquisition module provided with software
features such as screen lights or signals for the computer loud speaker or buzzer.
126
Le viseur solaire est fixé à l'extérieur du tube à diaphragmes, et dans son axe. Il est
constitué d'un tube de 15 mm de diamètre et de 20 cm de long. Dix diaphragmes de
1mm de diamètre sont fixés à intervalles réguliers à l'intérieur du tube. L'ouverture est
de 30', soit le diamètre apparent du Soleil.
Une photodiode est fixée à l'extrémité du viseur dans un tube coulissant facilement
extractible, en sorte que le viseur puisse être utilisé directement à l'oeil. Le signal de
la photodiode est envoyé sur un amplificateur opérationnel à correction d'offset. Le
dispositif a été conçu pour que le gain soit de 100. Le signal amplifié est envoyé sur
un voltmètre fixé au tube, dont la sensibilité est de 10 µV et sur deux amplificateurs
opérationnels montés en comparateurs délivrant un signal continu pour des valeurs du
signal d'entrée respectivement inférieur et supérieur à leur tension de référence
préréglée. La sortie de chaque comparateurs alimente une diode électroluminescente
respectivement rouge et verte. La tension de référence correspond au signal maximum
donné par la photodiode du viseur solaire placé dans la direction du Soleil.
A l'issue des réglages de parallélisme entre le tube à diaphragmes et le viseur solaire
un décalage angulaire de 15' est donné au viseur solaire dans le sens des aiguilles
d'une montre, grâce à une vis de réglage insérée dans la fixation du viseur au tube à
diaphragmes.
The solar sight is attached to the main tube and parallel to its axis. It is made of a
copper tube 15 mm diameter and 20 cm long.10 diaphragms 1 mm diameter are fixed
inside the tube. The opening is 30', that is to say the Solar apparent diameter.
A photodiode is fitted at one end of the tube within a grooving tube. This tube may be
removed easily so that the sight can be used directly. The output signal of the
photodiode is amplified by a FET operational amplifier mounted with a 100 gain. The
amplified signal is then sent to two comparators actuating respectively a red LED and
a green LED. The set value of the comparators is the amplified signal delivered by the
photodiode when placed towards the Sun.
When all settings are done, the solar sight axis is turned by a screw provided in its
support by 15' backward from the instrumented diaphragm tube axis.
127
PHOTOMULTIPLIER
Manufacturer :
Module series :
Serial number :
Sensitivity
:
Time resp.
:
Spectral resp. :
HAMAMATSU
H5784
840-331
127.0E+6 V/Lm control voltage 0.8V (1)
0.65 ns
300 to 650 nm
The module includes a voltage regulator, a high voltage Cockford-Walton
bridge power supply and a bipolar (min ± 11.5V, max ± 15.5V) amplifier
delivering a voltage signal (conversion factor 1V/µA).
note 1: Taking into account the nominal response curve, the sensitivity is 16
E+6 V/Lm for control voltage 0.6V
128
129
20 RESULTATS
21 Mesures
211 Mesures préliminaires
a) protections
Obturateur fermé, moteur en route, tube dans l'axe du Soleil, le signal du
photomultiplicateur ne dépasse pas l’offset, à la valeur nominale de contrôle 0.8 V.
b) vérifications
Des mesures ont été réalisées à l'aide d'une lampe ordinaire à fil de tungstène et à
halogène, à des distances telles que le signal corresponde à la valeur calculée pour la
couche K.
Les tensions enregistrées par l'ordinateur présentent des oscillations très importantes.
La polarisation provoquée de la lumière émise a pu être observée sur les courbes
obtenues malgré ces oscillations. Toutefois, elles rendraient entièrement impossible la
détection de variations d'éclairement quatre fois plus faible. Ce phénomène a été
attribué d'abord à un problème d'impédance et un adaptateur a été réalisé pour y
remédier, mais sans succès. Le constructeur devait d'ailleurs indiquer entre temps que
l'impédance de sortie du photomultiplicateur n'est que de 1 kΩ, valeur qui exclut tout
problème de ce genre.
20 RESULTS
21 Measurements
211 Preliminary verifications
a) protections
The photomultiplier offset value is obtained for a voltage control value of 0.8 V, when
the shutter is closed and the tube is turned toward the center of the Sun.
b) Other verifications
Measurements had been performed with tungsten wires and halogen lamps. The
distance to the lamps was such as to obtain a signal similar to the calculated value for
the K layer.
The voltage values registered by the computer are oscillating within an high
amplitude. The light observed through a polarizer fixed to one of the holes of the
rotating shutter gave registered values showing both maxima and minima per
revolution of the analyser, but the results show the expected values for the K layer
would be unusable when the amplitude will be reduced by a third. The first cause
found was a wrong balance of the impedances. An impedance adaptator had been
made with a FET operational amplifier, but without results. This could not be the
cause of the failure as the impedance value of the photomultiplier finally obtained
from the manufacturer is only 1 kΩ. With such a value, there cannot be any
impedance problem.
130
Le plus probable est que la lecture du signal par le convertisseur analogique-digital est
beaucoup trop rapide. Et comme le logiciel fourni ne permet pas de descendre endessous d'une mesure tous les dixièmes de seconde, il n'y pas de possibilité de
traitement statistique. L'enregistrement par l’ordinateur du bleu du ciel est totalement
inexploitable. La solution la plus simple devrait consister à placer un amplificateur
opérationnel monté en intégrateur avant le digitaliseur. Toutefois, à partir du 7
décembre 1996, il a été décidé d'interrompre les tentatives d'amélioration du dispositif
d'acquisition afin de tenter des mesures de la couche K au solstice d'hiver. La suite
des mesures a été réalisée au voltmètre.
Le bleu du ciel donne une tension maximale de sortie du photomultiplicateur de 25 à
30 mV à 90° de la direction du Soleil. L'ordre de grandeur de cette valeur est
conforme aux calculs qui ont conduit au choix du photomultiplicateur. La polarisation
atteint plus de 80%. La légère montée de la courbe obtenue provient du mouvement
de la Terre, le moteur d'ascension droite n'ayant pas été mis en route lors de cette
mesure.
signal PMC PMT signal
35
30
25
20
15
10
5
0
0
180
360
540
720
900
Polarisation du bleu du ciel: signal photomultiplicateur en mV, rotation
analyseur en ° en abscisse.
Sky blue day light polarization. y-axis Photomultiplier signal in mV, x-axis
analyser revolution in °
The cause of the problem may be more probably related to the very fast reading of the
signal by the AD converter. Unfortunately the software supplied with the acquisition
device is limited to 1 measurements every 0.1 second so that there are not enough
values for a statistical treatment. The values registered for the polarization of the blue
day light of the sky are fully unusable. The corrective action should be to insert an
operational amplifier fitted with capacitors as an integrator before the acquisition
module. Nevertheless it was too late and on the 7th of December 1996, it was decided
not to try to improve the acquisition mean any more and to use a multimeter instead,
so that the winter solstice measurements may be performed.
The curve here above is related to the sky blue day light. The maximum voltage value
of the photomultiplier signal with a control voltage to 0.8 V is about 25 to 30 mV.
The order of magnitude of these values are complying with the value calculated and
used to choose the photomultiplier. The slight ascending slope of the curve is a result
of the Earth rotation as the right ascension motor of the equatorial mount had not been
switched on.
131
L'absence de nuage le 7 décembre 1996, a été mise à profit pour vérifier le viseur
solaire. Cependant, l’impossibilité d’utiliser le dispositif d’acquisition, et en
particulier la nécessité de lire les valeurs de la tension du signal délivré par le
photomultiplicateur, n’ont pas permis d’utiliser le viseur solaire. D'autant moins,
d'ailleurs, que d'autres incidents allaient limiter son utilisation au règlage initial de
chaque série de mesures. Le viseur a été remis dans l’axe du tube principal dans ce
but.
212 Mesures de la polarisation de la couche K
Une première mesure du signal donné par la couche K a pu être effectuée ce même
jour. La tension est montée à 1400 mV à une distance de la surface du Soleil d'environ
1/5 de rayon, correspondant à deux fois l'ouverture du tube à diaphragmes. Cette
valeur est dix fois la valeur attendue, calculée d'après les données connues de
l'éclairement solaire: 1E+5 lux, la couronne ayant un éclairement 1E-5 fois plus
faible, encore faut-il penser que c'est là une donnée valable dans le plan de l'équateur
au moment optimal du cycle. Or, nous sommes cette année au minimum du cycle
solaire. De plus, par manque de temps et d'expérience, on s'est contenté de viser au
début de chaque cycle de mesures le point le plus à l'Est, et non un point dans le
prolongement de l'équateur solaire.
As there were no clouds on the 7th of December 1996, a verification of the solar sight
has been performed. However, as it was not possible to use the computer data
acquisition system, and mainly as it was necessary to read the photomultiplier
multimeter output, the solar sight has not been used as intended. In addition further
problems occurred so that the solar sight was only use for the initial verification of the
tube positionning before approaching the Sun surface. It has been set aligned with the
main tube.
212 K Layer polarization measurements
A first measure of the K layer signal was performed also on the 7th. The multimeter
output voltage value reached 1400 mV at a distance of the Sun surface not greater
than a fifth of its radius, equivalent to two openings of the diaphragm tube. This is ten
times the expected value calculated from the following data: Sun disk : 1E+5 lux,
coronal layer 1E-5 the Sun disk value. But these values are probably valid for the
upper part of the solar cycle and near the equator of the Sun. We are presently in the
lower part of the solar cycle. Moreover to save time, and also by lack of experience,
the tube was trained toward the most eastern part of the Sun and not toward the
equatorial plane.
132
Cette valeur dix fois plus élevée que prévu, ne peut résulter que de la diffraction et
des réflexions dans le tube. Ceci devait d'ailleurs être confirmé le samedi suivant, au
moment où le Soleil fut caché progressivement par une cheminée, puisque le signal
s'est mis à baisser beaucoup plus rapidement que du fait du seul éloignement du Soleil
résultant de la rotation de la Terre. La distance du tube à la cheminée est en effet
parfaitement négligeable par rapport à l'épaisseur de la couche d'air pollué traversée ;
la diminution de la diffusion sur une si courte distance ne peut atteindre une telle
proportion.
L'amélioration du tube nécessite son démontage. Il a été décidé de laisser les choses
en l'état pour tenter malgré tout la mesure d'hiver.
La suite de la journée a été consacrée à une initiation au fonctionnement de la
monture équatoriale. L'enclenchement du moteur d'ascension droite provoque un
déplacement du tube de plusieurs minutes. Un astronome amateur n'aurait
certainement pas eu de peine soit à éviter ce problème, soit à en tenir compte dans le
positionnement du tube avant enclenchement.
Le Soleil a été caché du 8 au 13 décembre. Un vent d'Ouest a dégagé le ciel, et le
matin du 14 décembre 1996 un léger vent d'Est a séché l'air. Il est probable que les
conditions ont été idéales pendant les deux heures de la matinée où les mesures sont
possibles, compte tenu de la situation du poste d'observation, malgré la présence du
boulevard périphérique de Paris en plein Sud, à trois cent mètres et les cheminées de
la centrale thermique d'Issy à peine au delà.
This value is as large as ten times the calculated value. This can only be the result of
diffraction and reflections inside the tube. This was confirmed the week after when
the Sun went hidden by the chimney. The signal starts decreasing far more faster than
the normal rate due to the revolution of the Earth. The distance between the tube and
the chimney is not significant compared with the thickness of polluted air beyond the
chimney ; the decrease of the diffusion cannot be so high for a so short distance.
The diaphragm tube shall be dismantled for any improvement, so that it was decided
to leave it as it was in order to be able to perform the winter measurement.
The rest of the days was devoted to an initiation to the use of the equatorial mount.
When the right ascension motor is engaged, the tube turns by some minutes. This
should not have been a problem even for an amateur either to avoid the problem or to
take into account the change in the initial positioning.
The Sun remains hidden from the 8th to the 13th of December. A strong West wind
cleans the sky and on the 14th in the morning a slight East breeze made it dry.
Probably such circumstances were the best as possible during some hours, despite the
high level of the pollution in this area. The Paris orbital is not farther than 300 m and
the Issy power generation plant is just in the background. In addition, the
measurements are only possible during the two first hours in the morning from the
location where they were performed.
133
Dès midi, des filaments nuageux apparurent. Une heure plus tard le signal devenait
erratique et atteignait 5000mV dans un nuage proche du Soleil. Ces conditions ne se
retrouveront pas avant le mois de février 1997. Ce sont donc les mesures du 14
décembre 1996 qui sont présentées.
Un manque total d'expérience en observation astronomique, autant sans doute que la
tension provoquée par la crainte de détruire le photomultiplicateur par une fausse
manoeuvre (au début de chaque série de mesure, le tube à diaphragme est pointé à
moins de 2 fois son ouverture de la surface du Soleil, le moindre faux mouvement
aurait provoqué la destruction du photomultiplicateur), n'ont pas permis de
comprendre immédiatement que le moteur d'ascension droite, insuffisamment bloqué
sur son axe, pour être enclenché le plus délicatement possible, allait à chaque reprise
des mesures, tourner autour de son axe et s'écarter de la roue dentée de l'axe
d'ascension droite au lieu de la faire tourner. C'est donc le défilement à l'aval du
Soleil, de la lumière solaire réfléchie qui a été mesuré.
A posteriori, cet incident a peut-être été une chance inespérée. Les mesures effectuées
en février avec le moteur d’ascension droite sont difficilement exploitables. Toutefois,
les mesures de février sans moteur d’ascension droite ont été effectuées en priorité
pour permettre les comparaisons avec les mesures de décembre, en sorte que les
mesures plus tardives avec le moteur ont été réalisées alors que de nombreux nuages
s’étaient déjà développés.
When the Sun came back after the chimney, mare’s tail clouds appears, just after
noon, and the photomultiplier signal went erratic. A tension as large as 5000 mV was
obtain on a cloud close to the Sun and the shutter was immediately turned closed. So
fine conditions were not found before the end of February, so that the measurements
performed on the 14th December 1996 are brought up.
A complete lack of experience added to the fear to destroy the photomultiplier in case
of a wrong operation ( at the beginning of each measurement series, the diaphragm
tube is trained at a distance of the Sun surface as low as twice the tube opening, the
slightest unexpected move should have destroy the photomultiplier), don’t allow for
understanding immediately that the right ascension motor was not correctly engaged.
The motor had been slightly tightened on its axes to allow for a smooth move. But in
fact it was insufficiently tightened and it turned around its axis instead of turning the
gear of the right ascension. The measurements are thus related to the passage of the
reflected light, after the Sun, while the earth is turning.
This incident was probably a good luck. The curves obtained from the measures
performed end of February, with the right ascension motor switched on, are even
worst. However, these measures have been performed after the measures with the
motor off, in order to allow for comparison with the curves obtained in December.
The bad results, with the right ascension motor on, may probably result from clouds
appearing at that time.
134
Les huit premiers essais ont été réalisés en prenant directement note des valeurs lues.
Le rythme d'écriture ne dépassait pas 1 par seconde. Pour augmenter le nombre de
points de mesure, le moteur d'entraînement de l'analyseur a été changé. C'est
seulement à cet instant déjà tardif que l'idée d'enregistrement sur magnétophone est
venue à l'esprit. Les 3 dernières séries de mesures ont pu être enregistrées avant que le
Soleil ne soit caché par une cheminée. La troisième de ces séries à été perturbée très
tôt par la cheminée et les résultats sont inexploitables.
La densité électronique de la couche K diminuant avec la distance à sa surface, on
s’est efforcé de commencer à viser un point aussi proche de la surface du Soleil ne
donnant pas un signal du photomultiplicateur supérieur à 1500mV. La lunette de visée
permet d'estimer que la distance correspondante à la surface du Soleil n'excède pas
1/5 de rayon solaire. L'enregistrement est arrêté lorsque la variation est inférieure à 1
mV.
22 Exploitation
Il s'en faut de beaucoup que l'exploitation des courbes soit aisée. Le phénomène
cherché ne représente que 1% de la tension maximale mesurée au début de chaque
série. Les séries de mesures enregistrées présentent des variations brutales de
plusieurs dizaines de mV. Les 8 séries prises par écrit, effectuées plus d’une heure
avant les autres, ne présentent pas ce défaut, probablement dû à des formations
brumeuses augmentant la diffusion.
Les courbes obtenues ont été corrigées d'une décroissance d'éclairement inversement
proportionnelle au carré de la distance à la surface du Soleil, mesurée par la position
de l'analyseur.
The eight first tests were performed while the signal values were directly registered in
writing. No more than 1 value per second was registered. In order to increase the
number of values, the analyzer motor was changed for a slower one. But at that very
late moment, the idea to register the value with a tape recorder went to mind. The
three last measurement series were recorded before the Sun went hidden by a
chimney. The third one was affected very early by the chimney and it is not usable.
As the electronic density of the K layer is decreasing with the distance to the Sun, the
series begin as close as possible to the Sun surface at a point where the signal is as
high as 1500 mV down to a point where the signal does not change by more than 1
mV. The optical sight allows to estimate that the initial distance to the Sun surface
was not exceeding 1/5 Sun radius.
22 Analyses
The analyses of the results is not easy. The effect searched is not larger than 1% of the
maximum voltage at the beginning of each series. Moreover, for the recorded series,
fierce voltage variations occurred; they may be several time 10 mV. As the 8 series
registered in writing are not affected by such variations, it is assumed that they result
from misty cloud development increasing the diffusion.
The curves obtained have been corrected by signal voltage decrease proportional to
the distance to the Sun surface to the minus 2. The distance is measured by the
position of the analyzer.
135
Pour les courbes enregistrées, cette correction a été faite par zone de décroissance
moyenne régulière entre les variations brutales mentionnées ci-dessus; seules les
parties des courbes comprises dans un intervalle intérieur à ces zones ont été
exploitées. Par chance, une partie des mesures de la première série enregistrée
présente une zone couvrant une fois et demie la rotation complète de l'analyseur. La
situation est moins favorable pour la seconde série; mais la zone allant de RS à 2RS de
la surface du Soleil est exploitable. Dans tous les cas, la chute très rapide du signal,
dans la zone comprise entre la surface du Soleil et une distance de 1 rayon solaire, ne
permet pas l’exploitation des valeurs relevées. Pour cette raison, seules les séries
prises par écrit numérotées 1, 5, et 6 ont pu être utilisées soit parce qu’elles couvrent
deux tours de l’analyseur soit parcequ’elles ont commencé assez loin du Soleil; les
autres relevés, ayant été interrompu trop tôt, ne sont pas utilisables.
Il faut ajouter que la rotation de l'analyseur n'était pas parfaite en raison de l'élasticité
de la courroie, non prévue pour une vitesse aussi faible. Cependant la révolution
complète est un paramètre incontestable, car constaté indépendamment de la vitesse
de rotation.
Bien que la période soit la plus basse du cycle solaire, bien que la saison soit la plus
défavorable, bien que la pollution hivernale du site d'observation soit parmi les pires,
bien que de nombreux problèmes matériels aient largement dégradé les conditions des
mesures, bien que l'opérateur manquât totalement d'expérience pratique dans tous les
domaines concernés, mais grâce sans doute à une matinée exceptionnelle, et grâce
aussi à la puissance du photomultiplicateur et à son excellente linéarité, l’ensemble
des courbes exploitables obtenues ne laissent aucun doute.
For recorded series, this correction is implemented for each zone where the light
decreasing is sufficiently regular so that the fierce voltage variations cannot affect the
interpretation. Only these zones have been used. By chance one zone of the first
recorded measurement series is longer than one and an half revolution of the analyzer.
The situation is not so good for the second series where only the zone included
between the distance RS and 2RS to the Sun surface has been used. In all cases, the
decrease of the signal between the Sun surface and a distance of 1 radius is so fast that
the measurements cannot be used. For this reason, the series registered in writing N°
1, 5 and 6 only have been used, either because they involve two revolutions of the
analyzer or because they begin far enough from the Sun; the other series cannot be
used.
It shall be added that the rotation of the analyzer is not regular as the stripe was not
intended for a so low rotation speed. Nevertheless the full revolution of the analyzer is
not doubtful as it was measured by an independent way.
Although the period was the lowest of the Sun cycle, although the season was the
worst, although the winter pollution of the site was among the worst, although many
technical problems lowered deeply the observation conditions, although the operator
recognizes a complete lack of practical experience in all the technical fields involved,
but thank to an exceptional weather and to the power and linearity of the
photomultiplier, all the curves obtained leave no room for any doubt.
136
Entre des distances de 1 à 3 rayons solaires de la surface du Soleil:
- l'amplitude maximale des variations de l'éclairement, mesurée pour chaque
demi-période, est inférieure à 20mV au lieu de 40mV. (Un premier calcul
effectué à partir des valeurs connues et de l’ouverture du tube à diaphragmes,
donnait 65mV pour une tension de contrôle de 0.8V, valeur qui a été portée à
40mV, pour une tension de contrôle de 0.6V, compte tenu du signal réellement
obtenu pour le bleu du ciel).
- des maxima et minima d'éclairement se retrouvent le plus souvent tous les 90°,
c'est-à-dire 4 fois par tour et certainement pas 2 fois seulement.
distance to the Sun surface
distance à la surface du Soleil
RS
2RS
3RS
40
30
20
10
0
180
360
540
sortie PMC en mV, rotation analyseur en degré en abscisse
PMT signal mV , anlyzer rotation in x-axis in °
1ère série de mesures enregistrée(14 décembre 1996 11:00)
first recorded measurement series (December 14th, 1996; 11:00)
Between 1 and 3 radius to the surface of the Sun:
137
720
- the lighting variation maximum amplitude, measured for each half period is
lower than 20 mV instead of 40 mV. (A first calculation, done with the values
given by books and with the diaphragm tube opening, lead to 65mV for 0.8V
control voltage. This value was increased up to 40mV for 0.6V control voltage
taking into account the effective signal given by the blue sky light).
- Light maxima and minima are observed the most often every 90°, that is to say
4 times per revolution, and in no case two times only.
Le 22 février 1997, des conditions semblables à celles du 14 décembre 1996 se
sont présentées, cependant un fond de brume subsistait et des nuages épars
perturbèrent plusieurs séries de mesure. On a reproduit ici la première, réalisée
comme le 14 décembre sans moteur d’ascension droite. La mesure commence à
plus d’un rayon de la surface du Soleil. Avec une tension de contrôle de 0.6V,
comme en décembre, la première valeur, à cette distance, peut dépasser
1400mV. On notera que les essais de vérification sans analyseur, effectués entre
temps, ont nécessité son enlèvement et qu’il n’a pas été recollé dans la même
position, en sorte que les maxima ne se produisent pas aux mêmes angles que
pour les mesures du 14 décembre. Toutefois toutes les courbes obtenues avec
les mesures du 22 février montre des maxima aux mêmes angles. .
On observe la polarisation normale de la couche K par le Soleil. La partie
gauche de cette courbe semble montrer que l’effet galactique n’a pas
entièrement disparu. Les autres courbes obtenues ne descendent pas aussi près
de la surface du Soleil et ne présentent pas ce maximum galactique. L’amplitude,
à distance égale de la surface du Soleil, est pratiquement le double de celle
constatée en décembre, mais une partie de l’augmentation résulte certainement
de l’augmentation de l’éclairement.
distance to the Sun surface
RS
distance à la surface du Soleil
2RS
3RS
40
20
0
0
180
360
sortie PMC en mV, rotation analyseur en degré en abscisse
1 ére série de mesures (22 février 1997 10:45)
PMT signal mV , anlyzer rotation in x-axis in °
first measurement series (February 22th, 1996; 10:45)
Similar conditions as on the 14th December 1996 were found on the 22nd
February 1997. Nevertheless the weather was misty and clouds disturbed
138
several series of measurements. The first curve was obtained with the right
ascension motor off. The first measurement is performed at a distance of the
Sun surface not smaller than one radius. The first value is 1400mV for a control
voltage of 0.6V as in December. The analyser has been removed in order to
perform the verification without analyser and is has not been sticked back
exactly in the same position, so that the angle of the maxima cannot be
compared with the curve obtained on the 14th of December. Nevertheless, all
the curves obtained on the 22nd of February show the maxima for the same
angles.
The normal polarization of the K layer by the Sun appears. The left part of the
curve shows that the Galactic effect seems not to have fully disappeared. But
this is not the case for greater distances from the Sun surface as shown also by
the other curves obtained from which none is going so close to the Sun surface
as this one. The amplitude is twice the one measured in December, nevertheless
a part of the increase is certainly resulting from the increase of the lighting.
139
30 CONSEQUENCES
31 La polarisation de la lumière
Les deux défauts de moment cinétique des électrons de la couche K, liés donc l’un à
la Galaxie, l’autre au Soleil, entraînent ainsi la polarisation de la lumière qu’ils
réfléchissent, dans les deux directions transversales perpendiculaires aux directions de
ces défauts de moment cinétique.
Le fait que la polarisation de la lumière correspond elle-même à une lacune dans la
répartition des moments cinétiques, que la lumière doit donc transporter, résulte de
l’unicité du phénomène. La polarisation de la lumière n’est qu’un seul phénomène en
ce sens que les polariseurs et les analyseurs sont interchangeables.
L’interchangeabilité n’est pas toujours facilement réalisable. Ainsi il serait difficile
d’utiliser la polarisation de la couche K comme analyseur d’une lumière
préalablement polarisée, car il faudrait soit une source polarisée de la même puissance
que le Soleil, soit polariser toute la lumière émise. Mais on verra plus loin que l’on
peut concevoir une expérience équivalente. En tout état de cause, aucune expérience à
ce jour ne montre le contraire. De plus, la proportion de lumière polarisée réfléchie,
dans la polarisation vitreuse, est telle qu’il est exclu que les électrons soient les seuls
agents de la réflexion et donc de la polarisation. En conséquence la polarisation n’est
pas une interaction de la lumière avec les électrons. L’unicité du phénomène entraîne
qu’il est impossible que la polarisation de la lumière soit d’origine électromagnétique.
30 CONSEQUENCES
31 Polarization of light
Both electron angular momentum defects, one linked to the Sun, the other one linked
to the Galaxy, produce as a consequence the polarization of light reflected by
electrons of the K layer, in the directions perpendicular to the angular momentum
defect directions.
The polarization of light is in fact a lack within the angular momentum distribution,
carried along by the light, as a result of the uniqueness of the phenomenon.
Polarization of light is one single phenomenon, that is to say that polarizer and
analyzer may be exchanged. The exchange is not always easy. For instance it is
difficult to imagine how the K layer polarization could be use as an analyzer for the
polarized source should be as bright as the Sun or all the light or the Sun should be
first polarized. But a similar experience can be imagined as it will be shown later in
this report. Any way there are no experience showing that the polarization is not a
single phenomenon. Moreover, the part of the polarized light by the vitreous
reflection is so high that electrons cannot be the only agents of polarization. As a
consequence, the polarization of light is not an interaction between light and
electrons. The uniqueness of this phenomenon makes it impossible that the
polarization of light could have an electromagnetic nature.
140
La polarisation est en effet parfaitement indépendante, non seulement des champs
électromagnétiques comme le montre la polarisation de la couche coronale K du
Soleil dans le plan de la Galaxie, mais également indépendante des charges
électriques comme on vient de le montrer.
La polarisation correspond à une lacune dans la répartition des moments cinétiques
transmis par la lumière. Chaque élément qui compose un pinceau lumineux transporte
un moment cinétique de direction définie, perpendiculaire à sa direction de
propagation. Il n’y a donc pas d’éléments de ce pinceau transportant un moment
cinétique perpendiculaire à la direction de polarisation, choisie historiquement comme
la direction d’éclairement maximal. En fait, les choses sont un peu plus complexes:
c’est la composante du moment cinétique de chacun des élements du pinceau
lumineux qui manque dans la direction perpendiculaire à la direction de polarisation
comme le montre la variation, en général elliptique, de l’éclairement derrière
l’analyseur.
C’est ainsi que la lumière qui nous parvient des étoiles de la Galaxie, principalement
situées à proximité du plan de la Galaxie, est partiellement polarisée par pompage
progressif, au cours de sa propagation, par la lacune Galactique de moment cinétique,
perpendiculaire au plan de la Galaxie.
Il doit en être de même pour la lumière qui nous parvient des autres galaxies et qui,
soit se trouvent à proximité du plan de la Galaxie, soit sont observées dans leur plan
principal.
Polarization of light is fully independent from any electromagnetic field, as shown by
the K layer polarization in the Galaxy plane, but, in addition, it is fully independent
from the electric charges as it has just been said.
Polarization is related to an angular momentum distribution lack carried along by
light. Each unit of a pencil of light carries along an angular momentum perpendicular
to the propagation direction, in a fully defined direction. There is not any unit of this
pencil of light carrying along an angular momentum perpendicular to the polarization
direction. This polarization direction has been historically chosen as the maximum
lighting direction. In fact, this is more intricate. The angular momentum component of
each unit of the pencil of light is missing in the direction perpendicular to the
polarization direction as shown by the elliptic distribution of the lighting after the
analyzer.
For the very same reasons, the light coming from stars, mainly located in the vicinity
of the Galaxy plane, is partially polarized by the progressive pumping of the angular
momentum component perpendicular to the Galaxy plane, of the light units by the
Galactic angular momentum lack.
The same pumping occurs for both the galaxies located near the Galaxy plane and the
galaxies observed within their main plane.
141
La conséquence fondamentale de cette nouvelle approche de la polarisation est de
remettre en cause le postulat de Maxwell. Si l’idée que la lumière et les ondes dites
électromagnétiques sont de même nature fut proprement une intuition géniale,
l’hypothèse que ces ondes sont en elles-mêmes de nature électromagnétique apparaît,
avec le recul, tout à la fois comme arbitraire et redondante. Il faut reconnaître qu’il
n’y avait guère de solution à l’époque. On était alors près d’un demi siècle avant que
la structure de l’atome ait commencé à être envisagée sous la forme que nous lui
donnons encore.
L’hypothèse est arbitraire, parce qu’elle attribue au phénomène une nature
électromagnétique, sur ce simple constat que les champs semblaient jouer un rôle
dans son émission et dans sa réception dans les domaines des fréquences
suffisamment faibles par rapport à la lumière. Il faut toutefois reconnaître que
l’hypothèse n’a rien d’absurde, elle est seulement arbitraire. Mais le plus grave est la
redondance. Elle n’est apparue que récemment. Le déplacement des électrons,
provoqué par le passage d’une onde électromagnétique, crée un champ
électromagnétique qui suffit pour expliquer les phénomènes observés. Il n’est
nullement nécessaire de supposer que les ondes sont en plus électromagnétiques en
elles-mêmes. Si une onde fait déplacer mécaniquement une charge électrique, il en
résulte un champ. Il n’est nullement nécessaire de supposer que l’onde soit un champ
par elle-même.
De l’arbitraire, de la redondance, et enfin de l’existence de la polarisation
indépendamment de tout champ électromagnétique et des charges électriques, il
résulte que le postulat de Maxwell n’est plus soutenable.
This new approach of the polarization of light has a main theoretical consequence. It
questions the Maxwell's postulate. Certainly it was a great idea to merge light with the
so called electromagnetic waves. Conversely this assumption that both are made of
the propagation of turning electromagnetic fields appears today to be fully arbitrary
and redundant. It shall be recognized that, at that time, there were not many solutions.
The structure of atoms, as they are still considered, had been discovered only more
than half a century later.
This hypothesis is arbitrary because it gives an electromagnetic nature to those waves,
within themselves, on this unique fact that electromagnetic fields seem to play a part
for their production within the frequency range low enough against light. Nevertheless
this assumption is only arbitrary and probably physically meaningless, but not absurd.
The redundancy is a more severe problem. It appears only with the discovery of the
structure of atoms. Wave may induce, only by a mechanical behaviour, the
displacement of electrons and an electromagnetic field is created by this
displacement; otherwise there are two electromagnetic fields in the meantime one
from the wave the other one from the electrons’ displacement. It is of no use to
assume that the waves are electromagnetic fields by themselves.
The result of arbitrary, the result of redundancy, the result independence of waves
from any electromagnetic field and from any kind of charge is that the Maxwell's
postulate is of no use.
142
On peut envisager une preuve supplémentaire, non pas de la nature même de la
polarisation de la lumière, car aucune expérience ne pourra jamais prouver une
théorie, seule l’erreur se prouve, mais de l’indépendance de la polarisation à l’égard
des champs électromagnétiques. Ce fut d’ailleurs l’objectif initial de la construction
du tube à diaphragmes. Au lieu d’observer la couche K, il était prévu d’examiner la
lumière réfléchie par un faisceau d’électrons dans le plan de la Galaxie. A défaut de
connaissance des ordres de grandeur, il était envisagé d’utiliser le faisceau d’un tube
de Perrin, acquis à cet effet, et des sources lasers. L’observation de la lumière
réfléchie était prévue à l’aide d’une monture spéciale permettant une rotation de 360°
dans un plan vertical contenant le centre de la Galaxie et de viser le centre de la
Galaxie. Une première série de mesure a été réalisée en vision directe avec des LEDs
fixées à l’obturateur du tube à diaphgrames. Aucune polarisation de la lumière émise
par les LEDs n’a pu être observée. L’idée d’utiliser la lumière réfléchie par les
électrons de la couche K dans le plan de la Galaxie est venue avant même de
commencer le montage du tube de Perrin.
L’observation de la couche K a permis de calculer le flux lumineux nécessaire pour
observer la polarisation dans le cas du tube de Perrin, puisqu’elle donne un ordre de
grandeur mesurable. Le résultat est qu’il faudrait un faisceau d’électrons dix fois plus
étendu et cent fois plus dense que celui du tube de Perrin, observé longitudinalement
dans un tube sous vide de dix mètres de long et avec un faisceau lumineux de mêmes
dimensions transversales que le faisceau d’électrons de 100 kW. Tout cela n’est pas
irréalisable, mais inaccessible financièrement.
The full independence of the light polarization from the electromagnetic fields could
be shown by another experience, although this would not be a proof of the nature of
light, as proposed by this report. No experience can prove a theory ; only errors can be
proved. The diaphragm tube was assembled in fact for this experience, and adapted
later on for the observation of the K layer of the Sun. It was intended for measuring
the polarization of light reflected by the electron beam of a Perrin's tube, bought for
that purpose, in the direction of the Galaxy center. It was easier to train the tube
toward the galaxy center than toward a point of the Galaxy plane. A special mount
had been erected for that purpose. A first series of measurements had been performed
with LEDs directly observed through the diaphragms. No polarization was found. The
idea to measure the polarization of the K layer of the Sun in the Galaxy plane came to
mind before the Perrin's tube was attached to the diaphragm tube.
The observation of the K layer allows for calculating the lighting needed to be able to
observe the polarization with an electron beam as it gives an order of magnitude of a
measurable phenomenon. The result is that the electron beam shall be ten times as
large as the one of the Perrin's tube, while it shall be hundred times as dense and as
long as ten meters. The lighting power shall be about 100 kW. All that is not
technically fully impossible but out an individual budget.
143
Il devient maintenant impossible de rester sur une position purement abstraite. Bien
plus les raisonnements enchaînés dans l’introduction n’ont qu’une apparence de
rigueur. On s’est efforcé de rester sur un plan purement conceptuel, mais sans support
physique de la gravitation et de la lumière la tentative est un exercice de haute voltige,
comme le sont tous les nominalismes et l’axiomatique en particulier. Avant de passer
au modèle physique sous-tendu par les développements conceptuels, une remarque
s’impose.
Pour des raisons psychologiques, semble-t-il, notre époque trouve dans Léonard de
Vinci un modèle, non seulement de l’Art, mais aussi de la pensée. Or son idée que la
connaissance procède d’abord de l’expérience est une absurdité insondable. Avec un
tel principe, nous en serions toujours à l’âge de pierre. On ne trouvera jamais des
tubes de Torricelli en haut du Puy de Sancy. Il fallait l’y monter, ce qui implique
l’objectif, la pensée d’abord. Dans le même ordre d’idée, on a observé la
décomposition de la lumière par les prismes pendant des siècles, avant que Newton,
par une série de mesures parfaitement structurée, sans autre dispositif qu’un tel
prisme, ne découvre la nature ondulatoire de la lumière. Or c’est l’objectif, la pensée,
qui a structuré l’expérimentation. La pensée précède l’expérience, toujours.
L’observation au hasard permet des découvertes, ainsi des galaxies et du décalage
vers le rouge par Hubble, mais il n’est jamais arrivé que l’explication sorte du hasard.
Les faits s’imposent, évidemment. A les nier ou à les contourner, on voile seulement
l’erreur. Mais la science n’est nullement une accumulation de faits. C’est d’abord une
démarche cohérente enchaînant les faits et les provoquant surtout.
It is now necessary to leave a purely abstract position to present some other
consequences. The reasoning presented within the introduction of this report have
only the appearance of logic. Remaining on a purely nominalist level, without any
physical support for light and gravitation, is purely a style exercise. There is no limit
on what shall be considered as unrealistic for the axiomatic development presently
considered as the only way for knowledge. In the mean time another way is also widespread.
For psychological reasons, Leonard da Vinci is taken as a model for both Art and
thinking. But this idea that knowledge proceeds first from experience is so far the
most unfathomable nonsense. With such a principle, the human being would have
remain at the age of stone. The decomposition of light by prisms was known many
centuries before Newton, using only such a simple prism, discovered the undulating
nature of light after a fully structured series of measurements. The series were
structure only by his mind toward his objective. Within the same range of
situations,Torricelli's tubes will never be found at the top of the Puy of Sancy where
Pascal's brother in law measured the atmospheric pressure decrease with altitude. He
first brought it there. And Pascal asked him to do so because he has first an idea, an
objective. Thinking comes always first, before experience. Some discoveries are the
results of observations made by chance, such as the discovery of galaxies and redshift
by Hubble. But the explanation never occurs by chance. Facts shall of course be taken
into account, otherwise there is only place for errors. Nevertheless, Science is not an
accumulation of facts. Science is first a consistent succession of facts linked to one
another and mainly emerging from an objective.
144
Mais, bien sûr, c’est une démarche sans issue, pour deux raisons : elle est
conditionnée par le contexte dans lequel nous vivons. A celui des mathématiques,
dans un cadre mécaniste, qui a prévalu sans partage pendant plus d’un siècle, succède
celui des modèles physiques dont le plus marquant est certainement la plasto-élasticité
dans l’étude du mécanisme de la rupture des matériaux. L’autre raison est qu’elle n’a
pas de fin. La cause première ne serait que la phrase qui l’énonce. C’est dire aussi que
l’on ne peut échapper au nominalisme. Il faut bien commencer par des hypothèses à
défaut de connaître la totalité de la Nature de l’infiniment grand à l’infiniment petit.
32 La gravitation et la lumière
Puisque la polarisation est considérée comme la propagation, si l’on peut dire, d’une
lacune de moment cinétique, qui peut être provoquée par une lacune de moment
cinétique liée au système de la gravitation, l’induction est que la lumière et la
gravitation utilise le même support. C’était l’essentiel du système de Descartes. Le
reste, qui a surtout été retenu, tient dans ses tourbillons. C’est un autre problème; il
convient toutefois de noter au passage que Descartes a consacré la moitié de sa
Théorie du Monde à expliquer comment ses tourbillons équilibrent la force centrifuge.
Aussi est-il très difficile de comprendre pourquoi Newton a condamné la théorie de
Descartes sur ce seul argument qu’il n’aurait pas équilibré l’accélération centrifuge.
Ironie de l’histoire de sciences, les tourbillons de Descartes ont une application
industrielle : les gyrocyclones. En réalité, Newton avait doublement tort car le
phénomène prévu par Descartes est encore plus fort qu’il ne le pensait puisque les
impuretés solides se rassemblent dans l’axe et à la base des gyrocyclones.
But, of course, the step has no end, and for at least two reasons. First, it is depending
upon the world we live in. After the kingdom of mathematics, where now something
appears to be rotten, it is the time for physical models. The most conspicuous of those
models is certainly the plasto-elasticity used for the study of the material breaking. No
doubt, other times will come. The other reason is that nobody knows the very nature
of things. That is to say that there is no chance to escape to the nominalism.
Hypothesis cannot be avoided.
32 Gravitation and light
As polarization of light is considered as the propagation of a lack of angular
momentum which may result from a lack of angular momentum within the gravitation
system, it is assumed that light and gravitation have the same support. It is the main
aspect of the Descartes' system. The other aspects are related to his whirls, and they
have only been kept in mind. As far as these whirls are concerned it shall be noticed
that half the World theory was devoted by Descartes to the balance of the centrifugal
force. Thus it is very difficult to understand why Newton has condemned the
Descartes' theory on this single fact that he would have not balance the centrifugal
acceleration.
By an irony of Science history, the Descartes' whirls have an industrial use in the socalled gyrocyclones. In fact, Newton was wrong. Even more, the phenomenon used by
Descartes is twice what he thought because the solid wastes are gathered in the axis at
the bottom of the gyrocyclone.
145
Continuer ici à accumuler les hypothèses, rendrait la démarche douteuse. En fait cela
résulte de l’ordre dans lequel on a voulu présenter les choses. Des principes et des
faits, on a déduit un certain nombre d’hypothèses enchaînées les unes aux autres.
Mais on peut procéder d’une manière plus simple: poser une hypothèse et lui
appliquer les principes en accord avec les faits. Cette démarche fut bien évidemment
la première. On s’est seulement efforcé de retarder autant que possible l’instant où, à
l’évidence, les yeux vont se fermer et les oreilles se boucher, si tant est que cela ne ce
soit pas déjà produit à la seule évocation d’un support pour la lumière et la
gravitation. L’Ether revient: c’est l’horreur!
Que ce nouvel Ether n’ait rien de commun avec celui de Lorentz, qu’importe, l’Ether
c’est l’horreur, exclue a priori.
Qu’importe si personne ne veut l’admettre. L’Ether remplissait l’Espace, comme l’air
l’Atmosphère, bien avant que l’homme n’existe, et le remplira encore bien longtemps
après sa disparition. Bien plus, sans Ether la matière n’existerait pas.
Qu’est-ce que l’Ether à présent? Un fluide composé de corpuscules. Ces corpuscules
se trouvent en état d’agitation brownienne, avec naturellement une vitesse moyenne
quadratique d’agitation exactement égale à la célérité de la propagation des ondes
dans cet Ether, mais ils ont aussi un moment cinétique réparti statistiquement
également, ce qui n’est pas le cas dans les fluides composés de molécules aux formes
plus ou moins complexes et dont la rotation propre se trouverait d’ailleurs gêné par
l’Ether.
It could be though that adding there a new hypothesis would jeopardize the attempt to
propose a new approach. But this is only the result of the way this approach has been
presented up to now. Starting from principles, several consequences, appearing as
hypothesis, have been deducted from one another. Nevertheless, there is another way
to present the new approach. That is to begin with one single hypothesis and to
proceed by deductions applying the principles in compliance with experiments. This
method was of course used first. The reasoning here above was intended to delay this
very moment where eyes will close and ears will stop, if indeed that has not already
occurred when it has been envisaged that light and gravitation need a support. Ether is
coming back! What a shame!
Although this new Ether is not a solid but a fluid and as such has no connection with
the Lorentz’s Ether, it does not matter. Ether is a shame; fully unacceptable! Those
who are so proud of the past changes, do not want now to change anything. Whatever
they say Ether fills space, like air fills the Atmosphere. In addition, matter cannot
exist outside Ether.
Ether is now a fluid made of corpuscles. Within this fluid, the propagation speed of
waves is exactly the quadratic mean brownian speed. In addition the corpuscles have
their own angular momentum, also statistically distributed. This is not the case of air,
water or similar fluids, where the molecules have intricate shapes so that spinning is
not possible. Moreover spinning would be impeded by Ether.
146
Toutefois, dans la propagation des ondes dans l’Ether, ondes nécessairement
longitudinales, seule la composante transversale du moment cinétique peut se
transmettre.
Ces ondes sont créées par le mouvement des corps dans l’Ether. Mais c’est
principalement les mouvements des électrons qui sont à l’origine de la plupart des
ondes de l’Ether utilisables pratiquement comme les ultraviolets, l’infrarouge ou les
ondes dites électromagnétiques. Les électrons peuvent ainsi provoquer dans l’Ether
des petits trains d’onde de même dimension transversale que la leur. Chacun de ces
petits trains d’onde transmet la composante transversale du moment cinétique qui lui a
été communiquée par son électron générateur. Chacun de ces petits trains d’onde
constitue ce qui a été appelé un photon, mais, dans l’éther, le photon n’est pas un
corpuscule, mais un train d’onde propageant une énergie ondulatoire bien définie et
fonction du mouvement de son électron générateur. La quantification des sauts
d’électrons suffit pour quantifier les énergies de ces trains d’ondes. C’est par là une
autre redondance des théories actuelles qui disparaît. Il n’est pas nécessaire de
quantifier les photons, qui peuvent naturellement transmettre le spectre continu de
tous les niveaux d’énergie. Mais justement, tous les niveaux ne sont pas possibles
puisque les sources, ces sauts d’électrons, sont quantifiées. En particulier, plus rien ne
s’oppose à la variation de l’énergie transmise, problème sans doute le plus irritant de
la quantification des photons eux-mêmes, quantification qui devient, il faut le répéter,
parfaitement inutile.
However, during the longitudinal wave propagation in Ether, only the transverse
angular momentum can be transmitted.
These waves are generated by the motion of bodies in Ether. Mainly they are
generated by electrons’ motions. UV, infrared, and the so called electromagnetic
waves, are a result of these electrons’ motions. Each electron generates a small wave
string with the very same length wave as the electrons' motion itself, with the same
transverse dimension and with the same transverse angular momentum. Each of those
small wave strings has been called a photon, but now the photon is no more a particle.
It is a wave string propagating longitudinally an undulating energy directly dependent
upon the frequency of its generating electron. Quantification of the electron’s jumps is
sufficient for quantifying the wave string energy. This was another redundancy of the
former approach. It is absolutely useless to quantify the photons as long as their
sources, the electron jumps, are already quantified. Thus photons may transmit the
continuum spectra. But all levels are not allowed because electrons’ jumps are
quantified. Moreover the energy transmitted by photons may change during the
propagation; this was a major inconsistency of the former approach including a
quantification of the photons themselves.
147
Au passage, on aura bien sûr éliminé l’objection possible sur la dispersion de ces
petits trains d’ondes. Il suffit pour cela de considérer que les corpuscules qui
composent l’Ether sont suffisamment sphériques pour propager leur ébranlement
seulement longitudinalement. D’un autre côté, ils ne peuvent être parfaitement lisses,
autrement, il serait difficile de concevoir comment il pourrait transmettre un moment
cinétique.
Contrairement à ce que pensait Descartes, les tourbillons ne peuvent être que la
conséquence d’un autre phénomène et non la raison de la rotation des astres. Pour
expliquer la gravitation par ce nouvel Ether, il faut tenir compte d’un fait qui n’a rien
changé aux théories actuelles malgré son énormité. Le fait que la matière soit
pratiquement vide n’est vraiment admis sans réserve que depuis soixante ans, c’est à
dire très largement après l’élaboration des grands postulats de l’optique et de la
gravitation. L’Ether remplit tout l’Espace, y compris entre les atomes et l’intérieur
même des noyaux des atomes qui doivent, de toute nécessité, être des bulles, seule
structure dont la masse, à épaisseur donnée,soit proportionnelle à la surface apparente,
là aussi condition essentielle pour qu’un fluide ait une action proportionnelle à la
masse.
Il faut que l’Ether se condense dans la matière pour que les corps soient comme attirés
les uns par les autres. Au passage, on aura noté aussi qu’ainsi l’Ether constitue la
matière où il se condense, du moins dans les lieux de l’Espace où nous observons une
condensation, car rien n’exclut qu’il y ait des lieux où, au contraire, la matière
s’évapore en Ether.
Many objections may come to mind. The first, certainly, is the dispersion of such
wave strings. There is an easy answer. The corpuscles of Ether shall be as similar to
spheres as possible, so that only the longitudinal motion is transmitted. On another
hand, they cannot be perfectly smooth otherwise they would not be able to transmit an
angular momentum.
Descartes’ whirls cannot exist and continue by themselves. They are a consequence of
another phenomenon and not the root cause of the rotation of all heavenly bodies. It is
necessary to take into account one of the most drastic change in the knowledge we
have about matter : it is nearly fully empty. This is now fully accepted for sixty years,
that is to say a long time after all the postulates related to light and gravitation were
proposed.
Ether fills Space from its most remote parts down to the inside of atoms and even
down to the inside of the atoms’ kernel. Atoms’ kernel shall be bubbles so that the
effect of an Ether stream on them can be proportional to their surface. This is an
essential condition achieving the proportionality of Ether streams action to the mass
of atoms.
Then Ether shall condense inside the atoms’ kernel so that bodies may be attracted by
one another. This is an appearance of course; they are not attracted, but pushed
reciprocally by the condensation stream of other bodies. Thus matter in made of
Ether. Such a condensation may not be general; there may be places in Space where
matter vaporizes in Ether.
148
Et c’est là que reviennent les tourbillons de Descartes. Pour les corps assez massifs, la
condensation provoque une mise en rotation. Le phénomène est en réalité très
complexe et on est conduit à constater l’existence, d’ailleurs justifiée par les
principes, d’une double série de tourbillons alternés autour de ces corps que sont les
astres.
La rotation de l’Ether dans les tourbillons conduit à expliciter le concept de défaut
potentiel de moment cinétique utilisé dans l’introduction. Le moment cinétique des
corpuscules de l’Ether en rotation dans les systèmes galactiques ou planétaires, doit
être compensé. Cette compensation se fait au détriment du moment cinétique propre
de ces corpuscules. Le concept utilisé pour une présentation aussi acceptable que
possible des choses, devient une réalité. Il y a un défaut effectif dans la répartition
statistique des moments cinétiques des corpuscules de l’Ether. Les degrés de liberté
des corpuscules se trouvent limités et ainsi les équations du mouvement modifiées.
D’une variation de la vitesse tangentielle en 1/√d, on passe à une variation en 1/d par
une zone à vitesse constante.
L’objectif ici n’est pas de développer entièrement ce nouveau modèle de la gravitation
dans son ensemble, mais d’en donner les éléments essentiels suffisants pour compléter
le tableau relatif à la lumière. On vient de répondre à tous les points soulevés relatifs à
la polarisation de la lumière, mais il y a bien d’autres conséquences pour la lumière.
Et d’abord la déviation de la lumière par les astres, phénomène prévu par Descartes,
mais le calcul montre que les tourbillons de Descartes ne donnent que la moitié de la
valeur mesurée, l’autre moitié résulte de la condensation que l’on pourrait appeler
l’effet Newton.
There whirls may come back. For heavenly bodies large enough, the condensation
generates a rotation. The phenomenon is quite complex and in fact, the equatorial
whirl is surrounded by two alternate series of whirls on each side. This is for
complying with the Hamilton principle.
Rotation of Ether within those whirls allows for explaining physically the angular
momentum lack as defined in the Introduction. The angular momentum of the Ether’s
corpuscles, shall be compensated. The compensation is done by pumping in the
proper angular momentum of these corpuscles. The concept use for the presentation
of the polarization of the K layer is now physically existing. This is an effective lack
in the statistical distribution of the Ether particles’ angular momentum. The freedom
degrees of these corpuscles are thus limited and the equations of the motion of Ether
streams are modified accordingly. From a tangential speed variation as 1/ √ d, it
becomes as 1/d, after a constant tangential speed intermediate zone.
This is not the objective of this report to develop a full model for gravitation, but only
to give the main issues necessary to understand the main properties of light.
Beyond the polarization of light; there are other consequences for light. First it shall
be deflected by heavenly bodies like the Sun. This had been foreseen explicitly by
Descartes, but his whirls only account for half the measured deflection. The other half
is a result of condensation and could be designated as Newton’s effect.
149
Puisque ainsi c’est l’Ether qui entraîne la Terre en rotation autour du Soleil, bien
évidemment il serait absurde de chercher à mettre en évidence un mouvement de la
Terre par rapport à l’Ether. C’est pourtant ce que Morley et Michelson ont essayé
avec leur fameux interféromètre, sans succès évidemment, mais il pensait à l’Ether
absolument immobile de Lorentz, un autre monde!
Par contre, si vous faites tourner la lumière dans deux sens différents grâce à des
miroirs disposés à la périphérie d’un disque, à la surface de la Terre, où l’Ether est
pratiquement au repos, bien sûr vous devez observer un décalage dès que le disque se
met à tourner. C’est ce que Harress et Sagnac ont tenté et ils ont réussi. Comble de
l’ironie de l’histoire des sciences, l’interféromètre de Michelson n’a aucune
application pratique dans le domaine pour lequel il a été conçu, alors que le disque de
Sagnac équipe tous les avions récents et les bas étages des fusées sous le nom de
gyrolaser.
L’interféromètre de Michelson a entraîné un immense bouleversement des théories
alors que l’appareil de Sagnac, largement postérieur n’a rien changé en retour.
Comble de malheur, son résultat reste totalement inexpliqué. Il y a eu de nombreuses
tentatives, toutes sont dédiées, la plupart comportent des hypothèses spécieuses sur le
mode de réflexion. Une seule est générale et admise par toute la communauté
scientifique.
A défaut de métrique non euclidienne représentant la répartition des accélérations
dans la rotation d’un solide, on a cherché à démontrer que la Relativité Restreinte
pouvait donner une première approximation du résultat obtenu par Sagnac. Le calcul
conduit d’ailleurs à une excellente approximation, mais il comporte une erreur fatale.
As the Earth is dragged by Ether it would be a nonsense to try to found any motion of
the Earth with respect to Ether. Nevertheless that had been tried by Morley and
Michelson with their famous interferometer. They failed of course, but they had in
mind that motionless absolute Lorentz’ Ether, another world!
Conversely, when light is turning around a disk fitted with mirrors for that purpose, in
the two opposite ways, of course a shift of the central fringe shall occur as soon as the
disk is set turning. The disk is turning with respect to Ether while Ether has no motion
with respect to the Earth. That had been tried by Harress and by Sagnac. They
succeeded of course. The most surprising is that the Michelson’s interferometer has
no use within the frame of its purpose although the Sagnac disk is used within all
recent planes and within lower parts of space rocket. It is known as the gyrolaser.
The Michelson’s interferometer negative result introduced a wide change within the
theoretical world, while the Sagnac positive result, coming too late, did not change
anything. But the worst of all is that the Sagnac’s result remains desperately
unexplained. All tentative explanations are dedicated and introduce generally a
specific hypothesis on the reflection not use elsewhere. It was not possible to found a
book in English where the only known solution is explained. Even worst, Sagnac and
Haress seem to be unknown. For this reason, reference is made here after to the
French publication "Les vérifications expérimentales de la Relativité Générale"
(Masson et Cie, 1964).
150
On se réfère ici à la présentation facilement accessible donnée dans l'ouvrage "Les
vérifications expérimentales de la Relativité Générale" (Masson et Cie, 1964). On
passera sur le fait que le passage d'un domaine d’intégration au domaine tangent, à un
instant donné, est un exercice de la plus grande audace, eu égard au fondement même
de la Relativité. L'auteur soi-même s'en excuse, en quelque sorte, quelques pages plus
loin, reconnaissant le caractère, au fond absurde, de la démarche, et affirmant
l'impérieuse nécessité d'avoir recours à une solution non-euclidienne qui, par malheur,
reste encore à trouver. Ne nous étonnons pas davantage, que l'auteur intègre une
différentielle de temps de parcours le long du contour du disque et non le long du
parcours de la lumière, ce qui confirme la confusion volontaire mentionnée d'abord.
Considérons seulement l'impressionnant développement qui s'étale de la page 68 à la
page 81, et allons d'emblée au résultat V-56. La différence de temps de parcours entre
les rayons lumineux parcourant la périphérie du disque en rotation, mais chacun dans
un sens, est de 4ωSi(0) /c2, c'est-à-dire proportionnel au double de l'angle d'intégration
θ=2π comme on peut d'ailleurs s'en assurer par V-52. En conséquence la différentielle
de la différence de temps de parcours doit être proportionnelle à la différentielle de θ.
Comment la différence des écarts pourrait-elle différer de l'intégrale de la différence
des différentielles? Or cette dernière intégrale reste désespérement identiquement
nulle comme on peut s'en assurer en regardant seulement l'équation V-50. En
application des principes relativistes, le signe du second terme de la parenthèse ne
peut changer en aucune circonstance. Le gyrolaser reste inexpliqué.
Il réserve en outre une surprise de taille. Placé dans un satellite géostationnaire, il ne
doit pas indiquer de rotation dans le plan équatorial de la Terre, par analogie avec
l'interféromètre de Michelson dans le nouvel Ether.
As a non Euclidean metric has never been found to accommodate the acceleration
distribution for a spinning body, the author proposes to find an approximation using
the Relativity, although she recognizes that only a non Euclidean solution is allowed.
Nevertheless, the result is perfect, unfortunately there is a fatal error in the
calculations.
Notwithstanding that it is not allowed by the Relativity postulates to switch from one
space to the tangent one at a given time, the author performs the integration of the
light course time differential along the disk circumference and not along the path of
light. But this is only to confirm that the relativity postulates are not implemented as
already said.
The result is that the difference between both light paths along the rotating disk is
4ωSi(0) /c2, that is to say proportional to twice the integration angle : θ=2π. As a
consequence , the differential of the difference between the light path duration is
proportional to θ. Now then, the difference between the time gaps is equal to the
integration of the difference between the differentials. The problem is that this
difference remains unfortunately permanently as low as zero as it is stipulated by the
relativist postulates. The gyrolaser remains unexplained.
Furthermore there is an additional problem. When fitted within a geo-stationary
satellite, the gyrolaser shall not show any rotation in the Earth equatorial plane. The
reasoning is similar to the Michelson’s result within the new Ether.
151
7
Les conséquences de la seconde variante
de la troisième
hypothèse du Professeur Allais
pour expliquer les résultats de
ses analyses des mesures
interférométriques de Miller
The consequences of Pr. Allais'
second option of his third hypothesis
to explain
the results of his analysis
of Miller's
interferometer measurements.
0ctober 1997, March 1998
152
Sommaire/ Summary :
E vidi gente , por lo vallon tondo ,
Venir tacendo , e lagrimando , al passo ,
Che fanno le letàne in questo mondo.
Come 'l viso mi scese in lor piu basso,
Mirabilmente apparve esser travolto
Ciascun dal mento al principio del casso :
Che nalle reni era tornato 'l volto ,
E indietro venir li convenia ,
Perchè 'l veder dinanzi era lor tolto.
153
Les résultats obtenus à l'interféromètre
de Michelson par Miller ont été
analysés par le Professeur Allais. Il en
résulte une incontestable corrélation
entre la vitesse mesurée et la position
spatiale de la Terre au cours de l'année.
The results obtained with the
Michelson interferometer by Miller
were analysed by Pr. Allais. The result
is an undeniable correlation between
the measured speed and the spatial
position of the Earth during the year.
Ce résultat est tout à la fois contraire à
la théorie de la relativité et à la théorie
de l'éther de Lorentz.
This result is contrary both to the
theory of relativity and theory of the
ether of Lorentz.
Le Professeur Allais a proposé trois
interprétations.
Pr.
Allais
interpretations.
La première associe d'une anisotropie
de l'espace avec l'interprétation de
Michelson et présente, comme la
relativité, l'inconvénient de ne pas
apporter de réponse à l'expérience de
Sagnac. Elle n'a donc pas été examinée
davantage.
The first involves an anisotropy of
space with the interpretation of
Michelson and, like relativity, the
disadvantage of not providing an
answer to the experiment of Sagnac. It
was therefore not considered further.
La seconde interprétation associe
l'anisotropie de l'espace avec une
anisotropie résultant des vitesses
cosmique et orbitale de la Terre. Cette
interprétation n’apporte pas davantage
de réponse à l'expérience de Sagnac. De
plus, il n'a pas été possible de
comprendre comment les anisotropies
peuvent compenser en permanence le
complément des vitesses mesurées par
Miller à la vitesse de la Terre, alors que
sa direction est variable au cours de la
journée et de l'année sidérale. Il faut
bien expliquer que 75% de la vitesse de
la Terre autour du Soleil reste
indétectable en permanence.
The second interpretation involves the
anisotropy of space with an anisotropy
resulting of the cosmic and orbital
speed of the Earth. This interpretation
does not answer to the experiment of
Sagnac either. Moreover, it was not
possible to understand how those
anisotropies may balance permanently
the part of the speeds measured by
Miller coming in addition to the speed
of the Earth, while its direction is
variable during the day and the
sidereal year . One must explain that
75% of the speed of Earth around the
Sun
remains
permanently
undetectable.
La troisième interprétation reprend la
nécessité de l'anisotropie astronomique
et comporte en fait deux possibilités :
soit la Terre entraîne l'éther, soit l'éther
entraîne la Terre. Bien entendu, ces
deux possibilités sont conformes à
l'expérience de Michelson, mais
également à l'expérience de Sagnac.
The third interpretation still involves
the need for an astronomical
anisotropy and an alternative between
two additional possibilities. Either the
Earth drags the ether or the ether
drags the Earth. Of course, these two
possibilities are consistent with the
Michelson experiment, but also the
experience of Sagnac.
154
proposed
three
En l’absence de matière, l’éther ne peut
être entraîné que par les champs de
gravitation ou magnétiques, dans le
cadre des connaissances actuelles. Il
convient de noter que les astres se
déplaceraient ainsi avec un nuage
d’éther, plus étendu, dans le cas de la
Terre, que l’atmosphère.
In the absence of matter, ether could
be dragged only by the gravitational
or magnetic fields, in the context of
current knowledge. It should be noted
that the stars would move with a cloud
of ether, greater than atmosphere in
the case of the Earth.
Dans un système planétaire, comme
dans les galaxies d’ailleurs, l’éther
serait entraîné par les astres dans leur
mouvement autour de l’astre central.
In a planetary system, as in the
galaxies, the ether would be dragged
by the stars in their motion around the
central star.
Deux cas peuvent alors être envisagés.
L’éther pourrait être entraîné également
en rotation. Le Soleil entraînerait ainsi
l’éther dans son mouvement orbital
autour de la galaxie, mais cette couche
autour du Soleil serait elle-même
entraînée en rotation autour du Soleil.
Dans ces conditions, l’expérience de
Michelson serait expliquée de manière
redondante, puisque la Terre tourne
avec l’éther entraîné en rotation par le
Soleil et par elle-même dans son
mouvement orbital.
Two cases may then be considered.
Ether could also be dragged in
rotation. The Sun would drag the ether
in its orbital motion around the
galaxy, but this layer around the Sun
would itself be dragged in rotation
around the Sun. In these conditions,
the experiment of Michelson would be
explained redundantly, as the Earth
rotates with ether dragged in rotation
by the sun and by itself in its orbital
motion.
Une conséquence nécessaire est
l’impossibilité d’un éther solide exigé
par
la
transmission
d’ondes
transversales comme le sont les ondes
électromagnétiques. Mais ceci est
commun à toutes les solutions basées
sur un entraînement ou un mouvement
de l’éther et la solution à ce problème
constitue justement un aspect essentiel
de toute approche de ce genre. Un autre
problème majeur est évidemment la
symétrie axiale des rotations de l’éther
avec les astres. Enfin, cette solution
pose le problème des frottements dans
l’éther, mais c'est un problème commun
à toutes les solutions comportant un
éther. Cependant, la redondance des
explications des résultats de Michelson
est probablement fatale à cette variante.
A necessary consequence is the
impossibility of a solid ether required
by the transmission of transverse
waves, as are the electromagnetic
waves. But this is common to all
solutions based on a drag or a motion
of the ether and the solution to this
problem is just an essential part of any
approach of this kind. Another major
problem is obviously the axial
symmetry of the rotations of ether
with the stars. Finally, this solution
raises the problem of friction in the
ether, but it is a problem common to
all solutions involving an ether.
However, the redundancy of the
explanations of the results of
Michelson is probably fatal to this
variant.
155
Le cas où les astres n’entraînent pas
l’éther en rotation, mais seulement en
translation, permet d’éliminer la
redondance
de
l’explication
de
l’expérience de Michelson, et la
symétrie axiale. Cette solution est aussi
acceptable a priori que la variante de
l’entraînement des astres par l’éther.
Elle n’a pas été retenue, car il apparaît
incohérent de penser que l’entraînement
ne peut exister qu’en translation et non
en rotation. A tout le moins, il devrait
exister une rotation sidérale, dans le cas
de la Terre, de 360° par an.
The issue not involving a drag of the
ether in rotation by the stars, but only
in translation, may eliminate the
redundancy of the explanation of the
Michelson experiment, and the axial
symmetry. This solution is as
acceptable a priori as the alternative
involving the drag of the stars by the
ether. It was not accepted because it is
inconsistent to suggest that dragging
can exist in translation only and not in
rotation. At the very least, there
should be a sidereal rotation, in the
case of the Earth, 360° per year.
Bien qu’aucun argument définitif ne
permette de rejeter d’emblée ces
variantes des interprétations des
mesures de Miller analysées en temps
sidéral, c’est la seconde variante de la
troisième interprétation proposée par
Professor Allais, qui a retenu par les
ingénieurs associés dans leur théorie de
l'Espace.
Although no definitive argument
allows for rejecting these alternative
interpretations
of
Miller
measurements analysed according to
the sidereal time, the second variant of
the third interpretation proposed by
Mr. Allais, has been chosen by the
associates engineers for their theory of
Space.
Il faut rejeter l’éther solide bien sûr,
mais l’entraînement des astres par
l’éther explique avec la même
simplicité l’expérience de Michelson et
celle de Sagnac, sans redondance.
One must reject the solid ether of
course, but the dragging of stars by
ether explains with the same
simplicity both experiments of
Michelson and Sagnac without
redundancy.
La nature transversale de la lumière ne
peut pas être expliquée par la
mécanique des fluides. Les ingénieurs
associés ont donc proposé une
modification radicale en introduisant
les fluides à six degrés de liberté. Le
moment cinétique des corpuscules de
l'éther permet de rendre compte des
propriétés transversales de la lumière.
The transversal nature of light can not
be explained by fluid mechanics. This
is why the associated engineers
proposed a drastic change by
introducing the notion of fluid with
six degrees of freedom. The angular
moment of this ether fluid corpuscle
answers the transversal issue.
156
8
The inversion of the
electron magnetic
property
and its implication.
May 2014
157
I wish to thank all scientists who help me making this paper as clear as possible.
Abstract : This paper is related to the definition of electron. A first
inversion was done when attributing to the electrical currents the
wrong direction. But a second inversion could have been done for
the magnetic property of the electron. An experiment is proposed to
confirm this inversion.
Résumé : Ce rapport concerne la définition de l’électron. Une première
inversion a été faite en attribuant aux courants électriques le mauvais sens
d’écoulement. Mais une seconde inversion aurait pu se produire pour les
propriétés magnétiques de l’électron. Une expérience est proposée pour
confirmer cette inversion.
Key words: Electron ; magnetic field ; magnetic moment ; cathode ray ;
electrical current ; electric field.
Contents
1. Electron definition inversion
2. The proposed experiment
3. Figures
4. References
158
1. Electron definition inversion
The electric current within conductors was first given the wrong direction
before discovering electrons. The same inversion could have been done when
defining the intrinsic magnetic property of electrons. The wrong solution would
have been choosen in both cases.
1.1
The standard model of the electron.
The electron is a lepton with a negative elementary electric charge. The
invariant mass of an electron is approximately 9.109×10−31 kg.; their electric
charge is −1.602×10−19 coulomb. They have an intrinsic angular moment or spin
of 1⁄2. In additions electrons has an intrinsic magnetic moment along its spin
axis. It is approximately equal to one Bohr magneton, which is a physical
constant equal to 9.27400915(23)×10−24 joules per tesla. The orientation of their
spin with respect to their magnetic moment defines the helicity. Electrons have
zero colour charge. Electrons have properties of both particles and waves.
The electron is considered within the standard model as a magnetic dipole,
characterized by its intrinsic magnetic moment. So that it is analogous to a tiny
bar magnet.
In line with the axiomatic approach, it is postulated by Quantum Mechanics that
the magnetic moment of electrons is always kept stochastic both within
conductors and within cathode rays which are said to be unpolarized. As the
magnetic moments of electrons are distributed at random, they would not create
any magnetic field.
159
If the existence and properties of electrons have been subjected to many
experiments, it appears that the Rowland’s disk is the only experiment used to
demonstrate that the motion of electrons is the cause of magnetic field of
conductors.
The problem of obtaining a magnetic field by moving charges along a straight
line is linked to the MHD (magneto hydrodynamics), the dream of the 1960 to
produce directly electricity without turbine. It fails. There are only few
experiments which succeeded to obtain a very small effect but they used
superconductors where, I am convinced, the main part is played by the atomic
structure and not by the single fact of moving the charges.
1.2
The Maxwell equations (in void)
r
div(ε 0 E ) = ρ
r
div( B ) = 0
()
r
r
∂B
rot ( E ) = −
∂t
r
r
 B  ∂ ε0E r
rot   =
+J
∂t
 µ0 
( )
Where J is the product of the charge density by the mean speed of charges:
r
r
J = ρ m × vm
( )
r
∂ ε0E
has been added by Maxwell to
The second term of the fourth equation
∂t
the Ampere equation. It was formerly call the displacement current. When there
160
r
is no current J is null. This is the Maxwell-Hertz equation. The main solution
of this equation is a wave function which Maxwell assume to be the equation of
r
light. The problem is that when there is no current J then there is no moving
( )
r
∂ ε0E
charges, there is no field variation either, so that the term
should also be
∂t
null. So that, the Maxwell-Hertz is presently no more considered as a
consequence of the Maxwell-Ampere equation, but as a postulate.
Nevertheless this term was maintained because it explains that capacitors being
charged have a magnetic field.
1.3
The inversion of the intrinsic magnetic moment definition.
An inversion could have occured in the definition of the intrinsic magnetic
property of electrons of the standard model of Quantum Mechanics. Unlike the
standard model of the electron in Quantum Mechanics, the intrinsic magnetic
property of the electrons, call the magnetic moment, could not be similar to that
of a small magnet.
The other solution is that electrons could have a magnetic field with the
structure of an electric current.
The wrong definition was choosen as a result of a long history. The magnetic
field of electrical currents is presently considered as an effect of the translation
161
speed of electrons within conductors. This effect was first discovered by Gian
Domenico Romagnosi in May 1802 and reported to the French Academy of
Science which didn’t register the discovery. The Danish scientist Hans Christian
Ørsted made the same discovery twenty years later and the Danish Academy
published immediately his report. Nevertheless, Ørsted recognized that he was
informed of Romagnosi’s experiment. The cathode rays were dicovered at the
end of the 19e century. In 1895, the Perrin experiment shown they are
negatively charged. Thomson made in 1897 the bold suggestion that cathode
rays were material constituents of atoms and it turned out to be complying with
all further experiments. At that time the Maxwell-Ampere law was modified to
take into account the existence of electrons. But the cause of the magnetic field
was not questionned. The electrical current was understood as a flow rate of
electrons the other way round, opposite to the conventional direction of the
electrical current.
The choice of the definition of the standard model of the electrons was
considered to be confirmed by the Rowland’s experiment performed in 1876.
Nevertheless, in the Rowland’s experiment the disk rotates. This experience
shows that the rotation of an electric charge produces a magnetic field, but a
rotation cannot be considered as a translation. It is fully inconsistent to reject the
explanation of the rotating disk of the Sagnac experiment as an effect of a
translation and, in the mean time, to accept that the rotation of a charge could be
equivalent to a translation. This link between charge rotation and magnetic field
seems to be in fact in line with the standard model of electron.
162
There is another possible explanation, but it is less satisfactory. This experiment
is very similar to Barnett’s experiment with a magnet. It could be deduced that
magnets and conductors do have similar Weiss structures inside, but in the case
of conductors, they are not correctly oriented so that normally there is no
magnetic field. The rotation of the disk modifies the arrangement of electrons
within the conductor by the Coriolis acceleration resulting from the centrifugal
acceleration of the disk.
The consequence of this change in the electron magnetic property is that the
magnetic field created by the intrinsic magnetic property of the electrons has the
structure of the magnetic field of the currents as well as of cathode ray.
Standard electron dipole field
Inverse electron field
Consequently, the magnetic field of magnets can not result directly from the
intrinsic magnetic moment of the electrons as in the standard model.
The magnetic field of the magnets results from the arrangement of electrons in
circular structures, such as rings or coils, which would create therefore a field
163
topology consistent with reality. The Weiss domains would be therefore
structures different from that implied by the standard theory.
This inversion of field topologies would allow to consider a quantitative
explanation of the Barnett’s effect which remains purely qualitative for the time
being. One might think that the electron ring arrangements that causes the field
of magnets are tilted by the Coriolis’ acceleration resulting from the centrifugal
acceleration, giving thus an important amplification factor not included in the
standard model of Quantum Mechanics.
The inversion of the magnetic field definition of electrons would have a
consequence on the cause of the magnetic field of electrical current and cathode
ray. There are presently two co-existing potential causes of the magnetic field
by an electron flows: translation speed and “intrinsic magnetic moment”. The
“intrinsic magnetic moment” of electrons is presently considered as a result of
electron spinning, so that motion of the charge is considered as the only cause of
magnetism. Nevertheless, within a conductor or within a cathode ray, electrons
have finally two potential ways of producing the magnetic field. This is against
the specific causal uniqueness principle applicable for basic concepts of physics
as well as for geometry. It is also against the Ockham's simplicity principle. The
inversion of the magnetic field definition of the electrons would make possible
to attribute the magnetic field of electrical current and cathode ray directly to
the electron magnetic property.
164
1.4
The modified Maxwell equations
The change is only related to the Maxwell-Ampere equation, this is the only
term involved in the change of the electron magnetic definition inversion :
r
div(ε 0 E ) = ρ
r
div( B ) = 0
()
r
r
∂B
rot ( E ) = −
∂t
r
r
 B  ρmm
rot   =
µ0
 µ0 
r
Where vector m is the Bohr’s magnetron as defined by the standard model of
Quantum Mechanics, i.e. with the magnetic property of a small magnet, and ρ m
the density of electrons taking into account their average orientation. The
r
induction B may have several components if there are several average
orientations determined statistically from the electrons distribution. It may be
added that the same apply to all particles owning what is call by the standard
model an “intrinsic magnetic moment”.
The term added by Maxwell disappears because the magnetic field is always
produced by the magnetic property of electrons. In the case of capacitors being
charged, the magnetic property of electrons is oriented in the capacitor plate
when they move within the structure of the conductive plate modified by the
electric field and they are included in this new equation exactly as all other
electrons in conductors and cathode rays correctly oriented. This is exactly what
occurs within the cathode of the cathode ray gun. But, as in the capacitor the
165
electrons are not ejected outside toward the anode, they loose their magnetic
property orientation under the effect of unoriented electrons of the plate.
The field created by a cathode ray should result directly from the intrinsic
magnetic property of the electrons, now inversed . This is the point which could
be verified experimentally.
2. The proposed experiment
Electric fields don’t change the magnetic property of electrons. If the cathode
ray is deviated up to 90 ° by an electric field, the intrinsic magnetic property of
electron remains as it was, so that the magnetic field of the cathode ray should
no longer be measured by coils placed in a plane containing the beam, unlike
what can be observed before deviation.
A similar result would be obtained by a deviation by a magnetic field.
2.1
2.1.1
The deviation of the cathode ray by an electric field.
In the standard model of Quantum Mechanics, the electric field has no
influence of the intrinsic magnetic moment of electrons as shown by Maxwell’s
equations.
So that after the bend of the cathode tube nothing is changed within this theory
and the second sensor shall measure the same magnetic field as the first sensor.
166
2.1.2
But if the definition of the intrinsic magnetic property of electron is
inverted, and accordingly if the magnetic field of the ray is produced
exclusively by the magnetic property of electrons instead of their motion, then
there will not be any magnetic field measured by the second sensor, because this
sensor is installed in a position enabling only to measure a magnetic field
structure as generated by an electrical current. The magnetic field of the
electrons is maintained as it was before the bend. It could be measured only by
coils with their axis aligned with the ray axis after the bend.
2.2
The deviation of the cathode ray by a magnetic field.
2.2.1 A magnetic field will also deviate the ray by 90°. In the standard model,
the electrons produce a magnetic field by their motion. They move with regard
to the second sensor exactly as they move with regard to the first sensor before
the bend. The magnetic field measured by the second sensor shall be exactly the
same as by the first sensor.
There is a difference with the deviation by an electric field which nevertheless
does not change the result of the measurement. In the standard model, the
electrons are not polarized before the bend. But the magnetic field deviating the
electrons should polarize the ray so that a complementary magnetic field should
be measured after the bend. But it would not be measured by the second sensor
because the overall intrinsic magnetic field of the electrons after deviation will
have an induction colinear to the magnetic field used to deviate the electrons.
167
2.2.2
But here also, if the definition of the intrinsic magnetic property of
electron is inverted, and accordingly if the magnetic field of the ray is produced
exclusively by the magnetic property of electrons instead of their motion, then
there will be also not any magnetic field measured by the second sensor, for the
very same reason.
The magnetic field used for the 90° deviation of the ray will change the
magnetic property of the electrons orientation, but the null result of the second
sensor will not be changed. The electrons are assumed not to be like magnets
but like currents, i.e. they have a magnetic field with an electric current
magnetic field structure. So that a magnetic field induction will tend to align the
electrons intrinsic magnetic field induction. As these inductions are in the
opposite direction for diametrically opposed points around the electrons, thus
the electrons will take an intermediary position. But this new state of their
overall magnetic field cannot be measured by the coils as installed in the sketch,
because the B vector are not in the right direction and furthermore it seems that
they could cancel each other.
That does not reduce the interest of the deviation by an electric field, but the
experience is much easier to achieve, because the magnetic field depends on the
intensity, so that a low voltage may be used.
2.3. Experiment protocol (see figure page 14)
The spherical portion of an original Perrin tube shall be replaced by a pyrex tube
of 40 mm diameter bent at the middle by 90°. It could be for instance a NARVA
168
PR2 tube still available in the Internet. Figures hereafter are suitable for this
kind of tube.
The magnetic field of the cathode ray will be measured in the straight part of the
modified Perrin tube and after the curved part.
The anode voltage of 400 V to crest will be two waves rectified, but not filtered ,
so that the current induced in the sensor can be detected by an analogue to
digital converter after an amplification by an Operational Amplifier with a
factor of 200. The anode voltage will be maintained
throughout the
measurement. Cathod heating will be supplied by a DC 6.5 V power supply.
The cathode ray deviation up to 90° will be obtained by an electric field
produced by two semi-toroidal plates placed on each side of the tube in its
curved part. They will be supplied with an adjustable DC power supply from
200 to 2000 V, rectified and filtered.
In this experiment, two identical sensors are needed with 12 coils of 1300 loops
each connected to a resistor 1MΩ and connected to an operational amplifier
with an amplification factor of 200.
One of the sensors will be placed before the curved portion of the tube and the
other one after. Both signals delivered by the AD converter will be sent to a data
acquisition module connected to a PC.
169
The first sensor should show the magnetic field of the cathode ray as it has been
checked by the author in May 2000 with a straight tube. For a straight tube, the
magnetic field of the cathode ray, measured by the author, had precisely the
value calculated from the amperage of the cathode ray measured at the top of
the tube in the same as in the Perrin’s experiment.
The second sensor, located after the 90° deviation of the cathode ray in the
elbow of the tube, should show nothing in both cases of a deviation of the
cathode ray by an electric field or by a magnetic field.
3. Figures
CATHODE TUBE
2000 V=
-
300 V∼
CATHOD
HEATING
6.5 VC
+
5 mA
COILS
COILS
OP AMP
X200
AD CONVERTER
170
5 mA
Example of a modified Narva PR2 Tube used by the author to measure the
magnetic field of an electron ray magnetic field. The tube of the new experiment
proposed by this paper shall be bent by 90° in the middle.
171
The coils and the overall experimental device for the straight tube.
172
4. References
Baigrie, B.
(2006).
Bernard Pr. M.Y. (1960)
Electricity and Magnetism: A Historical
Perspective. Greenwood Press.
Initiation à la mécanique quantique Editions
Hachette
Hennequin Pr. J.
(1970)
Electromagnétisme et
Editions Dunod
relativité
Keithley, J.F.
(1999).
Landau Pr. L.
(1966)
The Story of Electrical and Magnetic
Measurements: From 500 B.C. to the 1940s
IEEE Press.
Théorie des champs
Editions
Mir,
Moscou
Liboff Pr.R.L.
(1908)
Introductory Quantum Mechanics Holden-Day
Inc
Mach Pr. E.
(1908)
De la connaissance et de l'erreur
Flammarion
Marinov S.
(1978)
Rotating disk experiments. Foundations of
Physics 8 (1-2)
Editions
Pérez Pr.J.P.& alt (1997)
Electromagnétisme Editions Masson
Tonnelat Pr. M.A. (1971)
Histoire du principe de relativité
Flammarion
173
restreinte
Editions
9
Gravitation
Zonal effects
Physics Treaty
1985
174
Zonal aspects of gravitation
1. Zonal distribution of comets
The inclinations of comets with regard to the plan of the Ecliptic are distributed
randomly. The histogram of the inclinations of the comets with regard to the
equatorial plan of the Sun, shows that the direct elliptic comets are not
distributed randomly, but in zones. The retrograde comets are in the gaps of the
direct elliptic comets distribution.
Comet Orbits being practically plane, comets cross a number of Sun zones
increasing with the inclination of their orbit with regard to the equatorial plan of
the Sun. For this reason, the zonal effect is observable only for inclination not
greater than 30°. Over that inclination, comets are changing too often of zone
while turning around the Sun, so that the zonal effect is completely disturbed.
There are practically no comets and no asteroids or planets as well in the close
vicinity of the equatorial plan of the Sun.
12
10
8
6
4
2
0
1 : TEMPLE-TUTTLE
2 : HALLEY
The zonal distribution of comets
in X-coordinate: the inclination of the orbital plan of comets with regard to the equatorial plan of the Sun
in Y-coordinate: the number of elliptic comets per interval of 1°
175
2. Zonal winds of gaseous planets
Zonal winds have been observed, for a long time now, at the surface of gaseous
planets.
Latitude
+ 60°
+ 30°
0°
-30°
- 60°
-100
0
100
200
300
400
500
vitesses en m/s
(continuous curve: Jupiter, dotted curve: Saturn)
Zonal Saturn and Jupiter winds (VOYAGER satellites probes)
176
3. Oceanic currents
The Atlantic Ocean, the Indian Ocean and the Pacific Ocean are animated by a
direct equatorial current, W/E direction, therefore in the direction of rotation of
the Earth. They also have two tropical opposite currents each side of the
previous one, then two direct moderate currents. Finally there are two polar
direct currents. That of the Antarctic is perfectly well defined. Its flow very
largely exceeds the 40 million m3/s.
There would be thus three zones well identified in the Northern Hemisphere
against two in the Southern Hemisphere
The nature of the Arctic and Antarctic currents ruins any attempt of thermal
explanation of the oceanic currents. The temperature is constant. Salinity, and
thus the density, cannot explain this phenomenon either. It is also constant
throughout this current.
177
4. Other phenomena linked to the zonal effect of gravitation
4.1 Equatorial plane of the Sun (for comets see § 1)
All the planets are in the vicinity of the equatorial plane of the Sun. This
is the case of the Earth.
This phenomenon has no explanation recognised by the entire scientific
community.
Poincaré published in 1901 (Poincaré, Œuvres mathematiqes ©
Gauthier-Villars, 1951 Volume VII p 41 to 217) important papers on the
rotating fluid masses. He highlighted, mathematically, the appearance of
flattened figures in condensing and rotating fluids. This phenomenon
has been mentioned several times on a qualitative point of view to
explain the equatorial concentration of bodies. Yet the complex studies
of Cournot (addition to the translation of the System Herschel, on the
distribution of cometary orbits) seem to show, a gap for the Sun system
body inclinations compared with the Gaussian distribution.
4.2 Galaxies
All stars are in the vicinity of the main plane of galaxies. This
phenomenon has no explanation recognised by the entire scientific
community.
4.3 The rings of Jupiter, Uranus, Saturn and Neptune
This phenomenon remains unexplained.
4.4 The Sun and the planets turn around themselves. Their rotation speed on
themselves decreases with latitude. (for gaseous planets see also §2).
This phenomenon remains unexplained.
178
10
Le champ magnétique
des
faisceaux cathodiques
Electron beams
magnetic field
May 2000.
179
1 Introduction
Des électrons peuvent être en translation sans champ magnétique. Dans
les directions observées, à un instant donné, le champ existe seulement
lorsque le tube cathodique est orienté dans un sens, il n'existe pas dans
l'autre sens.
Quelle que soit la cause de la disparition du champ, la présente note
montre que le champ magnétique d’un faisceau d’électrons ne peut en
aucune manière résulter de la translation des électrons.
L’analyse de la cause de la disparition du champ magnétique des
faisceaux d’électrons fait l’objet n’une note séparée.
1 Introduction
Electrons might have a motion of translation without having any
magnetic field. For each direction of measurements, at a given time, the
magnetic field exists only in one way but not in the opposite way.
Whatever is the root cause of the cancellation of the magnetic field, this
report establishes that the magnetic field of electrons can, in no way, be a
result of their motion of translation.
The root cause analysis of the magnetic field cancellation is included in
another report.
180
2. Dispositif expérimental
Le tube cathodique utilisé est un tube de Perrin modifié en sorte que les
bobines du capteur soient le plus près possible du faisceau. La partie
sphérique du tube d’origine a été remplacée par un tube en pyrex de 40
mm de diamètre.
La tension d’anode est de +400V en crête, redressé deux ondes, mais non
filtré, en sorte que le courant induit dans le capteur soit détectable par un
convertisseur analogique numérique après une amplification de facteur
200 ; l’ordre de grandeur de la tension mesurée est de 10mV. La tension
d’anode est maintenue tout au long des mesures. L’arrêt du faisceau est
obtenu en coupant le chauffage de la cathode. Le chauffage est alimenté
en 10 V redressé 2 ondes et filtré.
Le capteur est constitué de 12 bobines de 1300 spires chacune montée en
série et débitant sur une résistance de 1MΩ connectée à un amplificateur
opérationnel monté en sorte que le facteur d’amplification est de 200.
Le signal délivré est envoyé sur le convertisseur analogique-numérique
d’un module d’acquisition connecté au port parallèle d’un PC.
2. Device description
The cathodic tube used is a Perrin’s tube modified so that the sensor coils
are the nearest as possible from the cathodic beam. The spherical part of
the tube has been replaced by a pyrex tube 40mm diameter.
The tube anode is at up to +400V to crest, 50Hz, two waved rectified, but
not filtered, so that the modulated magnetic field of the beam might be
detected by the sensor after amplification factor of 200. The order of
magnitude of the signal delivered to the analogic-digital converter is 10
mV. The anode voltage is maintained throughout the measurements. The
electron beam is stopped by switching off the cathod heating. The cathod
heating is powered by a 10V two waves rectified and filtered current.
The sensor is made of 12 coils 1300 loops each. The sensor is connected to
a 1MΩ resistance and to an operational amplifier mounted in such a way
that the amplification factor is 200.
The signal is then delivered to an analogic-digital converter of the data
acquisition module connected to the parallel port of a PC.
181
CATHOD
HEATING
COILS
CATHODIC TUBE
OP AMP
X200
AD CONVERTER
182
183
3. Faits observés
Les résultats obtenus sont présentés dans le tableau ci-dessous. Les
mesures ont été faites dans deux directions : Nord-Sud et Verticale et
dans chaque direction le tube a été successivement orienté dans un sens
puis dans l’autre (vers le Nord puis vers le Sud, vers le haut puis vers le
bas). Le sens correspond au sens de déplacement des électrons entre le
canon et la cible. Dans chaque sens, deux mesures ont été réalisées. Les
valeurs indiquées sont les nombres d’occurrences de tensions
supérieures aux valeurs indiquées colonne de gauche dans les 1000
points enregistrés.
3. Observations made
The results obtained by the associated engineers are given in the table
below. Two directions were used : North-South and vertical and in each
direction the tube was oriented towards the two opposite ways, (ie North
then South, up then down) the ways referring to the cathodic beam from
the gun to the target. Two measurements were performed in each way
for the related direction. The values given by the table are the number of
occurences of voltage higher than the values of the left hand column,
among the 1000 aquired values.
29/04/00
HL12:20
fais vers
Nord 1
Beam
towards
North 1
>17 mV
>16 mV
>15 mV
>14 mV
>13 mV
>12 mV
2 ème mesure horizontale Nord Sud
nd
2 horizontal measurement North South
1ère mesure horizontale Nord Sud
st
1 horizontal measurement North South
nul vers
Nord 1
w/o Beam
towards
North 1
1
8
13
41
83
124
0
1
1
2
13
40
fais vers
Sud 1
Beam
towards
South 1
nul vers
Sud 1
w/o Beam
towards
South 1
0
1
1
14
61
124
84
nul vers
Nord 2
w/o Beam
towards
North 2
fais vers
Nord 1
Beam
towards
North 1
0
1
7
28
81
137
1
8
13
41
83
124
>17 mV
>16 mV
>15 mV
>14 mV
>13 mV
>12 mV
0
0
3
4
17
59
-13
fais vers
Sud 2
Beam
towards
South 2
nul vers
Sud 2
w/o Beam
towards
South 2
2
5
12
32
72
143
2
2
7
26
75
142
65
1
Ecart
1ère mesure verticale
st
1 vertical measurement
fais vers
Haut1
Beam
up 1
>17 mV
>16 mV
>15 mV
>14 mV
>13 mV
>12 mV
Ecart
nul vers
Haut 1
w/o beam
up 1
7
18
33
68
107
172
2
2
3
13
37
85
87
fais vers
Bas 1
Beam
down 1
2ème mesure verticale
vertical measurement
nd
2
nul vers Bas
1
w/o beam
down 1
0
0
1
1
1
9
fais vers
Haut 2
Beam
up 2
0
0
0
2
3
11
>17 mV
>16 mV
>15 mV
>14 mV
>13 mV
>12 mV
nul vers
Haut 2
w/o beam
up 2
5
11
22
30
65
120
2
109
nota: fais = avec faisceau (cathode chaude); nul = sans faisceau (cathode froide)
note :beam= with electron beam (hot cathod) ; w/o beam = without electron beam (cold cathod)
184
fais vers
Bas 2
Beam
down 2
nul vers Bas
2
w/o beam
down 2
0
0
0
2
3
0
0
0
1
2
11
8
0
0
0
2
5
11
3
4. Conclusion.
Le bruit de fond résulte essentiellement du champ magnétique du circuit
de mise sous tension de l’anode et dépend donc de la position des cables
qui changent en fonction de la position du capteur fixé au tube
cathodique.
Le champ magnétique qui résulte du circuit de chauffage de la cathode
est très largement inférieur au seuil détectable par le capteur. A titre
indicatif, le champ magnétique résultant du circuit de mise sous tension
de l’anode +400V est lui-même indétectable lorsque le courant est filtré
au lieu d’être simplement redressé deux ondes.
Le champ magnétique du faisceau cathodique est en phase avec le bruit
de fond à 50 Hz.
On constate que l’écart entre les signaux mesurés avec faisceau et sans
faisceau dépend de l’orientation du tube. Il est pratiquement nul lorsque
à l’heure de mesures, le faisceau est dirigé vers le bas et vers le Sud.
Il est maximal lorsque le faisceau est dirigé vers le haut et vers le Nord.
4. Conclusion
The background noise results mainly from the magnetic field the anode
voltage circuit, and thus depends upon the cable routes which are
modified when changing the tube and sensor position.
The magnetic field associated to the cathod heating is several orders of
magnitude lower than the sensor threshold. This is confirmed by the fact
that the magnetic field of the anode circuit is not detected by the sensor
when the related current is not only two waves rectified but also filtered.
The gap between the signals measured with electron beam and without
electron beam is nearly nul when the beam is either oriented downwards
or towards the South.
This gap is maximum when the beam is oriented either upwards or
towards the North.
185
4. Conclusion (suite).
Il n’est pas concevable qu’un champ magnétique de 50 Hz, de signe
contraire à celui du faisceau, apparaisse au moment de la mesure avec
faisceau, le seul changement entre les mesures étant la mise en route du
chauffage de la cathode, mais bien plus qu’il n’apparaisse que dans un
sens dans les deux directions utilisées.
En conséquence, il existe des orientations de l’Espace dans lesquelles, à
un instant donné, les faisceaux d’électrons n’ont pas de champ
magnétique alors que leur mouvement de translation n’est en rien
modifié par rapport aux autres orientations.
Le champ magnétique des électrons ne peut donc en aucune manière
résulter de leur translation dans l’Espace.
La seule autre variable connue étant leur moment magnétique, et en
conformité avec les rapports précédents des ingénieurs associés, il est
donc établi que le champ magnétique des électrons résulte exclusivement
de leur moment magnétique. L’équation de Maxwell-Ampère est donc
fausse et il ne saurait y avoir le moindre problème de mouvement relatif
dans les phénomènes électromagnétiques.
4. Conclusion (continued)
It cannot be imagined how a 50 Hz magnetic field opposite to the beam
magnetic field, could appear just when the magnetic field of the beam is
measured as the only change is the switching on of the cathod heating.
Moreover the magnetic field should only appear in one way of the
selected directions.
As a consequence, there are orientations within Space where, at a given
time, the electron beams have no magnetic field although their
translation motion is not modified compared with the other orientations.
Thus the magnetic field of electrons can in no way be a result of their
translation motion in Space.
The sole known other variable is their magnetic moment. So that, in
compliance with all the previous reports of the associated engineers, it is
now clearly established that the magnetic field of electrons is only a
result of their magnetic moment. Therefore the Maxwell-Ampere’s
equation is definitively wrong. As a result, there is not the slightest
problem of relative motion within the electromagnetic phenomena.
186
11
The electron intrinsic magnetic field
is not a dipole.
The Rowland effect.
April 2016
187
1 PRESENTATION
The magnetic field of a conductor crossed by a current of 0 to 2.5 A,
rotating at speeds between 100 and 260 revolutions per second is five to
fifteen times higher than that produced by a current of the same
intensity in the motionless conductor.
The electrons are rotating with the conductor. The magnetic field results
from the Rowland effect. But if the magnetic field of the rotating
electrons was likened to the field of a loop, it could not be detected by
measuring coils which are parallel to the conductor.
Thus, this magnetic field can only result from the intrinsic magnetic field
of the electrons aligned in the conductor axis by precession effect.
But the position of the coils relative to the conductor does not allow to
detect dipole fields. The coils can measure the rotational magnetic field
of a variable current, but can not detect a variable dipole field.
It is therefore necessary that the magnetic field of the electron has a
rotational structure that can be measured by the coils.
188
2 DEVICE
The conductor is a copper tube 4 mm OD diameter and 280mm long, fixed at its
ends to two steel rods of 3 mm diameter. One of the rods has a length of about
700 mm and is guided by three ball bearings inserted in a 10 mm ID diameter
support tube. An electric motor 12 VDC, 25 A 15600 RPM is fixed to the other
end of the rod and fixed to the support tube. The other rod 50 mm long, is
guided by two ball bearings. The two steel rodss are fixed to the copper tube by
insulating connectors. The current is delivered to the conductor by carbon
contacts maintained by springs.
The power supplies are located more than one meter from the device. The motor
itself is 700 mm from the sensor located in the middle of the rotating tube.
The motor speed is measured by an UV door: 12.81 Hz for 20 wave lengths:
256 Hz.
189
3 SENSOR
The magnetic field of the pulsating current at 100 Hz passing through
the rotating tube is of the order of 10E-7 Tesla. It is measured by a sensor
with two coils in series situated on either side of the rotating tube and
contained in its plane.
Both sensor coils consist of approximately 1200
turns of copper wire 0.1 mm diameter.
The sensor signal is sent to an integrated linear
amplifier AD 820 The gain is 200. The offset is not
corrected. The amplifier is supplied with +15 V
regulated monopolar.
The rotating tube is supplied with 9V AC rectified
but not filtered. This results in a pulse at 100 Hz of
the current in the tube. This pulse induces a
voltage in the sensor coils. The sensor can not
distinguish the direction of the magnetic field as
the variation of the inductor current sign changes
at each half period. The amplifier allows only
positive changes to allow measurement with a voltmeter.
The amplified signal is measured by a digital voltmeter. It is also sent to
a digital to analog converter connected to a USB port of a computer with
a digital oscilloscope.
190
This curve shows the voltage delivered by the coils measured after
amplification as a function of the current in the motionless conductor.
191
4 MEASUREMENTS
The voltage is measured after the amplifier for the intensities of 0, 1 and
2A. Measurements were replicated dozens of times in each direction of
rotation and each direction of the current in the conductor.
In the absence of current, the rotation produces no magnetic field.
The starting and stopping the motor as well as the power supplies
needed for the experiment cause no voltage measured after the amplifier
when the conductor is not crossed by any current.
The intensity drop in the conductor resulting from the rotation is about
0.5 A corresponding to the increase of the contact resistance.
Instead of falling in proportion to the drop in intensity of the current in
the rotating tube, there is a very significant rise of the magnetic field
measured by the sensor.
The field is three to seven times higher than the field of the same current
in the motionless conductor.
192
The rotating conductor has been shortened to 140 mm in order to reduce
the vibrations and enabling measuring the rotation speed. The rotation
speed was much higher than in the previous configuration. It reached
256 revolutions per second. The field is fifteen times higher than the field
of the same current in the motionless conductor.
In this configuration, the sound frequency
of the device is more than twice that with
the first conductor 300 mm long. The speed
should then be less than half; say one
hundred revolutions per second. This
speed is confirmed by the fact that the
current in the motor increases from 10 to 20
amps. Under these conditions, the
magnetic field would be proportional to
the rotational speed of the conductor.
The vibrations were still significant so it
was not possible to verify systematically
the effect of the rotational speed on the magnetic field with this device.
193
5 ANALYSIS OF CAUSES OF THE INCREASE OF THE MAGNETIC
FIELD
The Tolman-Stewart effect that occurs when the rotation is sharply set
on or off can not be the cause of the observed phenomenon since it is
permanent. Moreover, the magnetic field that would be created by the
rotation of the electrons with the conductor can be likened to the field
outside of a solenoid of infinite length, so there is virtually zero.
The conductive tube is made of copper and is thus not magnetic. It can
not therefore be a Barnett effect.
In addition, in the Tolman-Stewart effect as in the Barnett effect the
magnetic field is found to be in the axis of the conductor. The sensor can
not detect such a field.
It can only be a Rowland effect. The rotation of the electrons around the
conductor axis causes a magnetic field. This phenomenon is the cause of
the very large increase of the magnetic field of the rotating conductor.
This effect obviously depends on the direction of rotation, but the
current in the conductor being rectified two phases and the sensor
amplifier being supplied with monopolar voltage, there is always an
increase of the induced voltage with the same sign.
This phenomenon would result from the rotation of the electrons with
the conductor like loops and as a result of small translations causing a
magnetic field.
However, this explanation of the Rowland effect by translation of
electrons is impossible. The field would be coaxial to the driver and
could not be detected by the sensor.
It is therefore a phenomenon of precession of the axis of rotation of the
electrons. It results from the Coriolis’acceleration. A body rotating
around an axis and set rotating around another axis has its own axis of
rotation pushed toward the axis of the rotation imposed.
But in the context of current theories, the magnetic field of the electron
has a dipole structure.
However, the position of the coils relative to the conductor does not
allow detecting dipole fields. Depending on the orientation of the dipole,
the field lines traverse the coils either in opposite directions and the
induced currents cancel or they do not pass through at all.
194
It is therefore necessary that the magnetic field of the electron has a
rotational structure to be measured by the coils.
195
The Rowland effect can not result from the transition of electrons as we
have seen. The translation of electrons can not be the cause of the
magnetism of the electric currents
The magnetic field of electric current thus results directly from the
intrinsic magnetic field of electrons. Their field is oriented in the
conductors so that the conductor produces a resultant magnetic field.
196
The old approach of electromagnetic theory, attempting to justify the
Biot and Savart law by a formal analogy with Coulomb's law could not
hide the lack of experimental evidence.
The axiomatic approach has been clarified. The most recent books took
as its starting point the postulate of the magnetic force between two
moving charges.
Unfortunately, not only this assumption obviously has no experimental
justification, but it now seems completely contrary to experiment.
197
6 MAGNETS
As a consequence, the magnetic field of the electrons can not be directly
the cause of the magnet fields.
The magnetic field of the paramagnetic,
diamagnetic, ferromagnetic, anti-ferromagnetic
and ferrimagnetic bodies has an overall topology
orthogonal to the conductors field. The only
structure allowing for constituting such a
topology with the elementary magnetic moments
is the Helmoltz torus. A similar structure is
obtained with a loop or a solenoid in the case of
conductors.
It must therefore be concluded that magnetism of
magnets and of bodies mentioned above, results
from the existence in matter of magnets of such structures whose
orientation conditions can account for different types of magnetism.
It is necessary that the electrons are organized into rings structures in the
matter of magnets.
Electrons of magnets include in such structures generate a magnetic field
similar to that of the magnets.
198
7 MAXWELL
The Maxwell-Ampere
equation is false. There
is not any vector
relationship between the
magnetic field of the
electrons in an electric field, so in translation, and the current vector J.
There is only a formal coincidence. The magnetic field of the current is
proportional to the average magnetic intrinsic magnetic field of electrons
itself proportional to the electric field in the conductor.
However, the current is itself proportional to the same electric field. But
the quantities have nothing to do with each other. And primarily, a
Galilean reference frame change does affect neither the electron intrinsic
magnetic field, nor the angular momentum.
If the charge must of course be conserved, even under varying conditions,
the direct relationship between the magnetic field and the current vector
is false as we have just seen. The magnetic field of electrical current is
the geometric sum of the magnetic fields of electrons in the conductor.
Without potential difference, so without an electric field in the conductor,
the magnetic fields of electrons are distributed randomly and the
conductor has no magnetic field.
An electric field in the conductor directs the rotational magnetic field of
electrons so that the driver has himself a rotational magnetic field.
Thereby by dissociating the direct link between the current vector and
the magnetic field, we introduced the ability to show a magnetic field
variation of the electric field without the conservation of charge is
questioned. The problem of the Ampere equation in variable regime is
therefore moot.
The displacement current Maxwell added to the Ampere equation to
cancel the mathematical divergence of the current in variable regime and
meet the conservation of charge, has now a very simple physical
explanation. It is simply the magnetic field resulting from the magnetic
field of electrons that do not enter into the current J. For example, in the
case of electrons that move transversely in a sudden enlargement of a
conductor. They produce a transverse electric field that orients
accordingly their magnetic field. Moreover, the equations are very
similar in the case of the sudden enlargement of a pipe in fluid
mechanics.
199
This is also the case of magnetic fields which
appear when you load capacitors. The electric
field in the plates themselves is zero, and
therefore produces no magnetic field. But
during charging, electrons are accumulating in
the plates and cause a momentary additional
electric field which orients their magnetic field.
The magnetic field of the electric currents result directly from the
rotational magnetic field of electrons. This field is totally invariant in a
Galilean reference frame change.
The problem of relative motion does not exist in electromagnetism.
8. REFERENCES
Baigrie, B.
Electricity and Magnetism: A Historical Perspective.
Greenwood Press. (2006).
Bernard Pr. M.Y. Initiation à la mécanique quantique
Editions Hachette (1960)
Hennequin Pr. J.
Electromagnétisme et relativité restreinte.
Editions Dunod (1970)
Keithley, J.F.
The Story of Electrical and Magnetic Measurements:
From 500 B.C. to the 1940s
IEEE Press. (1999).
Landau Pr. L.
Théorie des champs
Editions Mir, Moscou (1966)
Liboff Pr.R.L.
Introductory Quantum Mechanics
Holden-Day Inc (1908)
Mach Pr. E.
De la connaissance et de l'erreur
Editions Flammarion (1908)
Pérez Pr.J.P.& alt Electromagnétisme
Editions Masson (1997)
Tonnelat Pr. M.A. Histoire du principe de relativité
Editions Flammarion (1971)
200

FLUID MECHANICS AND PHYSICS

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