# THEORY OF THE MAGNETIC BOUNDARY LAYER

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CIA-RDP80T00246A007500560002-6

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6

Document Creation Date:

December 22, 2016

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July 21, 2009

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2

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Publication Date:

October 14, 1958

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REPORT

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CORY OF THE MAGNETIC BOUNDAF( LAYEI
by V. N. Zigulev
(Communicated by Academician L. N. Sedov, October 14) 1958)
Doklady,~Akademii Nauk SSSR, volume 124, No. 5 (1959) PP. 1001-1004
In this work, examples are cited which illustrate the phenomenon that-a moving
plasma is shielded from an external magnetic field and from the electric currents that
flow in it; the thickness of the shielding layer, called a magnetic boundary layer, has
the order for motions at large magnetic Reynolds numbers.
1) If we introduce the vector potential of the magnetic field W ( H = curl W
then the equations of magneto-hydrodynamics can be put in the form
aw
at
d/v W =0
V x carl W = -grad + -~,~ 7' W
div(~V~ = 0 ~~ _ _ + H aHa +
. ~t a xa 4~ ~ axe axe
p T dS _ r a v a + a ar + ~' S )
s )
dt ?~ axe aXx arc 47r
where is the velocity vector with components V
x 3) ;
grad _ E + C at ; E is the electric field strength;
J is the magnetic.viscosity, which,in the general case is a function of the
temperature T and the pressure ; / is the density of the medium;
N Z 2 i s the
is the magnetic field strength;
viscous stress tensor; ,$ is the entropy of unit mass; k is the coefficient
of thermal. conductivity; is the vector current density. The first equation
of the system as written resembles the equation of hydrodynamics in Lamb's form. The
electrodynamical part of the equations of magneto-hydrodynamics in their customary form
Russian title:
TEOP 1 MArHNTHOro TrorPAH v1 KHorO c.AoP
Author:
E.H. )K N rY1-EB
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and let the plate be immersed in a H x
can be obtained from the first equation of the system if one applies to it the curl
operation and sets 7~ = C o ns t in the flow.
2) Let there exist a semi-infinite3 plate (coinciding with the half-plane
X H , x > O ) along which there flows an electric current in the direction Oz ,
conducting fluid at rest, the
current in the plate being.: insulated
from the -MiiiA_
It is easy to see that there is p1g 1
introduced in the fluid a,magnetic field denoted by the vector /7 , parallel to
the plane )C~y . We now bring the fluid into motion at velocity LI in the
direction Ox , and we consider regions of the flow, sufficiently removed from
the edge of the plate in the direction of the y,- axis, so that the corresponding
magnetic Reynolds number Re l,rL-will be a large quantity; then, on the
basis of the preservation of vector lines in a medium of infinite conductivity, and
on the strength of the requirement Rem > j , the magnetic field vanishes in
the main flow and persists only in. the layer adjacent to the surface of the plate and
having a thickness of order LZV Re?, . We call this layer the magnetic boundary
layer of the first kind.
If we consider again the problem of a semi-infinite plate in a stream of
conducting fluid for the case /ee >> 1 , where this time a current flows on the
plate in the direction of the X -axis with a constant linear '(along 00 ) intensity
(the plate is insulated fromthe outer flow with the exception of the leading edge,
and regions removed from the beginning of the plate), then, on the basis of the
properties of the preservation of vector ?ines (2) , the electric current in the
fluid is localized in a layer.adjacent to the plate and having a thickness of order
L~ Re We call this layer
a magnetic boundary layer of the
second kind.
3) Carrying out, in the equations
of paragraph 1), estimates similar to those ng. 2.
made concerning the ordinary boundary layer (see, for example., 1)) for the magnetic
boundary layer of the first kind, we obtain the. following equations:
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aw + u aw +v a`./ _ aZw
a t a x a~ ~" C9
C)U au au a i aw a=w _ ';VV air/ a , au
at +~~` ax +%ov a - ax + 47r (a aX a ax a y) + a a
Z
a ~^ = o ~ f'T~at + a- + v a 5) -- ~a" (k aT 1 + 4 a W
where k' is the 2 - component of the vector /4 is the coefficient of the
ordinary viscosity. /
Analogously, the equations of the magnetic boundary layer of the second kind
take the form:
aH+uOH vaN+H
C au + av 1 _ aZH
at aY-
044 +VOU au +/V au _ -
a t ax ai
2
t at ax a a ( a ) 4vr ka )
In arriving at the equations for magnetic boundary layers of the first and
second kinds, terms have been retained, in the equations of paragraph 1) whose order
relative to those omitted. is
As is clear from the equations of magnetic boundary layers, the pressure
across them can change by many times (since. the quantity H~ = cohst across
the layer), something which markedly distinguishes the boundary layers under considera-
tion from the ordinary ones. This circumstance, existing as a consequence of the
electro-magnetic field forces, can be applied in questions of thermally insulated bodies:
it is sufficient to make iAyr _o > 4, ('see, f?r example, the two articles (2)),
since near the plate there appe rs a zone of cavitation, i.e., a zone, where
In the case of an incompressible fluid, the equations of the steady magnetic
boundary layer of the first kind take the form:
M-/ ; W) Cgs
D 95
D(x; y) D( ) ax 3
while the equations for the'steady boundary layer of the second kind are:
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D(HIVi) _ __ D( s % ~l __ ~ 2 ~, a3
o (x1 y) ~" ay. D (x; y) ax + '~ ~~3
where is the stream function, ) the kinematic viscosity coefficient.
4), For the equations of the magnetic boundary layer of the first kind for
an incompressible fluid, there exists the following class of similar solutions:
q1 A -T Y
z f (r) 14
i/2 ,/Z X a _ d
pm 0
where c 1A, Y lkrn o are certain constants.
The functions f and satisfy the system of ordinary differential
equations:
(a+~~~,,= Irf"-(a+Y).~'z 4~'q~u= dZ~~w f,a
Here d = (S-z)~/4- ~B =Y (d+-Z)0/4 In the case
where const
For the equations of the magnetic boundary layer of the second kind for an
incompressible fluid, the analogous class of similar solutions takes the form:
Vz 6z 1/Z.
f (~) and h (~) satisfy the equations:
~'h f'-,B h 'f = y J 2 + w c
Here o( = (e-2)/4 is an arbitrary constant; if ~m = cohst
=ate/
5) In the classes of solutions mentioned in paragraph 4+) solutions are to
be_found for problems of the flow over semi-infinite plates in the presence of a
magnetic boundary layer of both the first and second kinds, if the X - component
of the magnetic field 14, = co m s ~ = /-~ along the plate.
In the case of the magnetic boundary layer of the first kind, the problem
comes down to the solution of the system of equations: ~c c = -'/2 ;l~6 = i'='/Z S = o
ff,"=z0)1C?
with the following boundary conditions:
a)=O= 144
b)
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If &)4-41p that is, if the processes associated with the influence of the ordinary
viscosity are nono-eXistent, then conditions a) are replaced by:
O
In the case of-the magnetic tfiugdary layer of the second kind, the problem
canes down to the solution of the Blasius equation oC
l
iii
and the integral
ti(p)= C, + C'Z expJ,~ z f flt)
The boundary conditions take the form:
a) i?-0. f,..._c'_O i y=o~vn z
b)
I ~ z R S_ _0
If eJ