The steady flow of a weakly ionized gas between parallel plates assuming anisotropic conductivity

TBE STEADY FLOJVOF A WEAKLY IONIZED GAS

BETWEEN PARALLEL PLATES ASSUMING ANISOTROPIC CONDUCTIVXTY (USTANOVIVSEEESIA TECBENIE SLAB0 IONIZOVANNOGO HEZBDU PABALLELNYMI PLASTINAHI S UCEETOM ANISOTBOPII PBOVODIMOSTI) PYM

Vo1.25,

No.3,

I.

(Received

1961,

pp.

GAZA

473-477

B. CHEKMAREV (Leningrad) March

4,

1961)

It is well known that if the condition WT << 1 is not fulfilled in an ionized gas (where w is the cyclotron frequency of a charged particle and r is the average time between collisions (strong magnetic field, rarefied medium)), it is essential to take account of the fact that the transfer coefficients depend on the magnetic field, and they become anisotropic [I, 5-9 I, In the last few years several publications have appeared dealing with the flow of ionized gas, assuming anisotropic coefficients of viscosity and conductivity. Articles [ 2.3 1 give derivations of the fundamental magnetic plasmo-dynamic equations taking into account the influence of the magnetic field on the transfer process within a fully ionized gas, and a detailed study of Couette flow is adduced. Kaplan [ 4 I and Lighthill [ 11 1 have studied the effect of anisotropic conductivity on a magneto-gasdynamic shock wave and on the wave motion of a conducting medium. Article [lo 1 dealt with the effect of anisotropic conductivity on the longitudinal flow of a weakly ionized gas in a narrow annular channel within a radial magnetic field. This paper deals with steady flow of a weakly ionized gas between parallel non-conducting plates within a transverse homogeneous magnetic field B,. ‘Ihe following assumptions are made in order to simplify the problem: 1) the mean free path in the gas, X, is much less than the transverse dimension of the channel 2a, i.e. the conditions of a homogeneous medium are fulfilled;

2)the

degree of ionization of the gas is small;

3) the condition

OiTi << 1 is fulfilled for the ions, and this allows

701

I. B.

702

Chekmarev

the viscositycoefficientq to be considereda scalar quantity,and the effect of ion slip with respectto the gas can be neglected; 4) the followinginequalityis satisfied:R,.= up V,a << 1, where V, is the mean gas velocity in the channel; conditions allow one to use a stream function in the following j+wt(jxk)=o(E+vxB) (where k Is the unit vector 5) the physical constant.

i in the direction

characteristics

With these assumptions take the form

(21, (3) form:

or = Boez me ) of the magnetic

rotE = 0,

field)

of the gas (p, q, o, p, ~1 are assumed

the fundamental equations

~(vv)~=-v~+jxB+qAv, rot H = j, div H = 0,

and (4)

for

divv=O j+~~(jx~)~~(E+vxB) diwaB==@

this

problem

(1) (2) (3?

We will assume that the homogeneous external magnetic field B, = pHO whilst there are constant pressure gradients is parallel to the z-axis, along the x- and y-axes (~p/~x) = - P,, (dpfay) = - Pr and there is an external homogeneous electric field E, and Eay. All equations of the system (1) to (3) can then be satisfie d if we assume that the required quantities are functions of the transverse coordinate z only. The system therefore reduces to the following equations:

Flov

of a veakly

ionized

703

gas

Using (9) to eliminate the stream density from (41, we arrive components: following system of equations for the velocity

- ~-~2(Eov_l-OTEox) 11

@U --$L+os+= dz" d2v --dz2

N2 a~ D-cmY

(IO)

~u=-;+f2(E,,-orE,,) N2 =

The solution is of the form

of

Iv2

(11)

1 + WV

which is even with respect

(lo),

at the

to the z-coordinate,

u=

(12) aNz PNz + Clsiha* C‘psb - a cos a

V=-

sin aPNZ + il,

a

(13)

where zc:- -+1/~1+aW+1

p =

9

A, =

the constants

+ &2--

1/p?

A, = +&

9

If we determine

-&

(PV -

mP,)

1

(14)

-E%

(15)

Cr and C2 using the boundary conditions

u=v=o

for z=a

(1fi)

we find C = _ AlcosbaNcos PN + 1

c,

ApinhaN

A1 =

-

AlsinhaN

sin PN

9

sin PN - A.pxhctN cos PN . AI

A, = aidbz &V +

For volume. gas fluxes Qz and Q, in the directions axes, we obtain by calculation

Qx=

where

COST BN

(17)

of the x- and y-

i u&=2(1 (A, - "1":1:ay

CD1= asinbaM~~haN _t p sin pN cos PN (D2 = /3sfiaNc0shaN -

a sin

f3iV

co.5PN

(19)

I.B.

704

By

full

Cheknarev

use of Formula (9) we derive the following flows in the directions x and y:

expressions

for

the

--a

and u(z), by using (5) and (9), H, and Hy can be found. The in-

For known velocity components u(z) the induced magnetic-field intensities duced electric-field strength E, and the volumetric charge density 6 are determined from Equation (7). We now deal with two particular cases.

1) Assume that a given constant channel pressure gradient Pz along the x-axis maintains the steady motion Fig. 1. of the gas, whilst Q, = I,.= Eix = 0. ‘Ihis example corresponds to the problem of the flow of a conducting gas through an infinitely long channel with non-conducting walls whose width is large compared to the height. The conditions Q, = I, = 0 yield the equation -

‘%a + ‘ml MA 1

*‘-

It follows

from these

From system (21),

A

where

1

=

latter

using

_ o

expressions

(151,

B,$

that I,=

=0

(21)

0.

we find

MA, CM&-- @I) Pxa2 N2Az

A’2 = 01 (MA, -

Tbe flow in the x-axis

E,,-

,

direction

9

,4

=

2

’

MU’,2

pxa2

(22)

TV&--y-

CD,) f cD,(MA,oz

-

(Dz)

(23)

is given by the formula

QX = (MAI -Nf;\‘: +

a2.J

2p;n3

which for the case M = const, or + 0 leads to the well-known of the Hartmann problem with isotropic conductivity and total

(24) solution current

Flov of a veakly

ionized

705

gas

equal to zero: (25)

The graphs in Fig. 2 illustrate the relation between the flow OT given by Formula (24) for Q,” = QJ( 2P$ : q) and the quantity various values of the Hartmann number M, If only the intensity of the external magnetic field Ho is varied, the quantity y = ws/M remains

i%.

Fig.

2.

3.

constant and is a characteristic of the degree of rarefaction of the medium. A graph showing the relation between the flow Q,” and the Hartmann number is shown in Fig. 3. Daspite the fact that the total flow Qy and total current 1, are zero, because of the anisotropic character of the conductivity transverse flows and electric currents arise. 2) In the second example it is assumed that both the medium and the currents can flow freely in the x- and y-directions. For simplicity, put P = E,, = Eaz = 0, whilst the value of P, is assumed to be given. We t x en find

Qx=-

2P,a3(MA2 - @i) + ot@z Y*Al

’

I,=-I,=

--

2P,a mz & M& =xa Bo

MA1 MAI

tD1

(27)

1.B.

706

Cheksarev

or + 0, the expressions found transIn the limiting case M= const, form into formulas of another regime of motion in the Hartmann problem, in which the intensity of the electric field is equal to zero: QX _ zp;a’

Md$--

M

,

I, = _ T

““;~,;;~~~

(28).

A graph showing the ratio between the flow Q, arising because of anisotrogic conductivity of the transverse stream and the main flow Q, as a function of OT is shown in Fig. 4. Lateral and for

flow vanishes both when or + 0 large values of WT.

It follows from the illustrations on the graphs and from analysis of the formulas that at large values of or the solutions obtained tend asymptotically to the normal ones which correspond to the problem for a non-conducting fluid:

Thus the effect of a magnetic field on the main stream is weakened as the value of or increases because of the decrease in conductivity of the gas. On the other hand, anisotropic conductivity complicates the whole flow pattern, and this is expressed by the appearance of transverse flows, an induced electric field, and a volumetric electric charge.

Fig. 4.

BIBLIOGRAPHY magnitnogo polia na 1. Her&man, B.N. and Ginsburg, V.L., 0 vliianii konvektivnuiu neustoichivost’ v atmosferakh zvezd i v zemnoi ionosfere (The effect of a magnetic field on the convection-instability in the atmospheres of stars and in the terrestrial ionosphere). Astronor. Zh. Vol. 32, NO. 3, p. 201, 1955. 2.

Gubanov, A.I. and LWkin, Iu.P., Uravneniia magnitnoi plasmodinamiki (Magnetic equations of plasma dynamics). Zh. Tekh. Fiz. Vol. 30. No. 9, p. 1046, 1960.

Flow

of

a weakly

ionized

707

gas

3.

techenie v magnitnoi Gubanov, A. I. and Lun’ kin, IU. P. , Kuettovskoe plasmodinamike (Couette flow in magnetic plasma dynamics). Zh. Tekh. Fiz. Vol. 30. NO. 9. p. 1053, 1960.

4.

anizotropii provodimosti v magnitnom pole na Kaplan, S. A. , Vliianie strukturu udarnoi volny v magnitnoi gazodinamike ( The effect of anisotropic conductivity in a magnetic field on shock-wave structure in magnetic gasdynamics). Zh. eksp. teor. fiz. Vol. 38, No. 1, p. 252, 1960.

5.

KaUling,

T. , Magnitnaia

gidrodinanika

IIL.

(Magnetohydrodynanics).

1959.

obzor magnitnoi gidroA.R. and Petchek, G.E., Vvodnuii (Introductory review of magnetohydrodynamics). In Sb, “Magnitnaia gidrodinanika” (Collection ‘Magnetohydrodynamics”). Atomizdat. 1958.

6.

Kantrovits, dinamiki

7.

Landau.

L.D.

and Lifshitz,

(Electrodynamics

8.

9.

of

E.M.,

Continuous

Petchek, G. E., Aerodinamicheskaia sipation). In Sb.“Kosricheskaia Casdynari cs” ) . IIL, 1960. Spitser,

L.,

Completely

Fizika

polnost’in

Ionized

Gas).

Elektrodinarika Media).

dissipatsiia gazodinamika”

ionizouannogo

IIL,

sploshnykh

Gostekhizdat,

sred

1957.

(Aerodynamic (Collection

gaza

(The

dis‘Cosmic

Physics

of

1957.

anisotropii provodimosti na statsionarnoe 10. Chekmarev, I. B., Vliianie techenie neszhimaemogo viazkogo ionizovannogo gaza mezhdu koaksial’Wmi tsilindrami pri nalichii radial’nogo magnitnogo polia (The effect of anisotropic conductivity on the steady flow of incompressible, viscous, ionized gas between coaxial cylinders under the influence of a radial magnetic field). Lening. Polytekhn. Inst. i/r M.I. Ralinina. nauchnotekh. inform biull. No. 7. Section on Phys. and Mathem. Sciences. p. 81. 1960. 11.

Lighthill,

M. J.,

anisotropic 1014.

Studies on magnetohydrodynamic wave motions. Phil. Trans. Roy.

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and other

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No.

1960.

Translated

by

V.H.B.