Self-Superposable MagnetobydrodynamicMotions RICHARD R . GOLD

Communicated by C. TRUESDELL 1. Introduction

The formal concepts of superposability and self-superposability of nonlinear partial differential equations, in particular those equations of interest in hydrodynamics, have been examined in [11 and [21. Following a brief review of the literature and some remarks regarding the philosophy of the subject, superposabihty and self-superposability conditions were established for the equations governing motion of viscous fluids, compressible and incompressible, Newtonian and non-Newtonian. Corresponding conditions were determined for the equations of motion of magnetohydrodynamics and for the potential equation of gas dynamics. Tbe present paper is an extension of the basic idea.s set forth in Chapter 8 of [2]. Following these formal results, we now seek solutions of the resultant system of equations, i.e., we wish to construct self~superposable magnetohydrcdynamic motions. The appropriate equations are given in the next section. Solutions are obtained in Sections 3 and 4 for two and three-dimensional motions, respectively. Both the cases of steady and nonsteady flow are considered. The mathematical complexity arises, in part, from the nonlinearity of the basic equations resulting from the terms ([7. V) U and (B 9 V) B in the momentum equation and curl ( U • B) in the induction "equation. As a result of this difficulty a large number of the magnetohydrodynamic problems which have been treated with regard to such questions as stability, shock waves, magnetic fields, ahd.aerodynamic applications involved linearizing techniques of one sort or another. In the astronomical literature one finds that very often U is either ~dentically zero or is assumed to be known, in which cases the equations are linear in the remaining electromagnetic variables. Several authors have begun thr study of the full equations under simplifying assumptions which make the system more amenable to solution, as is noted in the discussion of Case 2 in Section 4. It is at this point that the introduction of the concept of superposability m a y be of definite use in exploring the character of the basic equations by proriding additional exact solutions and general properties. In general this is our primary purpose, and solution of specific boundary-value problems is not attempted in this work. Obtaining exact solutions of a particular system of nonlinear differential" .equations is but one of the potential applications of the concept of superposability. The pertinent conditions would allow one to examine general classes qf functions

Self-Superposable Magnetohydrodynamic Motions

383

which may have this property of additivity (see [1] and [3 l, for example). In view of the role that the additivity or superposability property plays in a]most all methods for solving linear differential equations, an examination of these methods applied to particular nonlinear equations in the light of generalized superposability results seems warranted. Finally, it is worthwhile to study the several formal mathematical questions which may be raised with regard'to a subclass of solutions of a nonlinear equation or system of equations (or, more generally, to nonlinear differential operators of interest) which exhibit this linearity.

2. Magnetohydrodynamic Equations. Superposability and Self-Superposability Conditions The motion of an electrically conducting fluid in the presence of a magnetic field obeys the equations of magnetohydrodynamics. The fluid is treated as a continuum, and the classical results of fluid dynamics and electrodynamics are combined to express the phenomenon. For a viscous incompressible fluid with constant properties (i.e., ~, I~e, e, e, v) the equations of momentum and of induction constitute the full magnetohydrodynamic system: ~U/3t + ( U . V) U - - ~ ( B . V) B---- -- Vs + v ~72 U,

;t = (4re e/le)-x : constant, a B / a t ---- V •

(U•

s = P/O + [2 + ~ B2/2, +~VZB,

= (4 ~ a ~1~)-1 = constant,

(2.t a) (2.1 b)

where the external force on the fluid other than the Lorentz force is expressed b y the potential function [2, U is the velocity vector, p the pressure, 0 the density of the fluid, ~ the magnetic permeability, e the dielectric constant, v the kinematic viscosity, a the electrical conductivity, and B the magnetic induction (the magnetic field strength or intensity is H---- B/~;1). Inherent in these equations are the following assumptions, often referred to as the magnetohydrodynamic approximations. The electrical component, 0r of the ponderomotive force in the momentum equation is small compared to the magnetic component, j x B. The displacement current in Amp~re's law and the convection current, 0, U, in Ohm's law are negligible compared to the conduction current, a E. A detailed discussion of these equations and assumptions is available in a number of references ([~'[ -- ~'6~). Both U and B are solenoidal vectors; hence V.U=O,

V.B=O.

(2A c, d)

System (2A) must be,s01ved for U, B, and p. The electric current density, j, is then readily obtained from Amp~re's law, j = (4~/~) -1 V• B. The electric field strength, E, is obtained from Ohm's law, E = j a - 1 - U • B = ~1 V• B - U• Clearly 1 7 . j = 0 , while the divergence relation for E, V . B = 4 ~ e - ~ 0 ~ , determines the excoss charge density 0~, which need not be zero in general. The superposability conditions can now be derived in the manner outlined in detail in ~1] and [2~. Briefl~, if (U1, B1, Sl) and (Ua, ]~z, so) are two solutions

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IlWH.XRI, R. GOLD:

of (2.1), what is desired is to determine sl-4-s.,_+~ so that (UI-FU2, B I + B 2 , sl+s.,-F~ ) is also a solution. Introducing the three solutions into (2Aa) and ,ubtracting tile latter result from the sum of the first two, one obtains -- ([1' V) ['z -- (['~" 17) L1 @ ~[(BI. V) B , -1- (B2. V) BI] = V=.

(2.2a)

Similarly, from equation (2.1 b), we have - - V x ( U t x B.,) -- V • (/-'~ x Bt) = O.

(2.2b)

For self-superposable flow, U , = U2= U, B 1--- 9 2 = B and equations (2.2) become -- ( U . V ) U + ~ ( B . V ) I ? = ~-A:z, [/X ( U x B ) = 0 ,

or,

UxB:=I7Z .

(2.~a/ (2.~h)

In order for tile system to be self-superposable, i.e., if (U, B) is a solution, (n U, n B ) is a solution, there must exist a pair of single-valued ~calar function~ and Z defined by (2.~). The conditions of integrability for (2.~) are curl [(U. V) U 1 = 2curl !(B- I7) B],

(2.4a)

curl (U y B) :- 0.

(2.4bt

On the assumption that the flow i.~ self-superpo.~able, ,ystem (2.1) m a y be simplified by replacing the nonlinear quantities with ~ and Z via equations (2. ~). Eliminating ~ from the resultant m o m e n t u m equation (2.1 a) by taking the curl and noting (2.t b), one obtains ~/g'l=pV2~,

~=V•

(2.5a) (2. ~ b)

bB/~l = ~l Vz B .

Equations (2.t c, d), (2.4) and (2.5) comprise the self-superposable system which determines U and B. The pressure would then be determined from (2.1 a). For steady flow (2.5) become V" ~ = 0 (v @ ,)), (2.6a) V" B -- 0

(r] =}=0).

(2.61))

Similar results were obtained in [21 as well as in [7]. For B = 0 the condition~ and equations noted above reduce to the nonconducting case discussed jn [1~. I t is of interest to note that for steady flow equation (2.5a) is satisfied for a perfect fluid (v = 0 ) and ~ need not be harmonic; thus (2.6a) no longer applie.-. Simi{afly, for a steady flow in which the electricat conductivity is infinite (~ = 0) equation (2.5b) is satisfied, and B need not be harmonic. I t m a y be remarked that the self-superposability condition for equation (2.1a) expressed as equation (8.2.2) of [2] (equations 2.3a or 2.4a above) can be written in the alternate form curl (U x curl U) = 2 curl (B x curl B) b y using equations (2.1 c, d) and the vector identity (F. V) F = ~ VF 2 - F x curlF.

(2.7)

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Force-free fields which are of considerable interest in the study of static equilibrium ( U = 0 ) in astronomical problems ([8] and [9]) are characterized by the condition B• For s (2.7) reduces to B • 1 6 2 which contains this result. The self-superposability relation is in fact a generalization of this force-free field condition to the case where U~: 0. Since B satisfies the diffusion equation (2.5b), in the present system there is an obvious analogy with the several electromagnetic and astronomical analyses of the magnetohydrostatic problem, where putting U = 0 in the induction equation (2.tb) results in this same diffusion equation ([4, pp. 4 and 5]; [13] and [1~]). 3. Plane Flow ( U independent of ~, and U. = 0)

Let us assume in this section that the magnetic field is restricted to the plane of motion, i.e., B is independent of z, and B,----0. Introducing the stream function ~(x, y, t) along with a comparable function ~(x, y, t) for the magnetic field so as to satisfy (2Ac, d) identically (U,=~0,y, Uy------~,~; B , = ~ y , By ------ r , where the comma in the subscript denotes partial differentiation), we reduce equations (2.4, 5) to the 1orm (V~ ~),e= v V2 v,o, (3Aa)

e(~,, v~,)lO(x, y) = ~ a(r v~ ~)io(x, y),

(3.t b)

9 ,,-- n v~ 9 + l(e),

(3.t c)

~(~, qs)]a(x, y) _-- 0.

(3.t d)

The vorticity ~, which is normal to the plane of motion (~,----~y=0), is ~,-----~p,,,--~,yy=--17~, where 1714=V~(V~). The Jacobian of the functions h(x,y,t) and g (x, y, t) is given by a (h, g)/a (x, y) ----h,, g,y--hy g,,, and/(t) is an arbitrary function of time. Equations (3.1) must be solved for ~p and q~, while (2.t a) reduces to the following equations for the determination of the pressure, Pie +~9:

(p/o+Sg),,=--V,,,,,--k[V,~,+~y],,+V,,,. V~V'--~-r ( P l o a L ~ ) , = ~ , , , , - 89[,p, -u ,p, ~]

V~r

+~,,,, 9 v;,p-a~. , 9 v , * ~ - ~ [ ,v ' ~ ] , , ,

(pie + Q ) . , = O.

(3.2a) (3.2b~

(L2c)

In plane polar coordinates, ~o(r, ~9, t) and # (r, ~9, t) are defined by the conditions U,=r-l~,a, Uo=--~p.,; B,=r-lr Bo=--q~,,. Equations (3.1) have the same form with the operator V~ now defined by PT~=9*/gr*+r-~/~r +r-*9*/~O a, and the Jacobian relations now given with respect to r and ~9. Equations (3.2) apply using this definition for V~ and replacing 9/?x by 9l~r, 9/gy by r-~/9~9. In order to solve system (3.1) for ~ and q~, let us introduce general orthogonal curvilinear coordinates, m and r, in which case by setting m =~V the Jacobian relations (3Ab, d) m a y be simply satisfied. The functions h~(m,fl) and h~(o~,fl) are the usual metric coefficients, (ds)*=h~(do~)*+h~(dfl)~. Equation (3.td) implies immediately, since ~0~= 1 and ~v,a=0, that 9 ,a=0.

(L~a)

The vorticity is given by ~, =

-- [712 ~o =

-- (h~ h~) -~ (h~ hi'),~.

(}.3b)

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RtCHA~D R. GOLD;

while, by use of (3.3 a), r 9r

V~q} becomes

- (hah~)-~ (h,h~ I # , ~ ) , . = -- h~' # , ~ + ~, ~,~.

(3.3 c)

The Jacobian relation (3.tb) reduces to ~,a----2~,~,,~, or, by use of (3.3a, c), (t -- 2 q~,L) ~,,a = -- 2(h~2),a qi,~ q~,~,.

(3.3 d)

Using (3.3a--c) in (3.1c) reduces it to the form ~b,, _~ ~/(ha h~) -a (h z h~~ #, ~),~ + / (t) ----B ha z qi, ~ ~ _ ~7~, ~, ~ + ] (t).

(3.3 e)

Applying O/Off to (3.3 e) and using (3.3 a) gives (hfZ),a q),~,~=~,.a q), ~,, and substituting this result into (3.3 d) yields 8,, ~ - o , ( 3 . 3 f) from which it follows that (h~z) ,a q),~ = O. (3.4a) By use of (3.3 f), (3.1 a) becomes ~z,t = v

(ha h2)-2 (h2 hi-1 ~:~,~),~ =

v h~ ~ ~, a ~ -- r

~ , ~.

( 3 . 3 g)

Applying 8/b13 to this result and using (3.3 f) yields (h~2),a ~, ~~ = 0.

(3.4b)

In view of (3-3a) equation (3.tc) would immediately require that r in which case (3.3f) follows directly from (3.1 b). ~In addition to being useful in a sugsequent application, the somewhat lengthier discussion given above emphasizes the fact that ~,a=_0 is required for all solutions q~, including, for example, the ease where the nonlinear terms are identically zero (~='0), i.e., q)=2-Gp. The significance of (3,3 f), namely, that ~ = ~, (a, t), is quite apparent. First it follows that (3.1 b) can be satisfied only if both sides vanish identically, so that 8 (% V~p)] O(x, y)-----0. (For steady flow [7~q~=constant, and (3.1b) reduces immediately to this result.) This implies that the vorticity (in addition to the magnetic field in view of (3.t d))As constant along the streamlines, where ~p= constant. Finally, equation (3Aa) along with 0(% [7~7,]/8(x, y) = 0 defines W(x, y, t) in the corresponding nonconducting case, obtained by KAMP~ DE F~RI~T (see the discussion in [1]). To this is now added the relatdd magnetic field for a conducting fluid and the corree~tion to the nonconducting pressure distribution. All the,solutions % ~6 of (%t) may now be obtained from equations (3.3), (3.4). The more interesting .ones are discussed in the foUowing outline. r (a) For ~,,t 4=0 equation (3.4b) is satisfied first by the condition (h~Z),a=0 '. One is thus led to the class of flows where the streamlines are parallel lines, =~o (x, t), or concentric circles, W= ~v(r, t). In the former case (3.t a) fs satisfied

by (x, t) = ca x 8 + c2 x~ + ~ (t~ x + F (x, t),

(3.5)

where ca and c2 are arbitrary constants, m~ (t) is' an arbitrary function of time, and F is any solution of the one-dimensional heat conduction equation t_

F,,----- K~ F,~,

(3.6)

Self-Superposable Magnetohydrodynamic Motions

387

for K 1= v. It should be remarked at this point that the reduction of our problem to" the classical equations of L A P L A C E , POlSSON, and HELMHOLTZor tO the well known heat conduction or wave equations, et al., completes the solhtion. In any particular problem the form of the solution will depend on the domain of the independent variables and the initial and boundary conditions to be satisfied, while the general solution may be obtained by separation of variables, the use-of the fundamental solution, or the employment of infinite series or various integral techniques. Since these will not be considered here, the reader is referred to the vast literature on such initial value and boundary value problems. Equation (3.t d) shows that # is also independent of y (but not necessaril3~ a function of time). The corresponding solutions for r from (3.1c) are, therefore, r

t) : c3 x~ + c4 x + m2(t ) + F (x, t)

(3.5 a)

where m ' 2 ( t ) - - 2 C 3 ~ = / ( t ) and F is any solution of (3.6) for K I = ~ , and (x) = k (2 7)-1 x2 + c5 x + c e,

/(t) = -- k.

(3-5 b)

The (i) and (k) components of equation (2.1 a) show that s , = s , = 0 (j) component reduces to (~ot -- v~v,,x),~= m~ (t) -- 6ClV = s~.; hence # / e + ~ = s (y, t) -

while the

89 A B ~ = [m~ (t) - - 6 c 1 v] y + ~n 3 (t) - - ~1 ,~r , , ,

(3.5 c)

where m~(t) is an arbitrary function of time and # is given by (3.5a) or (3.5b). For ~0=~v(r, t), V ~ = r - l ~ / a r ( r ~ ] ~ r ) , and equation (3.1 a) is satisfied b y ~0(r, t) ----[k~ r" + n~ (t)3 In r + k2 r 2 + G (r, t)

(3.7)

where G is any solution of G,, =/~'~ (C.,, + r-~ G , ) ,

(3.8)

for K 2= v. From ~(3.1 d) it follows that ~b,o----0, and the corresponding solutions for r are r (r, t) ----k 3 In r + k 4 r ~ + n 2 (t) + G (r, t), (3.7a) where n ~ ( t ) - - 4 ~ T k i = / ( t ) and G is again a solution of (3.8) (K2=r/), and (r) -----k (4~)-172 + k5 In r + kn,

().7b)

/(t) = -- k.

Using (2A a) in polar coordinates, one obtains, first, " ( ~ , t - - V V~ v2),,= r-l [n{ (t)

-

-

4 k i t ] = v -1 s.o

or

s(r, O, t)

= End(t) - 4kiwi 0 +

q(r, O.

Applying the result to the eo-component equation, one obtains s,, = q,, =

~-~( ~ , -

2 O,;),

and the pressure is therefore given by #/e + ~2 = [n,' (t) -- 4 k~ v] 0 + f r -1 (~v2r - - Jt {])2r)d r -- ~ 2 r

(3.7c)

In view of the fact that the pressure must be uniform in any region, it follows either that n ' l ( O - - 4 k l v ~ O , in which case the multi-valued potential function/~ Arch. Rational Mech. Anal., Vol. 6

27

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R88

R. GOLD:

must contain a compensating external force, or, if no such force exists, i.e., if .Q is single-valued, n 1 (t) = 4kl vt + n3 (t). In equation (3.7c) V/is the solution (3-7), while 9 is given b y (3.7a) or .(3.7b). (b) For ~,t4=0 the alternate condition specified b y (3.4b) is ~ - . ~ = 0 , i.e., ~, = - - V~ V/is a linear function of V/, V~ ~o = / q v / .

(3.9)

E q u a t i o n (3.1 a) reduces to v/,t = v V~V/,

(;.9.a)

v/(x, y, t) = e"K3' H (x, y),

(3.9b)

and it follows t h a t where H is a solution of

vim = K3H.

(~.9c)

G. I. TAYLOR [10] introduced this solution for the nonconducting p r o b l e m in his s t u d y of the decay of vortices. Corresponding to this solution for % since (h~2),a#O, equation (3.7a) shows t h a t q) is also linear in V/, say,

CD(x, y, t) = rnl (t ) V~ + m 2 ( t ) .

(L9d)

E q u a t i o n (3.1c) then becomes

,,~ v/,, + ,,z'~v / + ~ ; = ~ ~ t7~v/+ i, or, b y use of ().9),

v/,, = (~ K~ -- ml m~ l) V/+ ( / - - m~). Comparing this result with (3.9a), one obtains ~ h ( t ) = c eC.-,')~:,',

.4(t)

= l(t),

and

q5 (x, y, t) = cer m t H (x, y) + m 2 (t).

(3.9e)

decays according to the magnetic viscosity t/, while V/ decays according to v. In both cases the space variation H ( x , y) is a solution of (3.9c). If ~ = v = k , ,n 1 (t) = constant = c, m', (t) = / (t), and ---- c V/+ m 2 (t) = c ek'%t H (x, y) + m 2 (t).

(}.9f)

Generally, electromagnetic dissipation is more significant than viscous dissipation, t.e., v << r/. E s t i m a t e s based on a simplified kinetic theory approach for an ionized gas such as hydrogen (reference [5]) show t h a t v/rl= 2 • tO-'~/Q, where ~ is the degree of ionization and ~ is expressed in cgs units. For sufficiently low values of the density, which will occur in several stellar applications, it is observed that v[rl--+t. F r o m equations (3.2) and (3.9a, d) one readily obtains

P/Q + Q = ~- (V/~ + v/~r) -- ~-K3 (2 m~ -- 1) v/z + ma (t).

(3.9g)

For ) / = v = k, oh ( t ) = c, and if c = ;t-t, the nonlinear t e r m s in (2. $a) vanish identically ( x = 0), and (3.9g) reduces to

P / e + D = -- 89(v/y, + V/Yv) + m~(t).

(3.9h)

Self-Superposable Magnetohydrodynamic Motions

389

(c) If se,,~=0, equation (3-3g) becomes 2

~z,~=hl~.,~

....

(3.10)

from which it follows, using (3.4b), that (h~2),a h~ 8, 8,, ~ = 0. This is satisfied first b y the condition (h~),a ~-0, which now refers to the steady flow counterpart of (a), namely v/----~v(x) and ~p----~(r). For the former, equation (SAa) yields ~ ( x ) = cl x 3 + c2 x" + cs x + c 4.

(3.tt)

The corresponding solutions for q9 are given b y (3.5 a, b). Equation (2.4 a) now reduces to s,v = - - 6 ci v, so t h a t s (x, t) = - - 6 q v y + m 1 (t), and p/o + Q = ml(t ) - 6ClV _ ~_~ (/)2,~

(3.t I a)

gives the pressure distribution corresponding to # (x, t) as given by (~.5 a), while if q~ = q5 (x), equation (3.5 b) m a y be used, giving PIe +s

= r -- 6 q v y -- -~-,~(ktl-~ x + r~)2.

(3.1tb)

Notice t h a t in the last case the well known Poiseuille flow solution given b y (3.1 t) (parabolic velocity distribution) has the usual linear pressure distribution in y modified in this self-superposable example b y the magnetic pressure, which appears as a quadrafic term in x. For ~v=~(r) equation (LI a) gives ~, = k l r 2In

r + k 2 1 n r + k s r e.

(3.t2)

The corresponding expressions for q~ (r, t) and ~ (r} are given b y equations (3.7a, b). Note t h a t from (}.12) the vorticity is given b y V ~ v = 4 ( k l + k s ) + 4 k 11n r. If q~(r, t) is expressed b y equation (l.7a), r-ls,~=--v(V~,),,=--4klvr s. , = 7 , , = r -1 (~v~, - - ~ ~ ) , p/o+.O=--

"1,

s=--4klvO+q(r,t

).

and the pressure is given b y 4k~,,O-- ~,~qb2,,+ f r ~(W~,-- ,~q~2.,)dr.

(~.12a)

If q~(r) is given b y (3.7b), this becomes p/Q + Q ---- -- 4k, ,, 0 -- -.~-~ [k (2,1) -~ r + k 5 r-X~2 + f r-~(~,y, -- )~ q~2,,)dr. (~A2b) B y discussion similar to t h a t following (L7c), we see that /2 must be multivalued to assure the uniformity of the pressure in (}A2a, b). If this is not so, it is necessary to take k~----0, in which case the solution for ~v(r) reduces to the result [712~,= 4k s-- constant, which will be considered next. (d) The second possibility under (e) is ~.,~=0, which says t h a t the vorticity is constant throughout the fluid, i.e., V ~ = c~.

(}A3 a)

This is a solution of both the steady and unsteady flow problem such t h a t ~' ix, y, t) = a~ x -~= a., y" + b (t) x y + V(x, y, t),

(~.t 3 b) 27*

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RICHARD R . GOLD:

where b(t) is an arbitrary function of time, 2 ( a l + a ~ ) = q, and V is an arbitrary harmonic function (V~ V = 0), while in the steady flow case ~p(x, y) = a I x 2 + a., yg_+ a3 x y + V(x, 3').

(3.13c)

Equations (3.3 e) and (3.4a) give the corresponding solutions for r From (~.4a) the condition (h~2),a=0 refers to the previously obtained solutions for ~v(x, t), ~,(r, t), ~/,(x), and ~v(r), which now must be modified in view of (3.t3a) to give ~o(r, t) = na(t ) In r + n.,(t), (3.t4a, b) V,(.~,I) : Ct x~/2 4-ml(t ) X @~lto(l),

~V(X) :=Q X2/2 + C2x + ca ,

~(r) = c , lnr + Q .

(3.t4c, d)

lhe solutions for r are given by equations (3.5a, b) and (3.7a, b), while the necessary adjustments in (3.5c), (3.7c) and (3.tl a, b), (3.t2a, b) for the pressure are apparent. If r equation (3.3 e) becomes ~ . ~ = (cr/-X--cl~,~)h~ where ~:=--V~o=-q and / ( t ) = - - c . Upon substituting this result into (3.4a), we obtain as the only new possibility q ) ~ = c ( c l rl) -1, or r

:

C (el T]) -1 -[- C6 ,

(3.1 S)

This represents tile combination V ~ o = c x, V~q)=c~ -1 where vg(x, y) and ~ ( x , y) may be expressed by (3.13c) and (3.t5) respectively. Determination of the pressure depends upon specific forms of tile arbitrary harmonic function V(x, y). If one specifies c22=c~ ~ , the nonlinear terms vanish identically (.~=0). An alternate assumption for the magnetic field is B = B~. Equation (2.1 d) requires that B~ does not depend upon z. In this case (B. V) B == 0, and system (3.1) applie~ with q5 replaced by B~, [ (t)= 0, and (}.l b) becomes 8 (v2, V~ ~p)/8 (x, y) = 0. Both the vorticity and the magnetic field are normal to the plane of motion and are constant along streamlines. The solutions for ~/, determined above are still applicable, while the corresponding magnetic field (as well as the resultant values of E and j) now take on a new physical interpretation. This will have additional significance in subsequent investigations of particular boundary-value problem.~. Note that an immediate consequence of this assumption is that corresponding to a constant transverse magnetic field, B., one obtains the general cla,~ of motion., wherein the vorticity is everywhere constant or proportional to g, [equations (~.t3) and (L9)--(3.9c), respectively] in addition to motions corre.-ponding to ~?(x, l), ~o(r, l), W(x), and ~o(r), previously defined. The .~olutions outlined above for the steady" flow case are obviously valid for a perfect fluid as well ab one in which the electrical conductivity is infinite. For a perfect fluid 0 ' = 0 ) in steady flow, g,(x, y) need not be biharmonic, as is seen from (~.la). We require only that V,~0=/(W), and the equivalence of these sclf-.-ut)e~posability conditionb for plane flow with STOKES condition for ~0 to be the .,t~:eam function for steady flow of a perfect incompressible fluid has l~een previously noted. For a viscous fluid with infinite electrical conductivity (~/= O) in .-teady flow /3~ (or rib, if/(t) ~ 0 ) need not be harmonic. The stream functions derived above are applicable, while the magnetic field is specified only by the relation B: (or ~)--g0t,). For an inviseid fluid with infinite conductivity, the only requirement in steady flow is that the vorticity and magnetic field be constant along streamlines.

Self-Superposable Magnetohydrodynamic Motions

39t

4. Three-dimensional F l o w The assumptibn of plane flow in Section 3 requires that both conditions U =~ [7 (z) and U, = 0 be satisfied. R. BERKER ~11] observed that either restriction may be deleted, resulting in two classes of "pseudo" two-dimensional motions. To study further the character of the present system of equations we shall now examine one of these possibilities with regard to self-superposable magnetohydrodynamic flows. Because of the presence of the magnetic field, this type of motion, which is characterized in detail by BERKER for the nonconducting case, is found to have additional physical significance. Case 1. U 4= U (z~, U~:# O. For the same form of magnetic field, it is easily shown that the potential functions ~o(x, y, t) and qS(x, y, t) can be introduced in a manner similar to that of Section 3. The vorticity components, ~----V• U, and the components of ~ = V • B become

~ = i u,,~ - i u~,~ - i, v~ v,,

~ = i B , y - - j B,,~--hV~q~,

where V~ is the two-dimensional Laplacian operator. system (2.4, 5) can be written as follows:

The self-superposable

u,,, = ,, v~ u, ~- /~ (t) ,

(4A a, b)

~, (v,, v~ ~)/eCx, y) = a e ( ~ , v; q~)/eCx, y) ,

(4.2a)

e(v,, ~ ) l e ( x , y) = ~ e ( ~ , B,)l~(x, y),

(4.2b)

(v~ v,),,= ~' v~ v,,

*t=~lV~*+I2(t), e (v,, q')l~ (~, y) = o,

B...,=~TV~B:,

(4.3 a, b)

e (~, B:)le (x, y) = a (q,, U,)IC~(x, y),

(4.4a, b)

where [l(t) and [i(t) are arbitrary functions of time. Equations (4.t) are obtained from the reduced momentum equation (2.5 a); (4.2) represent the first self-superposability condition (2.4 a) ; (4.3) are obtained from the ~reduced induction equation (2.5b) and (4.4) from the second self-superposability condition (2.4b). Observe that equations (4. t) -- (4.4 a) are just equations (3. t), so that the functions ~ (x, y, t) and # (x, y, t) are the two-dimensional solutions derived in gectiorr % Equations (4A)--(4.4b) define the corresponding z-components, U, and /3,, for each twodimensional solution. The (i) and (j) component equations of (2.t a) are unchanged, so that in terms of the obvious 2D and 3D subscript notation it follows that SaD,x=S~D,~ and S,D,y=S~D,y. The (~') component equation reduces to

U,.t + ~vy U,,~--~v ~U,,v-- ~(O yB,.~-- q~,B,,y ) = -- s , + vV~V,. By (4.t b), (4.2b), this becomes s,~=-- /l (t) which, along with the results noted above, implies that s3~ (x, y, z, t) = - z h (t) + s~ o (x, y, t) + k (t).

(4.S a)

Since S3D= (p/e+Q)3D+ (~]2) (q~,~,+ , 2 y + B**), it follows that

(P/q +/2)30'= (P/q + t2)2D-- (~/2) B~ -- z/l(t) +/~(t), where (p/q+Q)~D is the pressure distribution determined in Section 3.

(4.5b)

392

RICHARD R. GOLD:

Introducing ~ and fl in the m a n n e r previously described, we obtain from (4.1)--(4.4b) the following additional equations: using (}.3a), from (4.2b) we have 9 U,~ = ;t q~,~ B~,~, (4.6a) and from (4.4b) Bz, ~ = q ~ U~,~. (4.6b) Combining equations (4.6), we obtain (1-- ~ ~,~=) U ~ = 0. (i) U , ~ = 0 . This implies B,,~=O from (4.6). Thus U, and B, are also functions of W, and (4.1, 3b) become U~.t = v (h1 h2)-1 (h 2 h~X U,~), ~ + / , (t) = v ha 2 U . ~ -- v ~:z U.~ + ]1 (t),

(4.7a)

B..,t = ~/(h I he)-1 (hz h~ 1 B,,~),~ = ~/h~ ~ B . . . . -- t/~, B,,~.

(4.7b)

Applying O/Off to (4.7) and using (t.60, we see that (h~2),~ U~,~ = 0,

(h~2),a B . . . . -- 0.

(4.8a, b)

Following the scheme of Section 3, for a representative number of cases the corresponding solutions for U~ and B, can now be obtained. (a) $~,t 4:0, (h~2),a=0. Corresponding to the two-dimensional solutions given b y equations (3.5), (L7) are the following results for U~ and B,: U, (x, t) = el x z + e,, x + m l (t) + F (x, t),

(4.9a)

where m'l(t)--2rlv=Jl(t), and F is any solution of (~.6) for kl=v.

B ~ ( x , t ) = r l B ....

or

B~(x)=eax-t-ea(rl~=O ).

(4.9b, c)

F r o m (t.5c) and (4.Sb),

P[O + Q = [m'~(t) -- 6q v] y + m s(t)

/t (t) z

(,~/2) ( .~ + B~)

(4.9d)

where 9 and B. are given by (l.Sa) or (3.5b), (4.9b), and (4.9c). U~(r, t) = h x In r + k s r ~ + ~!1(t) q- G (r, t),

(4.10a)

where n'x ( t ) - - 4 r k o = / l (t), and G is a solution ot (LS) for k.,--v.

B,(r,t)=rl(B

....

+r-lBz.,)

or

B..(r)=k~lnr+k4(rl~O).

(4.10b, c)

Equational (~.7c) and (4.5b) provide the corresponding pressure distribution. (b) ~,,t4:0, (h~),a4:0, ~. . . . = 0 . 'Equation (4.8a)indicates that U, is also linear in % ~ = c l ~0, (4.1t a) in which case for /l(t)=O (4.1b) reduces to (L9a). The vorticity components are ~ = q ~..~ = q G ,

~, = -

q

~, ~= q u , ,

~ =

-

v t v, =

-

~ @

v~.

If c~ : I - ks, ka< O, we note that this solution satisfies the condition =

V X U = C1 U .

(4.11 b)

Self-Superposable Magnetohydrodynamic Motions

393

The vorticity vector has the same direction as the velocity vector at every point of the fluid and is proportional to it. Mo,tions of this type (Betrami flows) have been treated by several authors [11, [151. The solutions for ~ (x, y, t) and q~ (x. y, t) are given by equations (3.9b, c) and (3.9d--f) respectively. Since B, is also linear in ~, it follows that = V• = c~B, (4.1t c) where the analogy with the previously mentioned force-free field condition of apparent. The remaining cases are readily derived from the results of Section 3. Let us refer now to the second possible solution.

CHANDRASEKHAR is

(ii) U, a =t=0, q~,~=2 -1. Thus, = ),-~ ~, + m2(t),

(4.t2a)

U.= ] 2 B~ + g(~., t),

(4.12b)

and from (4.6) i.e., U:-- ] 2 B~= function of ~o. Equations (4.t, 3 b) become (~.,-= V(hlh2) -1 [(h2h~ ~ [,~,~),~+ (hlh[lU~.ti),O~ + I t ( t ) ,

(4.t3a)

B~,, = ~-j(ht h~.)-~ [h a h~ ~ B:,~),~ + (hx h~~ B,.g).r

(4.t 3b)

Substituting (4.12b) into (4.13a) and using (4.13b), we obtain |,2(v~)-i(v_r~)B,t+h-i2g,~

_~.-lg.t.

(4.14)

Applying O/~fl to (4.t4) and using (~.3f) yields ]'2(r ~])-x(,' -- ~) B:.~t + (h~2),~g,~ =- O.

(4.t5)

Representative solutions for b: and B~ can once again be presented, using the scheme of Section 3. (a) ~,,t @0, (h~2),o=O. W e shall restrict ourselves to the case ~O=v/(x, t), 4== (x, t) in the following discussion. Equations (4A, 3 a) become ~o,~=v~v,~+p~(t) x + P 2 ( t ) ,

q),t = ~ q),~ +/,,(t),

(4.t6a, b)

where Pa and p~ are functions of time only and where the Jacobian relations (4.2, 4a) are identically satisfied. We now require, however, the following additional relations : U~= ] 2 B ~ + g ( x , t ) , ~ = ; t - ' ~ o + m 2 ( t ). (4.t7a, b) Substituting (4.t 7b) into (4.16b) we have ~o,,-----~/v?,**+ ] ~ (/~-- m,), and comparing this result with (4.t6a) yields (q --

or

v ) ~ o , , = pl(t) x + p~(t) -- ].:A ~j~(t) -- m~(t)],

(rl--,,)~p,~=tlp~(t)x+r~p~(t)

--1/J.[l~(t)--m'2(t)].

(4.18a) (4.t8b)

Substituting equation (4.t7a) into (4.tb) and using (4.3b) gives ll;t(r# --v) V~B~=VZlT-t(i 7 --v) B . . , t = v g , , ~ - - g . ~ + i ~ ( t ) .

(4.t9a)

394

RICHARD R. GOLD:

Applying OlOy to this result, we obtain ][~ r/-1 (~l -- ~) B,.~.t = 0. Now if r / = v = k , equations (4.18) require t h a t p~(t)=0, t h a t ~v(x, t) is now a solution of ~v,~= k ~v,~, + p., (1), and q) = 2 - ~ ~o+ m 2 (t), where P2 (t) = V~. (I., -- m~)

(4.19b)

p2(t)=[,t(/~--n4),

so

(4.20a) (4.20b)

defines q}(x, t). E q u a t i o n (4.t9b) is satisfied for B,,y, 4=0, and B,(x, y, t) is thus a solution of the two-dimensional conduction equation,

B . , , = k V~ B,.

(4.2t a)

E q u a t i o n (4.17a) defines U~(x, y, t) where g (x, t) is a n y solution of the onedimensional equation (4.21 b) vg,.. - g,,+/~(t) = o . If v ~ r / , integrating (4.18 a) twice with respect to x, differentiating the result with respect to t, and then comparing with (4A8b), we see t h a t Pl (t) = constant = al, P2 (t) -- ~ [/2 (t) -- m~ (t)~ = constant = a2, m s(t) = a 1 ~1t + aa, m4(t) = a2 ~/t+ ['it ( ~ 1 - - v ) [ f / : ( t ) d t - - m 2 ( t ) ] + a 4 = a ~ v t - t - ( ~ l - - v ) f p 2 ( t ) d l + a a . Therefore, v/(x, t) is defined b y

~p -'= al X3 + O.a .t2 + ms (t) x + m6(t ) .

(4.22)

This is the special case V~2 ~ o : q to be considered. The corresponding solution for ~ ( x , l) is given b y (4.20b). E q u a t i o n (4.t9b) now requires that B, y t = 0 ; hence B~ -----H 1 (x, t) + H 2 (x, y ) .

( 4 . 2 } a)

Putting (4.23a) into (4.3b), we see that H 1 , t - - t l H I , * * = ~ V ~ H 2 = H 3 ( x ), where H a is an a r b i t r a r y function of x. Letting H I = H I (x, t) + ha (x) in the equation H L t - - ~ H I , ~ , = H ~ , we obtain (for H a = - - t l h ~ ) HLt=~IIIL, .. Similarly, let H o = H 2 (x, y) + h, (x) m . r/V~~ H , = H~, so t h a t for Ha = ~ h,,, (t/=t=0), we have V~z H~ = 0. Note t h a t H 3 = t / h"2------~/ h"i implies t h a t h 2 ( x ) + h a ( x ) = q x + c ~. Thus B , ( x , y , t ) as defined b y (4.23 a) becomes

H I = I11 (x, t) + h~ (x),

(4.23 b)

I1,, , = r///1 ....

(4.23 c)

w h e r e / / 1 is a solution of ha is an a r b i t r a r y function of x, and H., -----//2 (x, y) + q x + c, -- ha (x) ,,

(4.23 d)

.qelf-Superposable Magnetohydrodynamic Motions

~95

where Ha is harmonic, i.e., V~//2 = 0.

(4.23 e)

U(x, y, t) is now defined by equation (4.17a), where g(x, t) is a solution of

v g , * * - g,t + /1(/) = V/~~]-a(~ -- r) ~[-Ia.t,

(4.24)

I t a (x,.t) having been determined from (4.23 c). (b) ~,,~=V0, (h~a),a ~ 0, ~,~ ~-----0---- q), ~~. The last equation requires that ~ be linear in v?. In Section 3 equation (3.9a) permitted the required compatibility of the solutions for ~/4:v by ~roper choice of the arbitraty function , ~ (t). It should be not~d that the condition ~. . . . = 0 , implying that the vorticity be linear in % can be satisfied only by (3.9)--(3.9c) since the choice of Ks equal to a function of time would ultimately lead to a contradiction. In view of (4.12a) we now require 1tl I (t) = ]--i in (3.9d). Substituting this into (4.3 a) gives ~vt----~lV2x~ + V~(/=--m'*) . For ~ l = v = k , v/(x, y, t) is given by

w=ekK*t H(x, y),

V~ H = K s H ,

(4.25)

while (4.12a) defines q~(x, y, t) with m'o(t)=]2(t). For ~=~v the only compatible solution possible from ~:~,== = 0 is V~ = q .

(4.26)

~(x, y,t) = ms(t ) + [(x, y),

(4.27)

Thus, ~p,t ----~ q + ]/~ (]2-- m~) and

where m3 (t) = ~ q t + V~ f ta (t) d t-- |/$ m, (t) and, in view of (4.26), V~ ] = ca. The velocity components are thus independent of time. For ~ = , , = k , B,(x, y, t) is any solution of B:,~ = k V~ B~. (4.28a) Equation (4.t2b) defines U,(x, y,t) where, from (4.t5), g , ~ = 0 implies that g=~cl(t)~p+na(t). In this case ~:-=--V~p-=--Is and ~0 is given by (4.25). From (4.t4), therefore, for n'~=/~, it follows that n~(t)=0, so that g = ca v + '**(t),

(4.28b)

n~ (t) - - / i (t).

For v@2/ equations (4.14), (4.15), which define g, show that for (h~a),~ @0 it is necessary that g, ~ = 0 and, further, that B,,at---- 0. Thus B,, t must be a function of % which merely expresses an integrability condition for g in (4.14); while B,(x, y, t) is in general a solution of (4.3b), in order that (4A4) define g it is further required that B,, t be a function of ~p. Since ~ , = - - [712~ = - - q is required, equations (4.26)--(4.28) and (4A4) yield, for na(t)--:c 2, the formula V-i (y - 7)-1~ t L , , = ca m's

(n', = q c, ~ + 11).

Thus V-~(~, - ~)-an B , = cants(t) + k(x, y), but

V-~(v

--

r~)-ar~[7~B,=V~h

~

V~.(v-rl)-2~B,,-=caTI - m3. ,

1

t

396

I~ICHARD R. GOLD: r

Hence ms (t) =constant = c3, i.e.,

,.~ (0

= ~ q + l/~ E/2(t) -

./2 (0j = c3.

As a result of the restrictions imposed on the solution 9, ~ for v4=~ by equations (4.I2a) and (4.26)--(4.28), B= is now given by B.=~/2-~(r--t]) '[c2(c3t+c4)+h(x,y)],

V~h=c2c3# -~.

(4.28c)

(c) ~ , , t : 0 . For (h~2),#=0 the solutions corresponding to ~0(x) and ~b(x) are readily obtained from (a). The second possibility, called (d) in the corresponding discussion of Section 3, is ~'~,,:0, i.e., V~ ~o= q.

(4.29)

The two-dimensional solutions for ~0 a~d ~b are given by (3A3)--(Lt5). In view of (4.12a) we now require that V ~ q b = 2 - 8 9 i.e., cZ~=c~/2 in equation (3.t5). For r = o = k , g is linear in 9, which is defined by (4.29) rather than by (4.25). B~ is given by (4.28 a) and U by (4.12 b). For v=~t/, the discussion in (b) applies. Case2. U =~ U (zg), Uo=~ O. The analogous problem of a "pseudo" two-dimensional motion with axial symmetry is now considered. Magnetohydrodynamic flows of this kind have already been studied somwhat, apart from questions of superposability. The basic equations will therefore be introduced in this section but, for the present, the discussion of the solutions of these equations will be restricted to several immediate results. Potential functions ~(r, z, t) and 9 (r, z, t) can be introduced so as to satisfy the solenoidal condition upon U(r, z, t) and B(r, z, t). Thus U=e,r

l~.:+eoU,~--e:r

iV,.,,

B=e~r-l~.:+eoBo--e,r-1~,,,

and :

-- e, U o , , + eo r-x D2~t , + e. r - l ( r Uo),,,

where D ~ : O~/Fr~ -- r -1 O/Or + 02/Oz2.

System (2.4, 5) becomes (D2~).,=,,D4~o,

(rUo).t=vD2(r[~) +/~(t),

(4.t0a, b)

,9 (~v, r -2 D2~o)/O (r, z) + 2 r 1 Uo Uo" = ~ [0 ( ~ , r -~ D 2 ~ ) / 0 (r, z) + 2 r 1 Bo Bo, :], (4.30c)

0(~,, r Uo)/O(r, z) = 2 e (~, r Bo)/O(r, z) , qs , = ~ DZ CI) 4- /2(t), O(~, ~ ) / O ( r , z ) = O ,

(r Bo),t= rl DZ(r Bo) .

(4.~0d) (4.30e, f)

O(~,,r-~ Bo)/c~(r,z) -~O(rI),r-~U)/g'(r,z). (4.30g, h)

For the magnetohydrostatic problem ( U = 0 ) CHANDRASEKHAR([8] and [9]) examined the case of a force-free field which, in general, is characterized by the equation B x c u r l B = 0. As previously observed, this relation is a specialization of the general self-superposability equation derived earlier. It is examined in

Self-Superposable Magnetohydrodynamic Motions

397

some detail for the axially symmetric problem presently being considered, and the reader is referred to these papers for the solutions when U = 0 . A non-zero velocity is accounted for in [12~, where, under the additional restriction that the fluid be inviscid, a comprehensive treatment of the full axisymmetric equations is given. Included are several interesting solutions of the full system, the derivation of a number of general expressions and integral relations, and a discussion of boundary condition considerations. In view of the obvious need for examining system (4.30) in its own right a detailed analysis will not be a t t e m p t e d at this time. Simplifications for the steady flow case are apparent, while some rather general remarks relevant to the corresponding two-dimensional case ( U o = 0 - - B o ) and to the immediate solution U = [ ~ B (note [12J) are better left for a detailed analysis. For the purpose of illustration, however, let us consider the following particular examples. (i) ~ = ~ b ( r ) . Equation (4.30g) shows that ~,=~,(r). The operator D 2 now assumes the form D~=rO/Or(r-l~/Or), so that [for v = 0 , (4.30a) is identically satisfied and the following is not required~ D4~0= 0 provides, after several integrations, the well known PoiseuiUe motion:

~v = ct rZlnr +

c~r 4

@ car S,

(4.31 a)

where the c, are arbitrary constants. From (4.30e), taking k~=--/~(t), we have D z ~ = kl ~-1, so that = k 1(27) -1 r e in r + c, rL (4.3t b) From (4.30d), ~',U'o.~=]t~,,flo,~ follows that

~t',, cP-'

while, from (4.)0h),

~. Bo ,. U[;.I

~,,Bo,:=ffP,~Uo,=.

It

Uo =B ~x

Clearly, there are two possibilities involved, namely, Uo.~=B o . = 0 or Uo,:~l/XBo, ~. In the former case, all the Jacobian relations (4.30c, d, g, h) are identically satisfied, and complementing equations (4.3t a, b) are similar expressions for r U o and rBo as obtained from the relations D2(rUo)=k2v -1 (k2=--/~(t)) , D2(r Bo)=O (t/:t:0) using D'~=r ~/~r(r-l ~/~r). Alternatively, Uo.~= l' ;t Bo. ~ and, therefore, ~ v , = V ~ t r From the latter it follows that ~ o = V ; [ ~ + c o n s t a n t , whence it is now required that q = k l V ] ( 2 ~ ) -~, c2=0, ca=V~ q . In addition,

~ = V~. Bo + g(,).

(4.~2)

Substituting (4.32) into D=(rUo)= k., v -1, we obtain

D ' ( r l ' 2 B o + rg) = l,'?r

Bo) + D"(rg) = D"(rg) = k.,~, ',

since, from (4.30f) for ~=4=0, D~(rBo)=O. Substituting (4.32) into (4.30c) yields

2r 1go Uo,,= 2 r - ' (~. 130+ g) l."2~B#., = 2 2 r -a B~ Bo,~+ 2 ]/]tr-lg ]3o,,= 2 2 r

i

Bo Be,: .

Since Bo,,~=9 for this case, we must have g ~ 0 (and therefore k2=O ), so that Bo(r, z) is a solution of D=(r B o ) = 0 .

Uo=V]Bowhere

398

RICHARDR. GOLD: Self-Superposable Magnetohydrodynamic Motions

(ii) # = # ( r , t). The previous discussion p e r t a i n i n g to the e q u a t i o n s c o n t a i n ing J a c o b i a n s still applies. W e require, therefore, t h a t ~ = ~ p ( r , t ) a n d t h a t e i t h e r U o , = B o , , = 0 , in which case all the J a c o b i a n relations are i d e n t i c a l l y satisfied, or Uo,,#: O, Bo,,@ O. I n the f o r m e r case t h e solutions for % q), L~ and B o are r e a d i l y o b t a i n e d from (4.30 a, b, e, f), using D 2= r O/Or(r-a O/Or) to express the r e s u l t a n t one-dimensional n o n s t e a d y flow equations. F o r Uo,~=t=O, Bo=#-0 we o b t a i n once m o r e Uo,,= l/2Bo,, a n d V',,= V2 ~ , , . F r o m the l a t t e r , ~0= ] 2 q~ + h(t), a n d t h e solution for ~o a n d q5 from (4.30a) a n d (4.30e) is t h u s r e s t r i c t e d in a m a n n e r a l r e a d y experienced, i.e., d e p e n d i n g on the c o n s t a n t s r/ a n d v. In a d d i t i o n , Uo-----V2Bo+g(r,t ) is specified b y ( 4 . 3 0 e ) i n t h a t g ~ 0 is required for Bo,,4: O. Referring to equations (4.}0b, f), a consistant solution is o b t a i n e d o n l y when v - - - r / a n d [l(t)=0. References [1] GOLD, RICHARD R., eft ~[. Z. v. KRZY~,VOBLOCKI: On superposability and selfsuperposability conditions for hydrodynamic equations based on continuum. I. J. reine angew. Mathematik 199, H. 3/4, 139-- 164 (1958). L2] GOLD, RICHARD R., & M. Z. v. KRZYWOBLOCKI: On superposability and sellsuperposabfllty conditions for hydrodynamic equations based oll continuum II. J. reine angew. Mathematik 200, H. 3/4, 140--169 (1958). [3] STRANG, J . A . : Superposable flmd motions. Ankara Universit6, Facult6 des Sciences, Comm. 1, 1--32 (1948). [d] COWLING,T. G.: Magnetohydrodynamlcs. New York: Interscience I'ubhsher% Inc. 1957. ~5] ELSASSER, \u ~[ : Hydromagnetic dynamo theory. Rex,. of Modern Phvsms 28, No. 2, 135--136 (1956). -6] SEARS, W. R.: Magnetohydrodynamic effects in aerodynamm flows. J. AeroSpace Sci. 29, No" 6, 397--406 (1959). ~7] KAPUR, J. N.: Superposabllity in Magnetohydrodynamics. Appl. Sc,. R e s , Sect. A 8. ~8] CHANDRASEKHAR,S. : On force-free magnetic fields. Proc. National Acad. ScL 42, No. 1, I (1956). i9] CHANDRASEKHAR,S., & K. PRENDERGAST: The equilibrium of magnetic star~. Proc. National Acad. Sci. 42, No. l, 5 (1956). -lOj TAYLOR, G. 1. : On the decay of vortmes in a viscous fhud. Phil. May. 46, 671 --674 (1923). -llJ R~xtB BERKER: Sur Quelques Cas d'Int6gration des I;;quatlons du Mouvement d'nn Fluide Vlsqueux Incompressible. Thesis, University of Lille, France, 1936. ~12] CHANDRASEKHAR,S.. Axlsymmetrlc magnetic fields and tluid motions. Astrophysical J. 124, No. 1, 232--243 (1956). kldt] ibad. pp. 244--265. [14] PAo, S. : Astrophysical J. 124, pp. 2 6 6 - 2 7 t . [l~j TRUESDELL, C, & R. TOUPIN: The Classical Field Theories. Handbuch der Physik, Vol. 3, P a r t I. I3erlin-G6ttingen-Heidelberg: Springer 1960. See w167 34. 112, 114. Engineering Division Hughes Aircraft Company Culver City, California

(Received August 27, 1980)