THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING EFFECT OF ELECTROMAGNETIC FIELDS ON GRAVITATIONAL STABILITY OF A VISCOUS FIELD Michael Bentwich A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1959 November, 1959 IP-398

Doctoral Committee: Professor Chia-Shun Yih, Chairman Professor Robert C. F. Bartels Associate Professor Chiao-Min Chu Professor Ernest F. Masur Assistant Professor William C. Meecham

ACKNOWLEDGEMENT I should like to thank Professor C. S. Yih and other doctoral committee members for their assistance, guidance as well as patience. This work is partially supported by the Office of Ordnance Research-U.S. Army, through Contract Number DA-018-ORD-15556 with the University of Michigan. The help of Miss Martha E.'Robb in proofreading and other matters is very appreciated. Associate Professor M. Uberoi was of great help to me in starting this work. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENT....................................... ii LIST OF FIGURES.v...*.................................... iv SUMMARY...,.........*............................ I. PRELIMIntARIES........................................... 1 2. Governing Equations.......*..2 3. Boundary Conditio nss....o.................... 4 II. VERTICAL MAGNETIC FIELD: FLUID BETWEEN WALLS............... 7 4. Reduction to Ordinary Differential Equations.*....... 7 5* Solution.*........*... 12 III. HORIZONTAL MAGNETIC FIELD PARALLEL TO WALLS.......... 15 6. Reduction to Ordinary Differential Equations.......... 15 7. Solution by Approximative Series...................... 18 IV. MAGNETIC FIELD NORMAL TO WALLS........................... 27 8. Reduction to Ordinary Differential Equations.......... 27 9. Exchange of Stability........................ 29 10. Solution and Approximations.......*....... **...,. 34 V. EFFECT OF ELECTROMAGNETIC FIELDS ON STABILITY OF A FLUID IN A CIRCUILAR TUBE.... * 0. 4 * *..* a.... #. *. *.... * 40 11. Splitting of the General Problem into Two Parts....... 40 12. Development for Sub-System I.................. 44 13, Development for Sub-System II..*..y*ste.m............. 46 14. Boundary Conditions and Solution...........*..... 48 VI. EFFECT OF VERTICAL ELECTRIC CURRENT ON THE STABILITY OF FLUID CONTAINED BETWEEN WALLS............. 55 15. Splitting of the General Problem into Two Parts...,.. 55 16. Decay of Disturbances Represented by System I......... 59 17. Solution for System IIse l........*............ 64 18. Other Solution and Comparison............... 72 BIBLIOGRAPHY.................................... 78 iii

LIST OF FIGURES Figure Page 1 Solid Fluid Bounda4ry.............e...... e.. 4 2 Q-R-a Relationship - H3 Imposed...,............... 14 3 Q-R'-a-b Relationship - H2 Imposed.......0o.........sd 23 4 &Q-R-a-b Relationship - H2 Imposed..................... 24 5 Q-R-a-b Relationship - H2 Imposed.. d 25 7 Q-R-a-b Relationship H2 Imposed......... 26 8 Q-R-a-b Relationship - H1 Imposed s.......... o........ 36 9 Q-R-a-b Relationship - H Imposed.*............. 37 9 Q-R-a-b Relationship - H1 Imposed....1................ 38 10 Q-R-a-b Relationship - H1 Imposed...................... 39 11 Rectangular Cross-Section of the Column..........O...... 65 12 Q-R Relationship -J3 Imposed.1..........*.. 77 iv

SUMMARY The effect of electromagnetic fields upon the stability of a conducti.ve viscous fluid with a negative temperature gradient is investigated. The numerical relationship between the imposed electromagnetic field and the critical temperature-gradient has been found. In most cases considered, convection is inhibited by the electromagnetic field, but the effectiveness of the inhi. bition has been found to depend very markedly on the mode of convection.

CHAPTER I PRELIMINARIES 1. Introduction The stability of a horizontal layer of fluid under a negative vertical temperature gradient was studied by Rayleigh(l1, Jefferys(2) Southwell and Pellew(3) and others. The stability of an infinitely high column of fluid with a negative vertical temperature gradient has been investigated by Hale(4) Taylor(5), and Yih(6) These authors relate the temperature gradient (f) to the dimensions of the fluid-container, the viscosity (v), the thermal diffusivity (K), and the expansivity (a) of the fluid, and to gravitational body-force per unit mass (g), at the point of instability. Chandrasekhar has found that the stability of an electrically conducting fluid can be affected by the presence of an electromagnetic field. In this work, the instability studied by Hale, influenced by imposed electromagnetic fields, is investigated. The relationship between the imposed magnetic field, the magnetic diffusivity (i) of the fluid, and the other pertinent variables affecting the stability of an infinitely high column of viscous fluid (of various cross sections) is sought. For the fluid contained between two walls, three cases will be discussed, with the imposed magnetic field being either in the direction of the (vertical) temperature gradient, or normal to the walls, or perpendicular to both of these directions. For the fluid contained in.a circular vertical tube, the imposed field is also vertical. For the fluid contained in a rectangular vertical tube, the effect of a vertical electric current on the stability of the fluid is investigated. -1

-2Instability of a quiescent fluid occurs, in general, if the loss of potential energy associated with a certain mode is larger than the energydissipation for this mode. The effect of viscosity is therefore to inhibit the instability of quiescent fluids. It will be shown in this work that an imposed magnetic field, which will give rise to electrical currents and therefore energy-dissipation in case convection takes place, also has a general stabilizing effect, which is different for different modes of convection. 2. Governing Equations For stationary fluid, the equation governing the x3- dependent imposed temperature (T) distribution is: 2= (1) in which V2 2 + 2+ 2 6x, 6x 6x 1 2 3 is the Laplacian, where Cartesian coordinates xl, x2. and x3 are chosen, with x3 axis being vertical. The equations of equilibrium for a stationary fluid under the action of an imposed c'urrent J and magnetic field H is: 0 = - VP + W x T+ p0 g (l-cT) (2) where p is the pressure and p0 is the density of the fluid at some reference temperature To The equation of magnetic diffusion [see Reference 7, Equations (1-10)is:

-3where u is the velocity of the fluid. Therefore, in the case of no motion, Equation (3) reduces to: a m ) H = (3) at If the perturbation scalar quantities are denoted by prime, p' and T', and perturbation vector quantities by small letters, j, h and u (unprimed or capital letters denote the mean quantities p, T, H, and J), Equations (1) to (3) become: ~~D 2 Dt (T+T') = V (T+T') (4) P [ a + (u7) - v] u (+p) + ~(J+j) x (H + h) + P g [1 -a(T+T')] ) (5) and a - (H + h) V= x [ ( x ( + h)] (6) Subtracting Equations (1) to (3) from Equations (4) to (6) and neglecting products of perturbation quantities and the effect of expansivity on inertia, one has ( = -'u (7) p ( - vV2 ) u = -p + Jx + jxH] - O AT' (8) and ( - n) h = V x (u x H) (9) in which, since T has only a vertical gradient _a5T and T T 0 ax3 axz ax2 The equations of continuity for velocity and magnetic fields are: V.u = O (1. V.h = O (]_'~)

and the Maxwell equation relating current to the magnetic field is: 4j = V x h (12) Equations (7) to (12) govern the phenomenon under investigation. 3. Boundary Conditions In all cases considered in this work, fluid is assumed to be contained within solid boundaries. At such a boundary, the condition of no slip demands that all velocity components vanish. Furthermore, continuity of heat flow, as well as temperature at the solid-fluid boundary requires (see Figure 1) that: T' = T'* k* * k ax 1 1 where the k's are the coefficients of thermal conductivity, and where asterisk denotes that the quantity involved is a function or a constant pertaining to the solid. 0 x flu id solid Figure 1. Solid Fluid Boundary.

-5If k* < < k, 6T' k* 6T'* Tx1 k 0 aT't* -X -being finite. axl If K* > > K, T' = 0T'* = O as throughout the wall T'* - O. It is worth noting that the boundary conditions T' = 0 or -= 0 can be imposed, in practice, not only by ax the choice of proper wall material, but also by varying the thickness of the wall. The boundary conditions to be imposed upon electromagnetic quantities can be formulated by the use of Maxwell equations. Due to the continuity of magnetic flux, h1 = A* h*, on the boundary. Continuity of tangential components of the magnetic field demands that: h = h* 2 2 and h3 - h on the boundary. The Iast two conditions imply that: Lgjr ( 2h= 3)(') C =4j* ax3 ax2 a3 ax2 or continuity of current. Since the velocity vanishes on the boundary, continuity of the electric field demands that: 1 j = j*

-6and 1 j= 1 j* C a*- 3 on the boundary. Phenomena under consideration will be accompanied by the accumulation of surface charge on the boundary. Surface charge, as well as displacement current, can justifiably be neglected in this work, as it is electrodynamical in its nature and deals mainly in time independent cases. Here again, if a's and i' s are of comparable size, variations of electromagnetic quantities would penetrate into the wall. But if u* >> > flux continuity is maintained by h* < < h. It can be shown 1 1 that generally the magnetic field h*? is not parallel to the boundary, and hence h > > h and h > > h or simply 1 2 1 3 h2 = h =0 2 3 on the boundary. Also, a* >> a implies j2= * j and j a nd3 2 C-* 2 3 at the boundary, while current continuity demands: jl = j* there. 1 Again, if j* is not parallel to the boundary: o2 3 = u on the boundary.

CHAPTER II VERTICAL MAGNETIC FIELD: FLUID BETWEEN WALLS The effect of imposed, uniform vertical magnetic field (H3) on the stability of a conductive fluid contained between two walls and heated from below will be investigated. The destabilizing factor, in this and other cases investigated in this work, is the negative temperature gradient. 4. Reduction to Ordinary Differential Equations Since the imposed magnetic field is uniform, the mean current is zero. The equations of motion are therefore, in this case: ( at- v) u = - (p' h H ah + (0,0, PO g ) + FL 4 (1)x3 and the equations of magnetic diffusion are: ( a ~TVP) h= H (4)-(6) bt 3 6x3 The equations of continuity are: 3u. i = 0 (7) ahx. hi =O (8) axi where i = 1, 2 and 3. The disturbance temperature is governed by ( - ) T' = - u3. (9) The boundary conditions for the velocity at the solid wall are: u = O at x = + d. i 1 - -7

-8Assuming the wall to be highly conductive electrically but insulative thermally, j2 = 3 = and =O at x + d. This is a rather artificial case, as walls which are very conductive electrically are not normally insulating thermally. It must be remembered, however, that instability under consideration is gravitational in its nature, and gravitational instability can be induced by an adverse density gradient caused by a variation of concentration of solute. The mathematical formulation for the problem in that case remains the same, but an electrically conductive wall impermeable to mass diffusion is now not at all artificial. For convenience, density gradient shall be considered as effected by temperature variation, bearing in mind that any artificiality in the boundary conditions may disappear when the density gradient is induced otherwise. A two-dimensional case will now be considered; it will be assumed that u2 = 0 identically, and that all variations of pertinent quantities with respect to x2 are zero. The boundary condition on j3 can thus be simplified to: 4J = 0 = ah2 - ahl = ah2 o- at x = + d 3,x a ax 1 1 2 1 The differential equation governing h2 is: a ( - lva) h2 = 0 A time-dependent solution for h2 is in general non-trivial. However, if the solution for h2 is of the form h2 = h~ (xl, x3) exp (7t), y will always be real and negative (see the more rigorous proof given later). Consequently, h2 will be taken to be identically zero.

-9Cross-differentiation of Equations (1) and (3) produces: a v2 ) Ul:u3 H3 a ~hl ah3 axi:3 3 - aT'P ga t (1o) - Po g -xl Since the fluid flow and the magnetic field are both two-demensional, the equations of continuity (7) and (8) permit the use of the stream functions * and X such that: t h1 aX and 3 ax 3 1 3 3 3 Substituting these values of hi and u. into Equations (4) and (6), one obtains: (7a TIN) ax a (at ) 3 3 3 and aT X 6=a ~ (6') 1 3 1 Equations (4') and (6') can be integrated to yield: (a- X = x 3 (11) In this operation, the cQnstant of integration can be omitted without affecting u or h Substituting the expressions for u. and h in terms i i 1 i of stream functions into Equations J9) and (10), one obtains:; gcsTi (10')

-10 - and 2 8~ ( - Kv ) T' = (9') Equations (9'), (10'), and (11) are the governing differential equations. These shall be non-dimensionalized with the aid of the following sub stitutions: x x3 1x z = and T= (Kt)/d2 d The transformed equations are: K 6 - K -2 @ a~ (9?) ( 2 T 2 V) T'= a x d d 2 2 in which -2 1 -2 2 In the following, one will assume the mode of convection to be: 4 = v f(x) cos(az ) exp(7 ( ), T' = (nd) (x) cos(az) exp(7' ), X = d s(x) si(az) exp( ). The corresponding solution is quite general, as a large class of x2-independent disturbances can be represented by: *! = Z v f (X) cos (aiZ) exp(7T) ij 1j i j and similar expressions for T' and X. In view of the linearity of the

-11differential equations, an analysis of a single mode can be applied to any disturbance expressible as a summation of the basic modes. Substituting *, T', and X into (9") to (11'), one obtains: [- -(D -a )] e(x) = p Df(x), (12) [Y - pr(D2-a2)](D2-a2)f(x) = a (D2-a2)[1 s(x)] + RD, (13) 7 _ (D 2-a )] (x) = a P f(x), (14) d v in which D denotes d Pr is Prandtl number v R is Rayleigh's 4 cix (Hi;2d2 number - ga and Q = VK pov The boundary conditions shall be expressed in terms of the 6Tt newly-defined functions. Since = 0 at x = + d, one obtains 1 ax]. DO = 0 at x = + 1 Conditions u1 = U3 = O at x + d, or = at x = + d, become 1 3 1 - Df af = 0 at x = + 1 From Equations (13) and(14), one notes that the magnetic field is unlinked with the differential system for f and 0 if a, the wave number, is zero. Thus vertical magnetic field has no effect on a zero-wavenumber convection mode, which is the most critical one; it doe's have effect on other modes. Hence, assuming here a / 0, boundary conditions on f are: f = Df = 0 at x = + 1

-12The condition j2 = 0 at x = + d yields 21 - vX = 0 at xl = + or in terms of s(x): 2 2 (D -a ) s(x) O at x = + 1 5. Solution The value of y in Equations (12) to (14) assumed complex (7 = r+ iyi, where Yr and yi are real), indicates whether the mode tends to grow (yr > O ), decay (yr < 0 ), or oscillate (yi / 0 ). It can be proved that when the mode considered is neutrally stable yi = 0 (see Chapter IV). A non-trivial time-independent solution for Equations (12) to (14) and the corresponding relatiornship between aQ and R will therefore be sought. Setting y = 0 and combining Equations (13) and (14), one obtains: (D2 a22 (P) = Qa2(p f) - RDO, (15) (D2 a ) (x) = D(Pf). (12) The boundary conditions are now: f = Df = 0 at x = + 1, DO = 0 at x = + 1 The last condition can, with the aid of Equation (15) and the boundary condition imposed on f, be written as: (D a2)2 f = at x = + 1 In order to utilize a solution obtained by Yih(6), the operator L will be defined: L D2 2 L~D -a

-13Combining Equations (12) and (15) and rewriting the boundary conditions governing f, one obtains: [L3 -(R-Qa2) L - Ra2] f O (16) f = Df = L2f = 0 at x = + 1 The differential system for the eigenfunctions in(6) [Equation (46)] is: (L3 - RL -Ra2) f = O (46) f = Df = L2f = 0 at x = + 1 In (6) the author deals with the case considered here, not accounting for electromagnetic effect. The symbols a and f have the same meaning as here. Designating quantities of the quoted paper by a bar, one is given R = R(a) for non-trivial solutions of Equation (46) satisfying the boundary conditions. Any pair (R, a), when substituted in Equation (46), would correspond to an eigenfunction. The same eigenfunction is obtained from Equation (16) if one sets: R -Qa =R -.2 Ra =R Thus the relationship between Q, R, and a is readily obtained from B = R(a) given by Yih. The results are plotted in Figure (2). Evidently, for a / 0, Q raises the Rayleigh-number required for marginal stability. As mentioned before (Section 4), for a = 0 there is no electromagnetic interaction with the flow. Also, if one utilizes R (a) of the analogous system and lets a approach zero, one gets R = R. Thus Q does not affect the critical (lowest) Rayleigh-number required for marginal stability corresponding to a mode independent of x3.

- Z 0 313 100 200 300 400 RAYLEIGH'S NUMBER, R Figure 2. Q-R-a Relationship - H3 Imposed.

CHAPTER III HORIZONTAL MAGNETIC FIELD PARALLEL TO WALLS The stability of a fluid contained between two walls and under a negative temperature gradient P, in the presence of a horizontal magnetic field parallel to the walls will be investigated. 6. Reduction to Ordinary Differential Equations As the imposed field H2 is uniform throughout the fluid, there are no imposed currents. The basic Equations (I.7) to (It8) become in this case, pLH h 2 a 22 PP -VV )u. = - (p' + ~ ~77. "-' 4 ) H2 ah. + (O~,~,po0 T) + (1) - (3) (at — V) hi = H - (4) -(6) ot 3~2 x2 ui O, (7) axi ax. 1 and 2 at- KV ) T' = -u3 (9) The same boundary conditions as in Chapter II will be assumed: T O' j0 3 = = O at x = + d. -15 -

In this chapter, it will be assumed that u2 = h2 = 0 identically. However, variations with respect to x2 will now not be neglected, for otherwise the magnetic field would neither affect nor be affected by the velocity field. Cross-differentiation of Equations (1) and (3) yields: 0 3 613 1d 2 3 h1 6T' - opo a-. (10) Again, if u2 = h = O, it is possible to utilize the stream functions.2 X and r: u = u = a 2 ax a x and 1 ax3 i 2 X H2 Rewriting Equation (10), one obtains: 2 6 2 2 IH 2 2 6V ( - vV ) V - = xE (V2 X) ga a (10') Integration of Equations (4) and (6) yields: (_a - x =) X_ 6t 6x, (11) where 2 a2 + a2 1 ax 2 a 2 Equations (9) to (11) with the boundary conditions, govern the phenomenon under consideration. Equations (5) and (2) are not implied in (9) to (11). However, Equation (5) is indentically satisfied. Equation (2) yields ipL = 0, and therefore need not be included in the differential as2 system, as no boundary conditions are given in terms of p'

-17 - With the aid of the substitutions, Xl x2 _ 3 ~t d = x, d = Y d = Z T = d2 = vf(x) sin(by) cos(az) exp(yr), X = a s(x) cos(by) cos(az) exp(yr) and T' = (Bd) 0(x) sin(by) cos(az) exp(yT), Equations (9),to. (11) become: [ - (D - -b2)] (x) P Df (12) r [- (D2 -2 b) (D2 _ a2) f(x) = - Qb (D2 a2)[ s(x)] + RDO, (13) and. [y - (D22 -a2 _b2)] s = b Pf(x). (14) The boundary conditions imposed on T' and u. now become: DO = 0 and. f = f = 0 at x = + 1. The requirement J = 0 reduces to ah a 2x O_ _ ___ at x= + d. o=~ x x -X x 6 a2 23 a3 or s,(x) = 0 at x = + 1 Similarly, the condition j2 = 0 reduces to (D2 -a2) s = 0 at x = + 1 In view of Equation (14)7 only one of the last two conditions should be impnposed.

7. Solution by Aproximative Series Equations (12) to (14) indicate that solutions are possible either for f(x) even, s(x) even, and 0(x) odd, or for f(x) and s(x) odd, and G(x) even. This observation shall be used to obtain two classes of solutions by expanding s(x) as an even series and 0(x) as an odd series, or by expanding s(x) as an odd series and 0(x) as an even one. In both expansions, the expanded series shall satisfy the boundary conditions imposed upon the functions these series represent. It can be proved (and will be done in detail) that when the fluid in neutrally stable with y = O., Equations (12) to (14) become: (D2 - c2) (Re) = - R D(Prf) (12) (D2 - c2) (D2 - a2) (Prf) = b(D2 a2) [Q a s(x)] - D(RO)~ (13) (D2 c2) [ s(x)]= - Qb(Prf) (14) where 2 2 2 c =a +b In view of the boundary conditions and the oddness or evenness of 0(x) and s(x), the following series are chosen to express these functions: 00.00 (-Re) = A' cos(ncx), [i Q s(x)] = Z B' sin(r n1,2,.. n n=123.. n (15' or 00 (-RO) 1 5 A'' sin(n.x), [a Q s(x)] 1,3,5... n 2 00 = Z B"cos(=B). 1,3,5... n (15''

-19 - Substituting these values into Equation (13), one obtains: (D - c ) (D2 - a2) (Prf) = - (b[(ni) a ]B 1,2,3 n + (nit) A) sin(nix), (16?) D - c) (D2 - a) (Prf) a) (b=(2 ) + a ]Bn (Z) At) cos() (16i?) One nrow def~ines, for convenience Mnl [(it)(~2+ 2 nit2 2 a + a] a [(-) +a], n' [() + c2] and M" = [(2) + c]. The general solution of Equations (16') and (16? ) is: 1M,2,3 (M Hc)(n'' + K' sinh:(ax) + L' sinh (cx) + M' cosh (ax) n n n + N cosh (cx)) (17') Prf(x) z b Bn'(Ma) - (n) A' ( cop(n) 1,3,5 (JII') (et'I) + K' sinh (ax) + L'' sinh (cx) + Mtcosh (ax) rnL n + Nn' cosh (cx)) (17t) Demanding that Equations (17') and (17'') satisfy f = D~ = 0 at x = + 1 one can evaluate K' - N' and - N' in terms of A and B. and - n n n n n n

-20obtain: Mn N' = O, Kn' = L'' = 0, n n n n - (nct) cos (nc) sinh (a) Ln a cosh (a') sinh (c) - c cosh (c) sinh (a) K' = (nm) cos (nm) sinh (c) Kn a cosh (a) sinh'(c) - c cosh[i(c) sinh.(a) t ) sin (7) cosh (c) Mn = c sinh (c) coshja) - a sinh"(a) cosh'(c) J n N- () sin () oosh (a) n;c sinh (c) cosh (a) - a sinh (a) cosh (c) By substituting newly-acquired values for (Prf) avs well as the expansions for (R-) and (QI s) in Equations (12) and (14), one obtains: (M) B' sin(nnx) b n (M ) + () A n=1,2.... i,2,n3x = Q ( - sin(nlx) + K' sinh (ax) n + L' sinh (cx)), (18') n=1,3,5 c n 2 1,5.. (Y) (T l) ( - cos(-t) + M"' cosh (ax) 2 n +NAT cash (cx)), (18,')

-21(nt M ( )A' cos(ntx) cos(nx) = - R E b(Ma) ( An 1,2, 3 c n 1,2,3 (Mn ) (M'a) - (nit) cos(nitx) + K' a cosh (ax) n + L' c cosh (cx) ), (19') and Z (Mn) Ant sin(n)" ='1,3,5. n 2 b -,Bt * - (nn > At -Rn b(M )Bn 2 n ( n) sin (323) 1, 3,,5 (Mne ) (Mn ) +M'' a sinh -(ax) + N' c sinh (cx) (19") n n The terms sinh (ax), sinh.(cx), cosh (ax), and cosh (cx) will now be expanded in terms of sines or cosines, and the coefficients of sin(nnx), cos(ntx), sin(n-), or cos( 2- ) on the two sides of Equations 2 2 (18'), (19'), (18''), and (19'") will be equated. The totality of equations linear in (A')s - (BA)s or (AA')s -(BA')s yields a secular equation stating that the determinant consisting of the coefficients of the A's and B's is zero. A good approximation is achieved by considering only the first harmonic, terms formed by sine or cosine of (itx) in Equations (18') and (19') and (2-) in Equations (18' ) and (19''). This amounts to equating the first element of the determinant to zero. The results are: +2 F') (20') R= 2 ( 1,DF (20?) it Ft 11 22

-22where F' = + a coth(a)- ccoth(c) E[7My 7M71 a C and Q' (Ml) (M 1-)2 2 R a )C(M (1 + FQ b' Ft) (20' ) F2 ('')2 /' where F'' = (1 +_ 2 (,/2)2 1 1 ) a tanh.() c ta () (M The effect of the imposed magnetic field on marginal stability has the following features: (see Figures 3-6): (a) For b=O, Q does not affect R, or convection is not influenced by electromagnetic effect if there are no x2 variations; (b) For b/O, the magnetic field raises R required for convection; (c) The modes for which the stream function V is odd with respect to x are more stable than the modes for which this stream function is even.

0 I — i I - //: STREAM FUNCTION, SYMMETRIC b:30 L1.5 0 1000 2000 3000 4000 RAYLEIGH' S NUMBER, R Figure 3. Q-R-a-b Relationship - H2 Imposed.

2 5: STREAM FUNCTION, ANTI SYMMETRIC 0 237.6 1000 2000 3000 4000 RAYLEIGH S NUMBER, R Figure 4. Q-R-a-b Relationship - H2 Imposed.

2t r: STREAM FUNCTION, SYMMETRIC b'= 24 a' 0O= --- 5 -JZ ~ ~ ~ ~ ~ ~ (=O -31.3 0 1000 2000 3000 4000 RAYLEIGH S NUMBER, R Figure 5. Q-R-a-b Relationship - H2 Imposed.

,,: STREAM FUNCTION ANTI SYMMETRIC 2 - 3237.6 0 0 10,000 20,000 30,000 40,000 RAYLEIGH'S NUMBER, R Figure 6. Q-R-a-b Relationship - H2 Imposed.

CHAPTER IV MAGNETIC FIELD NORMAL TO WALLS In this chapter, the stabilizing effect of a magnetic field normal to the two parallel walls containing the fluid is exarmined. The fluid is again under a negative vertical temperature gradient. Since the modes of convection of any wave length have a vertical velocity component u3, the term (u x H) is non-zero for all modes. Consequently, interaction between velocity and magnetic fields will always be present. This interaction will be shown to have a stabilizing effect. 8. Reduction to Ordinary Differential Equations Since the imposed field is uniform throughout the fluid, there is no imposed current. The walls are assumed to be relatively) very permeable magnetically and insulative thermally. This type of wall can in fact be achieved by a thin coating of insulative material on iron walls. If mass rather than heat diffusion is considered, no coating is needed. The boundary conditions would therefore be: aT' and h = h3- 0 at x = + d ax -=0 2 31 The basic equations for this particular case are: Po (PI - vv2) -- - (P' + 4 ) + (O,O,pgoa') -)K1 (ah 4 - (1)-(3) -27

-28 - C - IV ) ah H au. - o0, (7) ax. - 1 ahi. - o (8) Bx. and (at- T' = - 3 (9) One assumes here that u = O. Consequently, Equation (5) and boundary 2 conditions govern a time decaying h2 (see Section 16). h2, not appearing elsewhere in the system, will be taken to be zero. Cross differentiation of Equations (1) and (3) yields: p (p _ V2)) (aul u3) = 1H11 (6hl h3 )-p p a'.(10) at ax3 axl 4tx ax ax3 ax1 ax In view of the vanishing of u2 and h2, stream functions A and X can again be used: u =a at - hi_ x and h3 aX 1 =3 = -axl' Hi 3x3 H3 The thermal and magnetic diffusion are given respectively by ( - v2) T' - x () axl and at adx Equations (9'), (10), and (11) together with the boundary conditions govern the phenomenon under consideration. Equation (5), being satisfied, and Equation (2) governing variation in p', can be omitted from the differential system.

-29 - With = v f(x) cos(az) sin(by) exp(yT), T' = (pd) 0(x) cos(az) sin(by) exp(yT), X = d s(x) cos(az) sin(by) exp(yT), and xl X2, z x3 andt = Kt d d d d equations (9'), (10), and (11) become: [ - (D2 a2 - b2)] (x) - D(Prf), (12) [Y - pr(D2 a2 - b2)](D2:-a2)f(x 4 Q D(d2-a2)Q:six)' + RDO (13) and [y _ (D _2 a2 b2)] s(x) = D(Prf) (14) The boundary conditions are: DO = 0 at x = ~ 1 for the temperature disturbance. For u3 u = 0 or x - at1 3 at x1 = +t d, one has f = Df = 0 at x = + 1 In order to satisfy h2 = h3 = 0 at xl = + d, it is sufficient to have ax = at x = + d as h2 = 0 identically, or Ds = 0 at x = + 1 9. Exchange of Stability As stated before, when the complex growth rate Y( = 7r + i.i) has the following significance; y7r>O implies that the mode amplitude grows

-30with time, y <0 indicates a decaying disturbance. Further, Yi $ O implies that the disturbance oscillates. The differential system, in general, has a time-dependent solution, depending on a, b, Q, and R. It will be proved, in this section, that neutrally stable modes (having Yr = 0) are time independent altogether (having y = 0). Differentiating Equation (14) with respect to x and substituting the results into Equation (13), one obtains: [7 Y r( - a2 b2](D2D2 (D a2) f = Q[ - D2(Prf)+ (y + b2 )Ds] + RDO. (13') Multiplying Equations (12) by (x), complex conjugate of @(x), (13') by (x), (14) by s(x), and integrating each product in the modified equations from x =-1 to x = + 1, one obtains from Equation (12): ( + a + b 2) f(e.0(.)dx - f1D20e dx =f1D(Pf)(T) dx).(15) -1 -1 -1 r Integrating by parts the second term on the l.h.s. of Equation (15), one obtains: 2 D e. x = DO. )1 - f1Do = _f1lD 12Jdx, -1 -1 -1 -1 since DO. x)l1 vanishes, because of the homogeneous boundary conditions. -l Consequently, Equation (15) becomes: 2 2 1 (7 + a +b ) + +s = fjD(Pf) -()1x * (15) -1 Similarly, Equations (13') and (14) yield: -[Y + Pr(a2 + b2)] a2FO - [y + Pr(2a2 + b2)] F1 - PrF2 = QPrF1 + Q(y + b2)'l Ds.F)dx + R b.T'()dx, (16) -l -1

-31and [y + (Q)(a2 + b2)3 SO + = f'D(Prf) sx-)dx, (17) respectively, where: 0 =1 el2dx 1 1IDO Ixo12dx, a = D2e12dx, -1 -1 2 F = fllfl2 dx, F = f jDf dx, F2 = f ID2 f dx, -1 1 -1 2 -1 S = fls112dx, S = f1 Ds12dx and S = lD2s2I dx o -1 1'1 2 -1 Integrating by parts the last term in Equation (16), one obtains: f1. |)dx+6 f lo Df dx = - Jf f Df dx = (-f.Dfdx) -1 -1 -1 -1 Similarly: f Ds,' f)dx = -( 1 s D. ). -1 -1 Utilizing the last two relations, and inserting Equations (15) and (17) into Equationi (16), one obtains: -[Y + P (a2 + b2)]a2F - [ + P (2a2 + b2)] F - P F = Q P F 1 (;v lb) + ]+ a ] S + ( ) S1 (y + )b r K K 0 K1 r[(ya2 +2)a + b ] (18) Equation (18) can be separated into its real and imaginary parts: y7ia2F + F-'- ) [a 2S + S ] + = O (18). o1 Prr o]

-32and y ra F + F) [a S + S1] = Tr 1 PrK 0 P r o - a2(a2 + b2) p rF - (2a2 + b2) PrF - PrF2 - PF1 - + b 2S + ( )2 a2b2S + (/)2 b2S ]) + R[(a2 + b2) aO + 81]. (18)r To complete the proof of the principle of the "exchange of stabilities", one needs more relations between the quantities defined so far. Multiplying each side of Equation (14) by its complex conjugate and integrating from x = -1 to x = + 1, one gets: IY + 2 b212 S + ()2 a S + ()2S + 2(Yr + (1) b )(Q)S1 K0 o K 2 r K K + 22 ()S1 + 2[ r + () b2] a2(21)S P 2F. (19) 1 rK 0 r 1 From which two inequalities are obtained, so long as yr > 0: IY+ (2) b212 + a2b)2(b2)2S + b2 ()2S P 2F1 (20) and (2) S <P F (21) K 2 r 1 Integrating by parts, one obtains: f D2s sTF) dx = Ds. )11 - fllDs12d f1Ds12dx, -1 -l -1 -1 since the first term on the right-hand side vanishes. By applying Schwarz's inequality to the last equation, one obtains: S2.So >S (22) o1

-33 - Finally, one can prove that if fls(x)dx = 0, then: -1 f lls dx f llDs12 a -1 -1 or s % s, (23) where the equality sign holds only when s(x) = 0 identically. In this case, f s(x)dx is indeed zero, so long as y / O, as can readily be seen -1 by integrating Equation (14) with respect to x from -1 to 1. Therefore, Equation (23) holds. Combining Equations (21) and (22), one obtains: (a)281 K S1 Pr1 and, in view of Equation (23), (.) S P F (24) K 1i r 1 Returning now to Equation (17) and integrating its right-hand side by parts once, utilizing Schwarz's inequality, one gets: (1) S1 2 Pr2FoS1, (25) so long as y > 0. From Equations (.23) and-'(25),,it..fdllows that: (K)2 So p. (26) Combining Equations (24) and (26), one obtains: (T)2[a S + S] P[a F + F (27) when the equlity sign olds for the trivial case of no isturbance.

-34Equation (18)i can be satisfied by either yi = 0 or by the vanishing of the bracket coefficient of y7. If f and s are not zero identically, 1.i / 0 implies, because of Equation (27), that R is negative so long as / 4< 1 and 7r > O. However, if R is negative, the bracket coefficient of Yr in (18)r is positive. Since the r. h. s. of (18)r is negative, in view of Equation (20), 7r is negative, contrary to the original assumpQPr tion. "-Th'us Under'.the. assum-ptiont-ti~:, -f,:'y.7r0-is'possible.only if Yi=O. Consequently,. the,:-:m' de cor poding 6to neutr&l stability -is time-independent. It should be mentioned that the limitation up.on Q is independent of the wave length in the y or z direction. Physically, it means that one can, for any mode, impose an interfering magnetic field which would make the principle hold. This is not the case for the velocity fields considered in Chapters II and III. Nevertheless, one can prove the principle of "exchange of stability" subject to limitation upon Q involving wave lengths a or b. 10. Solution and Approximations The relationship between a, b, Q. and R shall be obtained for the time-idependent mode by the approximation method carried out in Chapter III. The governing equations in Chapter IV are of the same structure as those in Chapter III. In this chapter, however, i(x) and s(x) are either both odd or both even; therefore, the following expansions are chosen: (-RE) = Z A' cos(nnx), [21 Q s(x)] =l B' cos(ntx); 1,2,3.~. n 1,2,3. n or (-Re)= Z A" sin( 2) [LLQ s(x)] = z B' "ni-)x 1,3,5...n 22 1,3,5.. n

-35In these expansions, boundary conditions for both 6 and s are satisfied. By substituting i(x) and s(x) in Equation (13) and utilizing the remaining boundary conditions, f(x) is obtained. Then approximating f, s and 0 each by a single harmonic, one obtaines for f(x) odd and 0(x) and s(x) even: (M' )2 (M' F [1 + F], (20) where F' = 2 1+ - a coth(a) c coth(c) [1 - a c' and for f(x) even and e(x) and s(x) odd: R ( )( ( )M ) R" = [1.+ -, (29) (MV';2 where F" -- (~)2 _ 1 i 2" =( 1.-~- 2(ir/2)- L'12 tanh( a) - c tanh(c) (M ) a c The quantities (~l), (M1'), (M1'), and (M1'') have the same definitions a a C C as given in Section 7. Equations (28) and (29) indeed show that the least stable mode is that of zero wave length, and that for given Q and a R' R'. Hence, modes in which f is even are less stable. These equations also show that the effect of the imposed magnetic field is present for any wave length, as the term including Q is not zero for any wave length (including a = 0). In this case, the magnetic field would raise the critical Rayleigh number required for convection. These results are ploted in Figures 7-10.

l l Q' 2 -o t 4 it STREAM FUNCTION, SYMMETRIC 302 0 500 1000 1500 000 RAYLEIGH'S NUMBER, R Figure 7. Q-R-a-b Relationship - H1 Imposed.

2 3 zL S F... I UNCTION, AN1TI SIYMETRIC 0 1000 2000 3000 4000 RAYLEIGH'S NUMBER, R Figure 8. Q-R-a-b Relationship - H Imposed. -J 1

2 L: STREAM FUNCTION, SYMMETRIC b2= 24 a2 00 ~~~~~~~~~~~~~~~~~I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~I 0 0 500 1000 1500 2000 RAYLEIGH'S NUMBER, R Figure 9. Q-R-a-b Relationship - H1 Imposed.

0 50 I // rCI: ~~~~~~~STREAM FUNCTION, ANTI SYMMETRIC b 24 a' 0 0 5, ~~~5000 109000 15, 000 209000 RAYLEIGH S NUMBER R Figure 10. Q-R-a-b Rela~tionship - H1 Imposea.

CHAPTER V EFFECT OF ELECTROMAGNETIC FIELDS ON STABILITY OF A FLUID IN A CIRCULAR TUBE In this chapter, the stability of an infinitely high circular column of fluid, with an adverse temperature gradient and a vertical uniform magnetic field will be considered. Apart from the geometrical configuration of the container, this case is the same as that discussed in Chapter II. The purpose of this chapter is, however, mainly to demonstrate an analysis which is not used in Chapter II and is more general. 11. Splitting of the General Problem into Two Parts The governing equations, Equations (II.1) to (II.9), transformed to cylindrical coordinates, with (xl, x2, x3) indicating (r, 9, z), 6aU1.(u 8 ~ IrkH3 Hh. p v.. 2 u2 ul )] aXl (p + J3h )3 (1) P[ a2 2 2)] 4u ax 1 2 1 3 1 a (p' + H3),(2) x1 ax2 4i po(a - vV )u3= n- H a -3 -an (P + >3) + p goT', (3) 3 30 - __i_ 2 l 2 2 1 H3 a (4)3 X1 3 -40

-41ah- (V2h+ 2 1 - h2)= H3 au2, (5) 6t 2'x l2 X1 2 x 3 (a i)h3 = 3 H, (6) =x3 aX3 x1 6(xlul) 1 6U2 aU3 - + 0 (7) x -X x )x 6x i 1 3 1 (xlul) + 1 ah2 + ah3, (8) Xl 1 3 and at- )T'= - u3 (9) The boundary conditions, as in Chapter II, are: 6T' 0 and j2=J3= 0 at x1=d/2 However, here, unlike Chapter II, no velocity component or variation with respect to any coordinate will be assumed to vanish a priori. By cross-differentiation of Equations (1) and (2), (2) and (3), and (3) and (1) as well as (4) and (5), one eliminates the p' terms and obtains: 0P [ va(t - %1 2 a- 2 ) aXl 1 o2 xi 3 1 aT' + o~ 1 x6, (10) 19 2,dx

p [ v + 2 1l - j2) o at ( 2 2 a - = XH3 a33 x 2 x 3 6j p V2')t = 4aT'' (12) o 3 3 and (i- -r~V2)j - 3.(3 at -T )3 4i ax 3 where t1' g and t are the three vorticity components. 2 3 The system under consideration, consisting of nine equations and accompanying boundary conditions, and governing eight dependent vaiables shall be split into two systems. Each such system shall consist of fewer governing equations, fewer boundary conditions and governing fewer quantities. It can be shown that any velocity field u. satisfying Equation (7) can be split into fields u 1) and u.2) such that: u(1) + u(2) _ u 1 1 1 u(l) + u(2) = u 2 2 2 =(1) U 3 3 6u(2) u(2) a (2) 1 + +1'X1 2 0 1 x1 X1 6X2

-43and a(xlu)) 1 2, This split is possible, as the five differential relations govern five newly-defined quantities. Since ui satisfies Equation (7), so does u(2). Consequently, any incompressible flow can be split into two, flow I for which 53 = 0 and flow II for which u = O, where flows I and II 3 3 are both incompressible. This conclusion can in fact be shown to be a particular case of Helmholtz's Theory. The separation procedure will be preformed for all pertinent variables. From Equations (9) and (6), T' and h can be shown to depend upon u or u(l) so that T' = T'(1) and fl) (2) h3 = h) However, from Equations (12) and (13), j3 =, as it depends on 3, and in flow I is zero. All other quantities will be associated with both flows: j = (l) + j(2) l = j(l) + j(2) 2 2 1 1 1 h = h(l) + h(2) and h = h(l) + h(2) 2 2 2! 1 1 Since the governing equations are linear, it is sufficient to have each of the set of quantities, ( (1) and ( )(2), satisfy these equations separately, so that the sum [( )(1) + ( )(2)] would satisfy them, too. In the spirit of former chapters, one seeks a time-independent solution assuming: U 2 (2) U(r) sin(nO) cos(az), U2 = (2) V(r) cos(n) cos(az), d u = (-)W(r) sin(nO) sin(az), 3 d

h = H X(r) sin(nO) sin(az), 1 3 h2 = H3 Y(r) cos(nO) sin(az), h3 = H Z(r) sin(nO) cos(az) 3 3 and pd/2 T' 0 e(r) sin(ne) sin(az), where X. x x2= and z = 12. Development for Sub-System I The continuity of the flow and the nagnetic field requires: +u~ n +a (l) w'-(l) Du() + U() _ n V() + a W1) = o (14) r r and Dx(l) + x n) y a z(l) (15) r r where d dr As j3 = 3 = 0 in this flow, the corresponding differential relations are: D(rV(l)) = nu(1) (16) and D(rY ) = nX) (17) Since each of the three componentst fields is governed by two differential relations, fields can be uniquely defined by scalar quantities.

-45 - With: (rV()) n'V(r), (rY(l)) = n X(r), U( ) =D 4(r), X(1) = D X(r) and W(1) = ( a + = )X a n I where Ln = ( DrD -a ) and (L + )- ( DrD 2 r r Equations (16) and (17) are automatically satisfied. Currents as well as vorticity components will be expressed in terms of the newly-defined functions as follows: ~(l) = _1'y n (Lh,) cos(nO) sin(az), (18) 1 a (ad/2).(1) E1 3 n 1 Ja- ( ) r (Ln X) cos(nh) cos(az), (19) (1) 1 2 = a(.[/22 D(L W) sin(nG) sin(az) (20) and J2 = - 1 (4-/) D(LnX) sin(ne) cos(az). (21) Inserting Equations (18)-(21) into Equation (12), and setting to zero for time-independency, one obtains: (L _ ). ( 1 v ) n (Lt 0 (d/2)2 n a (d/2 r n + P 2 )2 D(l) X):+ P ( —& -a/)2 D(yv): -i LH (4aM) n (LX) -+ PogO n. (22) a r

-46 - Since for any function of r (L f r ) ) = - D f(r) + 1 Lnf (23) n 2 r r n r r Equation (22) yields. LnLn (Pr) = a Q a LnX + Re, (24) where p2a(H3 )2(d/2)2 V _4(/2 p' r s and R It should be noted that Equation (24), though derived from Equation (10), implies satisfaction of Equation (11) too, since $ and X uniquely define velocity and magnetic field respectively. The dependency of hl) on u.l) can also be uniquely determined by inserting the values of h(l) h l) and uil), expressed in terms of' and X, into Equation (5), thus yielding: ___ x -nDX H K (d/2)2 A r K (rEI) ()(d/2)2 r 3 -a i3 V nr (25') &d/2 T(d/2) r which, with the aid of Equation (23), becomes: (~) LX = a(PrV). (25) Finally, Equation (9) yields: -Ln (r) = (Ln + a2) (pI) (26) 13. Development for Sub-System I Since this flow and the associated magnetic field have two components each, conventional stream function will be used to define

-47these fields: (rU(2)) = ncp(r), V(2) Dq, X(2) = na(r) and y(2) D= so that continuity is automatically satisfied. The vorticity and current components are accordingly: = (2) - v (L + a2) cp(r) cos(ne) cos(az) 3 (a/2)2 n (2) = 3 (L + a2) (r) cos(ne) sin(az), J3 3 4id/2 n (2) _ v (&ant) sin(ne) sin(az) 2 = (d/2)2 r j(2) (1 a ) sin(nO) cos(az) (2) v 1 (d/2) (anrp) cos(ne) sin(az) and j(2) = _ ( ) (a Da) cos(ne) cos(az) Inserting the appropriate values in Equation (12), one gets: 0 = Ln (Ln + a2) (Prp) + a Q(L + a2) (I.). (27) Inserting the appropriate values of hi and ui in either Equation (4) or Equation (5), utilizing Equation (23), one obtains: 0 = Ln( a) - a(Prp). (28) In this flow, temperature effects are not being dealt with, and Equations (27) and (28) govern only two components of velocity (both horizontal) and two of the magnetic field. Physical considerations may lead to the

-48conjecture that c = p = 0 identically, as buoyancy force does not appear to play its part. This indeed.would be the case if the boundary conditions governing ( )(2) quantities were homogeneous. However, the boundary conditions contain quantities of the form [( )(1) + ( )(2)]. Therefore ( )(2) quantities, being linked to convection terms via the boundary conditions, do not necessarily decay with time. 14. Boundary Conditions and Solution Combining Equations (24) to (26) as well as Equations (27) and (28), one obtains: [Ln3- (R- a2Q) Ln Ra (Pr) = (29) and [L L + a2Q] (L + a) = 0 (30) n n Equations (29) and (30) can be rewritten: (Ln + ~a) (L + C3) (Ln + c?) (Pr), (29') (Ln + n 05 n('3 r 2 2 where a. and Pi are functions of R, Q, and a. The general solution for t and cp are of'the forms: 3 3 Pr= AZ i Jn (air) + Z Bi Y (i r) and 2 2 Pr p= Z C. Jn( ir) +.Y+ En rn + Ep r+n i=l i=l n i n

Wishing to avoid singularities at r 5 0, one sets Bi = Di = E = 0 so that only six, out of the original 12, constants are left to be determined. The condition of no slip at the boundar& requires: 1 = 2 = u 0 at xi =d/2 or DS * n_~ 0 ~ at r=l, (31) r r n_+ + ID, 0 t r=l (32 and (L +a2) = at r=l. (33) An electrically very conductive wall requires: at x1 d/2 Therefore.(2) J3: E3 = o 3 or (L + aa) 2 = O at r=l From Equation (27), this boundary condition can be written: L (L + a) = O at r=l (34) f n Also: (1) (2) 2 =j2 )+(2) it xl =d/2 implies 1 D(LnX) + an: 0 at r=l. a n r Equations (25) and (28) will be used to express the last boundary condition in terms ofn and p. Differentiating (25), one~ gets: D(LnX) ='a D(P= ) (35)

-50Since (Ln + a2) Q O0 at r=l Equation (28) yields a Q - (Prcp) =O at r=l. Therefore, the condition j2 = 0 yields: D*I + n_ = 0 at r=l r This condition has been required for ul = 0 (Equation (31)). The duplication is due to the physical requirement that j3 5 0 on the boundary, implies: the vanishing.g f j2: ther&e, which is explainab:le.in.the fol-: lowing fashion. If one lets the vertical potential drop vanish on r=l for any angular position 0 or height z, as one does by considering the mode of the form f(r) cos(nO) cos(az), one makes the container an equipotential surface. The physical conidition imposed upon the temperature disturbance is:'0 at x = d/2 6xl or Di-= O0 at r=l Differentiating Equation (24) and utilizing Equation (35), one obtains for this condition D(LnLn + a2Q) (Pr*) - 0 at r=l. (36) Equation (30) when expanded is: D[(1 a DrD - n a2) DrD - a + a2Q] (L + a2)(P ) = r r r 2 n r [(L + a2) - (a/-+ i )2][(Ln + a2) - (a -+)2] ( + a2)(Prcp) = O (30')

Hence, in (30'), pi = P2, both being complex for non-zero a and Q. Therefore, (plr) = (P2r) and J (plr) = J(p2r). If one requires cp to be real, one should have: C2 + C1 0 O (37) Thus, one has six homogeneous boundary conditions and six undetermined constants. Two more conditions are required at the origin. One requires 0(O) = 0 because T' is expressed in the form i(r) sin(nO) cos(az) and is single-valued at the origin. Also demanding u2 to be finite at the origin: ru2 = O at r=O Therefore: r(V(l) + v(2)) = r(+ Dp) = at r=O (38') and e(O) = (L L + a Q)* = 0 at r=O (39') n n by virtue of Equations (24) and (25). These conditions are automatically satisfied if n ) 1 as Jn(w) vanishes at w=O for n n 1. A solution for n=l can also be constructed, however, by setting p = = O. In this case, Equation (34) would.not be imposed, there being no j3 in this case. Equation (37) is automatically satisfied, as cp = 0 identically. However, Equations (31), (32), (33), and (36) will remain (in a somewhat simplified form due to the vanishing of cp), as well as (38') and (39'). With six boundary conditions, the six coefficients of the Yl(cxir) and the Jl(aoir) can be determined, not assuming Bi = 0. In general: () = J (w)ln(w +H(k+l),wN(2k+l) r~~~~~~~~~~~cw,~~~~~~~~~~~~~~ /J(~l() 2'2 W k~ k! (k'+~l):

-52Thus at the vicinity of the origin Yl(w) behaves like -(w)-1 as the first r.h.s. term as well as the power series vanish there. Jl(w) vanishes for w=O too. Therefore, Equation (38') when p=0 reads: = 0 at r = 0 (38) or = lim (B1 + B2 1 + B 1 ) r, O oalr c2r 3'3r Consequently: 3 B. Z - =0 i=l ai Equation (39'), by virtue of Equation (38), reduces to: LnL Q = 0 at r=O. (39) In similar fashion, this yields: 3 B Z (q2 )2 =Bi i=l - ci The other four boundary conditions at r=l yield four homogeneous algebraic equations linear in the A's and B's, whose coefficients are the Bessel functions 1 (i) and Yl(i)s. For the axisymmetric case, n = 0, solution can be obtained for c: = 0o again. Further, if n = O, definitions of V and Y yield: v = v() = n = and Y= ) = nX = This is understandable, since this flow is axisymmetric with 3 and j3 equal to zero.

-53The governing equation for (Pr1) is the same as before, namely (29) with n = 0. The boundard conditions are: D, = O at r= 1 (31) and (Ln + a2) = O at r = 1. (33) The thermal and electromagnetic boundary conditions are both implied by: D(LnLn + a2Q) = 0 at r= 1 (36) All the boundary conditions are expressed in terms of first or higher degree derivatives of', but the function' itself has no physical meaning when n = O. Differentiating (29) by r and setting n = 0, one has: 2.2 2 D(L + ) (L + )(L + ) (Pr) = (L + A) (L +c2) (L + Ca) [D(Pr)] 0 as L D1 Dr - a2 r The solution for D(Pr'V) would be, in general, in terms of six Bessel functions of order 1. The linear combination of the three J's is the solution sought. Again in this case there are just three boundary conditions. According to the results in (6) for an infinitely high circular cylindrical column, in the absence of an electromagnetic field, the governing equations and boundary conditions for the eigenfunctions are: L3Y = R(L + a2) g E = DY = L 0 = 0 at r = 1, where L in (6) is equal to L1 here, and where axisymmetry is assumed.

-54The governing differential system here is: L1 [D(PPr)] = [(R-a2Q)Ll + R a2] [D(Pr*)] D(PrV) = (Lo + a 2) (Pr*) = D(LoLo + a Q)(pr*)= 0 at r=l The second boundary condition can be simplified to: D[D(PrV)] = 0 at r=l. Similarly, the third boundary condition, utilizing the first one, reduces to: L [D(Plr)]= O at r-l Finally, E and D(PrI) have essentially the same physical significance, both being proportional to u2. Thus, both systems are, in general, similar and identical for Q = 0. The solutions of the differential systems have now been reduced to homogeneous algebraic equations linear in the constant multipliers of the Bessel functions or the powers rn. From these, one can find the secular equation in the usual form. Thus the general three-dimensional solution for the pertinent differential equations satisfying the boundary conditions can be found for any integral value n. Results for axisymmetry were found to be compatible with other work done in the field. Also, if a = O, the effect of the magnetic field is nil. Consequently the least stable mode, being independent of x3, is unreflected by the presence of the magnetic field. For n i O, the solution requires the handling of a 6x6 determinant, and has not been carried out numerically.

CHAPTER VI EFFECT OF VERTICAL ELECTRIC CURRENT ON THE STABILITY OF FLUID CONTAINED BETWEEN TWO WALLS In this chapter, the effect of a vertical current on the stability of a conducting fluid contained between two walls, with a negative vertical temperature gradient, will be investigated. What happens inside the walls is also considered in this chapter. Numerical results, however, have been obtained for simpler circumstances. 15. Splitting of the General Problem Into Two Parts The fluid and the containing walls shall be assumed to be under a constant uniform potential gradient in the x3 direction, so that the imposed current density is in the fluid (-J3) and inside the walls (-J3 a*). Accompanying magnetic field will be x3 and time independent too.in the fluid 3 J =H2 -H, 4,(-j3) 1a 6x2 will hold.. And in the walls, one requires: H* aH* 4(-J3 a ) = - a al aX2 A solution of the form: H = (4J3)x2 H* = (4tJ3 ) 2 321 )2 3 2 and H H*= = H = H* = 0 2 2 3 3 is possible, provided the boundary conditions are satisfied. -55

-56 - Assuming no surface current in the fluid-wall boundary, one has: H H* and H H at x = + d -2 2 3 31 Preservation of magnetic flux requires: LH1 = S*H* at x = + d Equality of electric potential gradients along the boundary requires: a(r x ax3) a* ~(o- x ) 1 3 1 3 and 1 aH2 t1 1 2H2 a Cf axr ax a( ax ) 1 2 1 2 at x = + d As only H1 and H* do not vanish, both being x3 independent, boundary conditions to be satisfied are: H1 = L*H~* at xl + d and 1Hl 1 Hl at x =+d a ax - a* - This is possible if: a = I*a* or Oa = *. The type of field under consideration (H1 x2) can, however, be produced by other means without assuming q = *. Seeking the least stable mode, which, according to former results is likely to be independent of x3, variation with x3 will be neglected. Consequently, Equation (I.8) now assumes the following

-57forms: p(- vV2) u2 iP (1) oat 1 ax 2 3 Po at v 2 3x2 1 3 +'Hi3 (2) and Po ( - vv ) u3 = H 2 + Pog- c (3) Obt 3 12 0 Unlike former cases, due to the non-uniformity of H1, Equation (I.9) reads: ( t- _rV)h -a (U2H1) (4) 6t u2 1 ( a)h2 a(u2Hl),(5) and (-T r )h3=H1 aU3 * (6) Also, continuity requires: au! + u2= 0 (7) 1xl 1 x2 and ahl + ah2 = 0 (8) ax ax Xl 6x2 Temperature disturbance distribution is governed by: (at - KV2)T =- (9) 6t~~ =3 ~

Inside the wall, disturbance quantities T'*, h 2Z h, and h* are governed by: 3 (t -.- ) h*- o, (4)*-(6) (.~ *V2) T'* O, (9)* and ah*l ah* + = O (8)* ax ax 1 2 since velocity terms vanish inside the wall. Boundary conditions are: ui =O at x =+ d assuming no slip at the boundary. Temperature disturbance as well as heat flow continuity requires: T' = T'* and k L= * at x -+ d ax =d 1x 3x1 Continuity of flux or magnetic field components require: h h* and h =h* at x =+ d. 2 2 3 3 - and Lhl=,&hj at x + d. Finally, by equating electrical potential on both sides of the boundary, one getsh h* ah ah ah ah)h (2 2 and 1ah 1 ah ( 3) =,- (-'h) at xl = + d'. ax a* 6x 1 1

-59It is possible to split the large differential system governing many variables into two differential systems each consisting of fewer equations, fewer boundary conditions, governing fewer variables and therefore easier to deal with. System I consists of Equations (1), (2), (4), (4)*, (5), (5)*, (.7), (8), and (8)* with the relevant boundary conditions. System II consists of Equations (3), (6), (6)*, (9), and (9)*, and appropriate boundary conditions governing u3, h3, h, T', and T'*. 16. Decay of Disturbances Represented by System I In dealing with cases of stability, one encounters a set of homogeneous governing differential equations accompanied by homogeneous boundary conditions. It is physically understandable that if such differential system lacks a term representing a motivating force but does include terms representing energy dissipation, it would govern solutions representing decaying disturbances. For example, the system governing h2 in Chapters II and IV, as magnetic diffusivity is dissipative in its nature and there is no motivating term in (II,5) or (IV.5). Another example is System I here. Decay of a disturbance governed by this differential system will be proved, for once, rigorously. Cross differentiating Equations (1) and (2), and utilizing Equation (8), one obtains: (_ V22) _) = -1 3 (10) at. aX2 aX P 6 2 1.o 1

-60 In view of Equations (7), (8), and (8)*, one can make use of the stream functions an,,,and X* U -~ h h - U1 - x2 2 h*= X* and h* -* ~1; 2 ax Combining Equations (4) and (5) as well as (4)* and (5)*, one obtains: (S -a Tl )X = -H and (a- rp) X*= O Rewriting Equation (10) in terms of the stream functions, one gets: - va,), v ='g; (fX) (10') o 1 Jrith = v;(x,,y) exp(yT), X = (4tJ3d ) X(x,y) exp(yT), X* = (4 3d ) X*(x,y) exp(7y) X 2 _ X= - y and r= -t d d 2 d one obtains. [Y - (~)v12 X = o, (1,,)) V 1 (Y -I V 2) V2r = (a)2X), ( d') 1 1 V 3

where 4 i2 22 and V 2 2 + Pot ax ay The boundary conditions in terms of the stream functions are: I = i = 0 at x= + 1 ax i o0 as IYI -~ for velocity components. Electromagnetic quantities are governed by: 1a l = 1dva* 1a = * and ax _x*: - = aX at x =+ 1 ax E by X* O as Ixl -)0 and by X"*_,X o as lyl -. At the boundary (x = + 1) the r.h.s. of Equation (11') vanishes; thus, subtracting (11") from (11")* (1*), one gets: - [(1) V 2X - (flX) t V2X*] = O Since the vanishing of one bracket term in the last equation implies the vanishing of the other, and since at = a*k*, either pd* X* = AX at x = + 1 or 1 a, x =+

Also ax ax* _-x x at x= + 1 -x ax Multiplying Equation (11") by X, Equation (11'')* by X* and Equation (10o'') by V, integrating throughout the volume within which these functions are defined, and utilizing Green's theorem, one obtains: y X + (() X1 - f an = Y x d, (12) f X +Jff i ~(12)* X*0 + (2) X*l - r. X*, (ln) *d* - 0 ( and 7:'Y -'2 - = Q () f f Y (X) v (13') 2 V V x Substituting Equation (11'') into the r.h.s. of Equation (13'), one obtains: 1' 2 Qffv y [x - y dv, (13) where the direction of n is defined to be perpendicular to the surface of integration, fffv'. dv indicates integration of the Volume including fluid, ffffv*'' dv* indicates integration over volume of walls, fIt... d and ff&... d 4 indicate corresponding respective surface integrations. In Equations (12), (12)*, and (13): =X _fffvIXIvd = fff v I f2d 0 X*0 = v ff Ix*12dv* x1 - ff *f = IVX*12dv, 2dv I f2dv f fffv 1,1P* 12'dv.0 f ffv 1*1 dv 2 1 - r IVV I 0

-63and Combining Equations (12) and (12)f by utilizing the boundary conditions, one obtains: 7[Xo+,X +. [X +x if+ffX*y a +dbt (14) 0 0 v 1 ax Rewriting Equation (13), integrating y a- V with respect to x once, ax and using the boundary conditions for. and,one gets: Y - 2 3 = r v y a = d QY r(f ff y r d) = vQ 3hT77d ) x (13') -QY(ffv Y dx Combination of Equations (13') and (14) yields: -~ - w=Q['3 - 1712(x + —x*) -r(Y) (x +Lx*)l (15) 1 2 3 Y o 1 1)] (5) Multiplying each side of Equations (11? ) and (111 )* by their complex conjugates, and integrating over their respective volume, one gets: 1H12 X2 + 27r(X) Xl + ()X2 X2- 7) f a) ndn 3 and 2' X*.+ 27( x +f (_x*)2 X*:-Y(D)f X* an. -(t)'r', X.a*.adS'~ = ~ (17)

When the last two equations are combined by using the boundary conditions, it follows that: 3= 1Y12(Xo + p X*) + 27r (p) (X1 + * 1) + ( (x2 + ) (19) V 2 1 2 Equation (15) can be split into its real and imaginary parts: Y1 = +&P [- |12(Xo+ a1"it x 7) -r (x + x x)] (15) and 2~Q 3 071\o 4LI rv 1 V 1 r and o=~ i [1 -Q(R) (X + tX*)]. (15)i Substituting from Equation (19) to Equation (15)r, one gets: -r Yl = + Q(a)2 (x2 + * Q() (*) +r Q() ( + * *) r 2 v 2 2 r v 1 Since T1V,2 etc. are positive definite, 7r < 0 and 7r = 0 only when = X = X*= 0 identically. Hence, if there is a two-dimensional, this disturbance will decay with time. 17. Solution for System II In the spirit of former chapters, the non-trivial time-independent solution of system II shall be sought. With xl x X1 X2 =x - = y d d T, = v (ad) e(x,y), h = v (41J3d) cp(x,y) c 3 Tj 3 and u3 = v f(x,y) 3d

-65one gets: -V1 f = Q y a - Re, (20) 2 (21) -V1 p = y (21) ax and Ve = f. (22) So far, what happens inside the fluid and walls has been considered, and the fluid has been considered to extend to infinity in the direction of x and y. Numerical solutions will be obtained for the case in which the fluid is bounded at x = + 1 and y = + b; under some circumstances, one may let b become relatively larger than 1. What happens inside the wall will not be considered. Consideration of a column (or slab) extending to infinity is somewhat artificial. Furthermore, in practice, disturbances are likely to be concentrated over a finite part of the fluid domain. Y I i b Figure 11. Rectangular Cross-Section of the Column'

-66A magnetic field varying:linearly- with y can be formed in several ways. Such a field can be formed by having r = ri* (Section 15) at the boundary walls at x = + 1 extending to infinity in the y direction, or by having iron masses behind thin glass walls at x = + 1. In this case, walls at y = + b shall be considered thermally as well as electrically very conductive, and walls at x = + 1 shall be considered thermally and electrically conductive. This choice is artificial, but the main purpose of the calculation is to shao the difference in the effectiveness of electric current in inhibiting different modes of convection. The boundary conditions to be imposed upon disturbance quantities are therefore: = O or -0 at x= + 1 xl ax T' = O0 or = O0 at y = + b h3 =0 or p =0 at x = + 1 ah 0O or -ag O at Y =+ b 3x2 by &nd u =0 or f =0 at x = + 1 and y = + b. The function f can be expanded as an infinite series: 2n -1 by [a] f = ZZ Amn sin(mnix) cos( 2 ), m. n or [b] f=Z Z Amn cos(" 2 x) sin (.-') mn 2b

-67These are not the only types of series expansion of f that satisfy the boundary conditions at x = +1 and y = + b. However, one need not consider the case in which f is even both x and y-wise, since on physical grounds (continuity) one must require: b b ffu dxdy = O, or f dxdy = 0,.Kb 3 b so long as one deals with the stability of a column contained within a certain volume. Also, from previous knowledge, it can be expected that the velocity field described by f qdd in x and y is likely to be a relatively stable one. Therefore, only the situation where f is in the form [a] and [b] will be investigated. For case [a], with 0zm = mi and Pn = 2n1 b', Equation (22) becomes: n 2 b V. -= Z Amn sin(a mx) cos(Pny) - 1 rntl,2,3.. m=l, 2,3 the solution of which is: e - ZAmn, [Z-.. s(a x) cos(3ny) n=l1,2 m=l,2 2 2 am ~+..[n +. Ln sinh( x) cos(ny) (23) In this solution: e = 0 at y = + b is automatically satisfied. By adjusting the complementary solution to satisfTy the boundary conditions, L's can be evaluated. Since: n DeI =O ~Z A Amn a am( cos(y)- x=+l amn m 2 2.. L am + Pn + Z L B cosh(gn) cos(Bny), n nn

-68one has: L= 1 ~, Amn (-1)moC% (24) n n cosh(n) am 0 + n2 Substituting assumed f in Equatiot (21), one gets: -v = Z ~ Aannxm) y cos(amx) COS(Pny) m n thus = =ZZ Ammn(nm) [- cos(amx) y cos(ny) m n i2 n 2 + 2An ccos(alx)sihh( ny)] + Mm coa mx ) sinh(amY 22 - mm mm ara + p m n +Z K cosh(n x) sin(pny) (25) n n Again with the aid of the given boundary conditions, Mm and K can be evaluated. Since: a-_ O at y = + b. ay Amn (cm) n4-l o = S Z 2 [nb(-l) ] cos(aZ) 2 [b(-1)' ] O's (amX) + - Mm (am) cosh(amb) cos(omx) thus n14 M = - 1 E Am(~nb) (-)l (26) mm ) (26) Losh(amb) n (a2 + pn ) Letting x = +1 in Equation (25), multiplying by sin(pny) and integrating

with respect to y from y - b to y = + b, one has: O = Kn cosh(n) + Z M cos(o)a + k — Acos(am) [bn + 2k 8 k (27) mk 2 + n 2 2 n' m a + k m k in which a = f sinh( my) sin( iy) dy, -b bi = b y cos(py) sin.y)dy, -b and i 6o. = Kronecker delta Finally, the newly-evaluated cp and 0 will be inserted in Equation (20). By expressing all quantities on both sides of the equations in terms of sin(omx)cos(Pny), and equating all sin(agx) cos(Pny) terms on both sides of Equation (20), for every pair (m,n) one obtains: Amn I m2 + n= mk 2 k (a 2 Amk. km 1Kk/mk m m+ M.neMn n) + R/2L e (28)

-70in which b m n y sinh( my) cos(.iy) dy = fmn, sinh(p x) sin(a x)dx = e -b 1 _b 1 b =ay and b08(p = c.n Iby sin(pny) cos(piy)dy = d and j y cos(b y) cos(i y)dy = Equations (28), (24), (26), and (27) yield an infinite set of homogeneous algebraic equations governing the A':s. By setting the determinant of the coefficients of A's equal to zero, one gets the desired secular equation for Q and R for non-trivial solution of the system. By letting m run from 1 to m', and n from 1 to n', a (m'x n') determinant, an approxlmation to the desired relationship between Q and R, is obtained. Letting m' = 1 and n' = 2, one gets: oCL 1 l tanh(1 2 22 O All R[l + 2 2 2 ] ( + p' i 2 2b2 __ 22 ]__ (+ 1 ) -Q % [ - 2-, 1o(1+1-,2( + -, 3 (2.al +,. 2. _ tanh(P2) 2p1 P2 ) t2 + h ~22( l 2" 2p,. B 2 2 2 2 22 p2 1 +2 2 + tanh(alb) (alb) a 2.A.2 +Ci2 -'-Ce~~a -2) 2 8 a 2a2 2 a + At2 Q (2' B 1 2 -22 12 2 2 2 (2 22) 2 2 ) (P2 (~a1 P2. - P

(~~~~~~~~~~~~~~~~~~~C CM CCI 04 04~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 04-, CCL (M ca. cq'1' H + H+ 01 0 HL H Co. +. Crl r-i rA(M c q I ca ca cq cq C~~~~~~~~~~~~~~~~~~~~~~~~~~lc. r-i t3 OI C o_ - -\i C 1,(~ ~ r —!C~ r-! ~ ~~~~~~~~~~~~ ~(O, C~J rU -i -I - ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ d r-[ - i 04~~~~~~~~~~~C 044 O'l r- Ca- r - I-i I0 H -C CD- CUj Cd % —.X C\1 4- -101) ~v m~ c ~ Ca ri Ca A C~j c0 +a04 i-i I CM 01 rH H;CDL r - dH+ HD C 04 01 + H C C~ CL j CL C\ C. C~j CD Co. r-I C~j 04+ +C~ C~~j ca Co. cq cq cq C~~~~~~~~~V c rl C~~~~~~~~~~~~~~~~~~~~~L ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~f~~ ~ 04. caj 04 + Hl tf r- r i o_ CM C r CU 01C4 +' H 0104 CH + ai C CUL + + P-I ca r-i Cj r- r r-I Co_ r-I b01 Crj Cl -i r-i q rj Cl \Irc04 _ ~~~U (V (U C4 CU~~~~~~o- c c Cr- c ri rf + rl rlC r-,+ %C r- C Co Co_ CL Co_ Co. o. C~ + C. N~ c4 r-i 1 —l Clj r0 0 04 r-I C(j r-i c0 04 o. C\0 r1 ca c4 a0 Col dH 0 + + +H0 00 rl ~ ~ ~ ~ 0 04 H 4 I 61 c4 + 61.~~~~~~~~~~~~~~~~~~~~~~0 H4 01 04 CQ1cv 1~~~0 01T Hl 01 04I H- +- 0 ~04H0 ~1 ~ 01~ 0

-722f 2 2 1 ]2 + (alb) tadh(Ob) 2 + 22 2 22) (29)a in which All All/(a + 2) and A2 A12/cd + 22) Special attention should be paid to the first two terms in the bracket coefficient of the diagonal elements. R(l1 tanh( ) 2 I4 2nl 2 2 (30)a 1+ (....2n)2 n 2b if Q = 0, the determinant reduces to one having only diagonal terms of the form of (30. Such a determinant would vanish by having one of its terms of the form (30, equal to zero. Vanishing of the nth diagonal term yields an approximation for Rn the Rayleigh's number corresponding to the convection mode having u3 varying as cos(2. cty), when electromagnetic interaction is neglected. By setting term (30k, equal to zero, having n = 1 and b very large one gets R1 -4/3 = 32.5 which is in good agreement with 31.3 obtained in (6) 18. Other Solution and Comparison If f is now.a-ssusmed:to. b.e,of expaneilon..form:[bl: f = Z Amn cos(2m'l fix) sin(-2Y), mn 2 b

-73in which X = (22-1_x) ar)nd = I m 2 b Resulting algebraic equations are: -ot 2 2 11 1 +.1 21 23 2 2 2 1 cothwm,2 (2X1 0 2L 2X1 2 2 (X+222 + ~22 2'02 (x 1+ c2 2 cothc 22 2X 1.2 + 2. ( 2~2 )2 2X12 u 2X 2_ O'2'0 2'Di-2.12 2 2' +.012 2 2 + 41 1, 2 2 2 22 X1 ~(1 2 4x12' u22 cothz)1 2X1' ~ X2 20) 2 _2 M.! 2 2 2 Xi'+ + u % (x + )) 1 2 i1 cothwl 2X12 h, -ot 12 22 X2 + 12 02 2 a 2 1 1 2 -1 2 X1+) 20:.'

-744)_______ 20-) tL 2 ) ~(x12 +~'% + (x+b) coth(Xb) I COth2 2X12'1'2 COth1D2 2X12 1 2 2 2 2 O=GAM 11222 +2 + 2 21 2 2 2 2 c1 212 1 1 1 2X - A'`U (X - 2+.. "2 k 2 2)2.+ (lb) coth(Xkb) o2 "2' 1 +'L2 ml 2,. 1'. 2 2 2 2o2 (2 211 XX1ct (-X +.0) 1 _ 1 2(+ 2M 2 1Q ~2 (i2 COthw 2X10)l )2 2w1U 2 2- 2 2 2 2 2;1 1 e1 2 ooth%' 271.l )2'2%'~2 _ (.1:~ )2+"('2' ~-~-"2 2 2 2 2 0.) - X k +, +M 1 2 1 1 1 1 1 2

-75 - + (Xlb) coth(X lb) 2 (29) Again diagonal elements have terms of the type 22 coth22o R[l+ 2 1 2 I ip (3[)b i which yield the critical R for convection modes having u3 vary as sin() by setting the (nth) (30)b term to zero. Since Equations (29)a and (29)b have been arrived at by approxiImation, the limitation upon utilizing these to derive numerical results should be carefully observed. Equation (23) shows that insofar as 0 is concerned, no approximation with respect to y is involved in the ealculations. By setting (30) to zero, an approximation of R, is obtained because of the representation of sinh(c {) by sin(alx) term. However, Equation (25) shows that y i yields the terms y2 cos(Pny), y sinh(mXy) and y sin(Sny) in Equation (20). Hence, there is also an error in these calculation arising from approximation in the y direction, when one considers cp as represented by only a few harihonics. This error is compounded with errors arising from approximation in the x direction. However, the y-error is of a relatively higher significance than the x-error, especially iif b is large. Thus, in order to solve the system for large b, many y harmonics should be taken into cmsidexation. The use of oly two y harmonics is chiefly to

-76denmonstrate the method, and can be utilized with confi-dence only for small b. Equating the determinants of the- coefficients in (29)a and (29)b to zero, one gets the Q-R quadratic relationship for b = 2: 33.6Q2 + Q[2932-R(46.2)] + [R(1.53)-236] [R(2-562)-119.5] = 0,(31) a and 3.6Q2 + Q[874-R(18.9)] + [R(1.695)-24.4] [R(1.128)-152] = 0 (31) In Figure (12), the relationship between Q and R has been plotted for both flows. The solid curves represent valid approximations, the dotted lines represent the expected contributions if more x harmonics are used in calculations, and the dashed dotted lines do not represent valid approximations. The Q corresponding to asy Rn of each of the flows is zero. Figure (12) also indicates that the imposed electric current does affect convection inasmuch as there is interaction between velocity and magnetic fields. In both flows, the term (u x A) is not zero, for y = 0 and s in fact j(u3H1), a vector in direction y. The term 7x(u x H) contains therefore a term L (u3 H1) which represents the amount of interx1 action of velocity and magnetic fields, and is clearly of bigger magnitude in flow [a] than in flow [b], for b = 2. Consequently, flow [b] is less stable with as well as without the imposed current. In view of the reasonable results obtained for b = 2, it is hoped that the method used can be applied to any b.

400 0 0 40 80 120 160 RAYLEIGH'S NUMBER, R Figure 12. Q-R Relationship - J Imposed.

-78BIBLIOGRAPHY 1. Rayleigh, Lord, "On Convection Currents in a Horizontal Layer of Fluid When the Higher Temperature is on the Under Side." Scientific Papers, 6i Cambridge University Press, (1916), 432-446. 2. Jeffreys, H., "The Stability of a Layer of Fluid Heated Below." Phil. Mag., (7), 2, (1926), 833-844. 3. Southwell, R. V. and Pellew, A., "On Maintained Convective Motion in a Fluid Heated From Below." Proc. Roy. Soc., A176, (1940), 312-343. 4. Hale, A. L., "Convection Currents in Geysers." Monthly Notices Roy. Ast. Soc. Geophys. Supplement, 4, (1937), 122. 5. Taylor, Sir Geoffrey, "Diffusion and Mass Transport in Tubes." Proc. Phys. Soc., B67, (1954), 868. 6. Yih, C. S., "Thermal Instability of Viscons Fluids." Quart. Appl. Math., 1959. 7. Cowling, T. G., "Magnetohydrodynamics." Interscience, 1957. 8. Chandrasekhar, S. "On Characteristic Value Problems in High Order Diff. Eqn. Which Arise in Hydrodynamic and Hydromagnetic Stability." Am. Math. Monthly, 61, No. 7, (1954), 32-45. 9. Ostrach, S. "On the Flow, Heat Transfer, and Stability of Viscous Fluids Subject to Body Forces and Heated from Below in Vertical Channels." 50 Jahre Grenzschichtforschung, Verlag Friedr. Vieweg und Sohn, Braunschweig, Germany, 1955.

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