Vol. 17, 1966 Kurze Mitteilungen - Brief Reports - Communications br~ves 639

Then,

0 = aij t i j = 2 6 (4 + ~2)-1/2 (tll _ t22 _ ~ t12)

+ 2 [ 2 + 1 2 (1 1 \1121-11~[t13 { 1 (1 1 \1/2'/231

1 ~ ( 1 + 1 1/2 -1/2 { 1 - - (1 1 } t23] ,

and since ,~, e, /, are completely a rb i t ra ry we obta in RIVLIN'S universal relat ion

t l l - - t2. 2 = ~t12, together with tla : t~3 = 0 .

5. C o n c l u d i n g R e m a r k s

We have dealt here with universal relat ions for isotropic compressible elastic bodies. The method used m a y be extended immedia te ly to isotropic incompressible elastic bodies. In tha t theory however, non-homogeneous deformations are allowed E5]. In consequence we m a y have universal relations holding only at a point in the body. Equa t ions (11) remain unal tered bu t added to them is the incompressibi l i ty condi t ion A~ ~2 ~3 = 1.

R E F E R E N C E S

[11 J. L. ERICKSEN, J. Math. Phys. 34, 126-128 (1955). [21 R. S. RIVLIN, Proc. 2nd Tech. Conf. (London, June 23-25, 1948) Cambridge: Heffer. [3] C. TRUESDELL, and W. NOLL, Handb . der Phys ik I I I /3 , w (1965). [47 C. TRUESDELL, and R. TouPIN, Handb . der Phys ik I I I /1 , w (1960). E5] J. L. ERICKSEN, Z. angew. Math. Phys. 5, 466-486 (1954).

Zusammen/assung

Es wird eine Methode zur Erha l tung allgemeingiil t iger Beziehungen ftir Verzerrungen kompressibler, isotroper, elastischer K6rper dargestellt.

(Received: April 12, 1966.)

T h e S w i r l i n g , R a d i a l Free J e t in t h e P r e s e n c e of a

T r a n s v e r s e M a g n e t i c F ie ld

By NORMAN RILEY, School of Mathematics and Physics, Univers i ty of Eas t Anglia, Norwich, Great Br i ta in

The work described in the present paper is an extension of previous work b y RILEY ~1] 1) and MOREAU [2, 3 7. The swirling axi -symmetr ic radial je t flow of a conduct ing fluid in the presence of an axial magnet ic field is discussed.

1. I n t r o d u c t i o n

When a conduct ing fluid flows in the presence of a magnet ic field the effect of the force, of electromagnetic origin, act ing on the fluid is such as to inh ib i t the flow of fluid across the lines of force. MOREAU ~2, 3 7 has demons t ra ted this quite s t r ikingly in his s tudy of a

1) Numbers in braekets refer to References, page 642.

640 Kurze Mitteilungen - Brief Reports - Communications br~ves ZAMP

two-d imens iona l j e t in a t r a n s v e r s e field a t low m a g n e t i c Reyno lds n u m b e r . The so lu t ion ob ta ined , wh ich is qu i te ingenious, m a y deve lop a s ingu la r i ty a t a f in i te d i s t ance in wh ich case m o t i o n in t he or ig inal d i r ec t ion across t he l ines of force is a r re s t ed a n d t he f luid flows ou t a long t he d i rec t ion of the m a g n e t i c field. The presence, or otherwise , of th i s s ingu la r i ty depends u p o n t he d i s t r i b u t i o n of the field.

RILEY [1] showed t h a t a s imi la r i ty so lu t ion of t he a x i - s y m m e t r i c b o u n d a r y - l a y e r equa t i ons w i t h a swir l ing c o m p o n e n t of ve loc i ty m a y be o b t a i n e d in t he absence of a p ressure g rad ien t . This so lu t ion was used to discuss t he p roper t i e s of va r ious r ad ia l j e t - t y p e flows. I n t h e p re sen t no te i t is shown t h a t , despi te t he a d d i t i o n a l compl ica t ions in t he equa t ions , a s imi l a r i t y so lu t ion is sti l l ava i l ab le for rad ia l f ree- je t flow w i t h a swir l ing c o m p o n e n t of ve loc i ty w h e n t he f luid is c o n d u c t i n g and the re is a n axia l magne t i c field. I t is shown t h a t , d e p e n d i n g u p o n t h e r ad ia l d i s t r i b u t i o n of t he axia l field, a s ingu la r i ty m a y deve lop as in t he two-d imens iona l case. I n such cases m o t i o n across the l ines of force ceases, all t h e f luid f lowing ou t in the axia l d i rect ion. I t is also shown t h a t the p ro jec t ed s t r eaml ine p a t t e r n , up to th i s s ingular i ty , is una f fec ted b y t he field.

The t y p e of j e t flow env i saged here m a y be p roduced b y a f ini te disk r o t a t i n g in a t r a n s v e r s e field. The flow over t he d isk i tself has been considered, for example , b y I { A K U T A N I [4].

2. The S imi lar i ty Solution

The b o u n d a r y - l a y e r equa t i ons gove rn ing t he flow descr ibed in w 1 are

02u Ou Ou w ~ #2 a H ~ u + v - - u ~ + v Oy x e OY ~ '

(1) 0W 0W U W #2 O~ 02W

u O x + v Oy- + x e OY 2 '

0 ( z u ) + 0 ( x v ) = 0 (2) o~- o y �9

I n these equa t ions /~ t he pe rmeab i l i t y , a t he conduc t iv i ty , Q t he dens i ty a n d v t he k i n e m a t i c v i scos i ty of t he fluid, are all a s s um ed cons t an t . The coord ina tes x and y are in t he r ad ia l a n d axia l d i rec t ions respect ive ly . Axia l s y m m e t r y ha s been assumed. The ve loc i ty c o m p o n e n t s u, v are in the rad ia l a n d axia l d i rec t ions a n d w is t he swir l ing c o m p o n e n t of veloci ty . The m a g n e t i c field H 0 = H o ( x ) is in t he y -d i rec t ion a n d t he m a g n e t i c Reyno lds n u m b e r is t a k e n to be suf f ic ien t ly smal l to ensure t h a t t he field is u n d i s t o r t e d b y t he f luid mot ion . I t is a s s u m e d t h a t t h e r e is no e x t e r n a l l y appl ied electr ic field. I t can also be shown t h a t , w i t h i n t he f r a m e w o r k of b o u n d a r y - l a y e r theory , t he i nduced electr ic field m a k e s no c o n t r i b u t i o n to these equa t ions . W i t h s y m m e t r y a b o u t t he p l ane y = 0 t he requ i red b o u n d a r y cond i t ions for th i s f ree- je t flow are

Ou Ow v Oy Oy - - O , on y = 0; u , w - > 0 as y + oo. (3)

The c o n t i n u i t y E q u a t i o n (2) implies t he ex is tence of a s t r e a m func t ion ~o such t h a t

X ? A - - X V ~ - - Oy ' Ox "

Fol lowing RILEY [1] we seek a s imi la r i ty so lu t ion of E q u a t i o n s (1) a n d (2) in t he fo rm

W = ] / ; A x gl(x) I ( v ) , V = ~ x - l g2(x) y , w = A2 e x-~ g3(x) /'(V) . (4)

Here t he q u a n t i t i e s A and e are c o n s t a n t s and /(7) is t a k e n as

1 / = t a n h ~ ~/. (5)

Vol. 17, 1977 Kurze Mitteilungen - Brief Reports - Communications br&ves 641

T o c o m p l e t e t h e s i m i l a r i t y s o l u t i o n t h e q u a n t i t i e s gl(x), g2(x), a n d ga(x) a re to b e d e t e r m i n e d s u b j e c t to t h e c o n d i t i o n t h a t for e = H o = O, gl = [1 + 13/x3J1/8 a n d g2 = [1 + I3/x3]-2/3. [n t h e s e e x p r e s s i o n s t h e c o n s t a n t l is a c h a r a c t e r i s t i c l e n g t h a s s o c i a t e d w i t h t h e j e t d e v e l o p m e n t .

I f we s u b s t i t u t e (4) i n t o E q u a t i o n s (1) we h a v e , w i t h a p r i m e d e n o t i n g d i f f e r e n t i a t i o n w i t h r e s p e c t to ~ a n d a d o t t h a t w i t h r e s p e c t to x,

74o2 : (6)

g, g, : x g , L g, g~ '

w h e r e ~ 0 ~ -- #s a / A = 0 H~. F o l l o w i n g t h e i deas i n t r o d u c e d b y 1V[OREAU [2j w e n o w w r i t e

X s (g, : g q = 1+ d g~(*)' .gs- + g~ : (71

g-~ \ x g ~ g ~ : g~ & J ' & - ga : '

a n d s u b s t i t u t i n g i n t o (6) t h o s e e q u a t i o n s m a y be r e d u c e d , b y v i r t u e of (5), t o

g s / / " + g , / ' S + g 4 1 ' = O , g s / / " + & / ' S + g 4 1 ' = O . (8)

C l ea r l y we r e q u i r e g5 = 7% g4, g6 = g7 = ks g4 a n d , u s i n g (5), t h e c o n s t a n t s k 1 a n d ks a re d e t e r m i n e d as k~ = 1, k~ = -- 2. T h u s , t h e e q u a t i o n s s a t i s f i e d b y t h e fou r q u a n t i t i e s gi(x) (i = 1 . . . 4) m a y b e w r i t t e n as

74~ : g~ -- g4, (9)

(fflg2 "~ ~X-<I I g 2 ] = 1 + g4 ' gsg~1-{ 1 + \ (-xegag' g2 ]I' "%" < ' * g : ' l = g s -) 1 - - 2 g , ,

g l (lO) g L ( 1 - x g a _ ] = 1 - - 2 g , . gs ga ]

U s i n g (9) t o e l i m i n a t e g4 in E q u a t i o n s (10) we f i n d t h a t s o l u t i o n s of (10), w h i c h g ive gl = [ 1 + la/x3] 1/3, & -- [1 + la/xn]-s/a w h e n e = H 0 = 0, a r e

g~ = x - ~ [ ( : - :)s/, + zs]~/s (1 - F?m, | g s = x ( x s - e s ) l / s [ ( x s - e s ) a / s + l s ] - s / s ( 1 - F ) ~ / = , ] (1i) g . = x [ ( : - : ) , i s + lnj-ws (1 - F ) .

w h e r e x

f ( x ) = / 2 ~ X (x 2 -- 82)-I/2 [(x 2 -- E2)3/2 q- /8]1/8 ~X" (12)

I n (12) t h e c o n s t a n t a, l ike l, is a s s o c i a t e d w i t h t h e d e v e l o p m e n t of t h e j e t a n d m u s t be c h o s e n so t h a t a > e if l 4= 0. W h e n e = 0 t h e r a d i a l a n a l o g u e of M o r e a u ' s s o l u t i o n is o b t a i n e d a n d w h e n H 0 = 0 R i l e y ' s s o l u t i o n is r e c o v e r e d .

T h e f o r m of E q u a t i o n s (11) a n d (12) i n d i c a t e s t h e p o s s i b i l i t y of a s i n g u l a r i t y o c c u r r i n g in t h e so lu t i on . As a n i l l u s t r a t i o n we t a k e t h e s i m p l e s t case p o s s i b l e in w h i c h a = l = 0 a n d t h e m a g n e t i c f ie ld is c o n s t a n t so t h a t ~ = c o n s t . F r o m (11) a n d (12) we t h e n h a v e

g2 = X (::2 e2)-i/2 (1 -- ~].~o ~ X2)I/~, g~ = x (:: - :)-,/~ (1 - 7402 :).

(131

ZAMP 17/41

642 Kurze Mitteilungen - Brief Reports - Communications brbves ZAMP

Clearly t he so lu t ion b r e a k s d o w n w h e n ~/0 ~ x = 1, m o t i o n across t he l ines of force is de s t royed a n d t he f luid is def lec ted in t he y -d i rec t ion para l le l to t he l ines of force. W h e t h e r or n o t b r e a k d o w n of th i s t y p e occurs depends u p o n t he r ad ia l d i s t r i b u t i o n of axia l field.

I t shou ld be n o t e d t h a t , as w i t h all s imi la r i ty solut ions, t he scope of so lu t ion (4) is l imi ted a n d i t m a y on ly be expec t ed to offer a desc r ip t ion of t h e flow a t rad ia l d i s t ances c o m p a r a b l e w i t h a d i s t ance w h i c h is n o t less t h a n t he m a x i m u m of l a n d a.

The e q u a t i o n of t he p ro j ec t i on of a s t r eaml ine on to t he p lane y = c o n s t a n t is unaf fec ted b y t h e presence of t h e m a g n e t i c field as long as m o t i o n across t he l ines of force persists . For , if t is t he angu l a r co-ord ina te , comple t i ng a s y s t e m of cyl indr ica l po la r co-ordinates , t h e n t h e p ro jec t ions in ques t ion h a v e e q u a t i o n s g iven b y

d l w e or

- - s i n ( 4 0 - 4 ) ,

where ;~0 is a c o n s t a n t . W e m a y note , f inal ly, t h a t w h e n a m a g n e t i c field is p resen t , t h e r e are no obv ious

i n t e r p r e t a t i o n s for t h e c o n s t a n t s A a n d e in t e r m s of t he f lux of m o m e n t u m a n d angu la r m o m e n t u m in t h e jet .

T h e a u t h o r is i n d e b t e d to Dr. H. K. MOFFATT for d r awing his a t t e n t i o n to the work of M. MOREAU.

R E F E R E N C E S

[1] N. RILEY, Quar t . J . Mech. App. Math . 15, 435 (1962). [2] R. MOREAU, Comptes R e n d u s 2 5 6 , 2294 (1963). [3] R. MOREAU, Comptes R e n d u s 2 5 6 , 4849 (1963). [4] T. KAKUTAI~I, J. Phys . Soc. J a p a n 17, 1496 (1962).

Z u s a m m e n ] a s s u n g

Diese A r b e i t i s t eine E r w e i t e r u n g f r t iherer A r b e i t e n yon RILEY [1] u n d MOREAU [2, 3]. Der wi rbe lnde a c h s e n s y m m e t r i s c h - r a d i a l e S t r a h l e iner l e i t enden Fl t iss igkei t in G e g e n w a r t e ines ax ia len m a g n e t i s c h e n Feldes wird u n t e r s u c h t .

(Received: April 19, 1966.)

R e m a r k s o n t h e B 6 n a r d a n d R e l a t e d S t a b i l i t y P r o b l e m s

B y R. L. SANI, Div. of Chemical Eng ineer ing , U n i v e r s i t y of I l l inois U r b a n a , Il l inois, U S A

I n i nves t i ga t i ng t he s t ab i l i t y of a th in , h o r i z o n t a l f luid layer h e a t e d f rom b e n e a t h t h fol lowing e q u a t i o n arises f rom t he m e t h o d of smal l d i s t u r b a n c e s for m a r g i n a l l y s t a b l s t a t i o n a r y d i s t u r b a n c e s (cf. [ I ] : ) , or [4])

M 3 w = ~ a ~ w . (1

Here M ---- - ( D ~ - - a2), D = d / d x , 4 is t he Ray le igh n u m b e r (a r ea l -va lued p a r a m e t e r a n d a > 0 is t he wave n u m b e r of t he d i s tu rbance . The b o u n d a r y cond i t ions a t x = 0 a n

1) Numbers in brackets refer to References, page 645.