omputing an

pp e Ma ematics enter

AEC ! ESEA! CH AN! ! EVELOPMENT ! EPO ! T

T I ! -LLSOO N i co

1333 Ed e PHY S ICS

NOTE S ON MA! NETO -HY ! ! O ! Y NAMICS

VI I

F L UI ! ! Y NAMICAL ANAL O ! IES

by

A 0 A 0 Bl a n k and Haro ld ! rad

Ju ly 1 5 ,1958

Inst itute ofMathemat icalSc ien ces

N E ! Y O ! ! UN I V E ! S I T Y

N E ! Y O ! ! , N E ! Y O ! !

The general s e t o f e qu at ions wh i c h i s under cons iderat ion

i s

p div u 0

p22 gr ad p

l- c ur1 B x Bd B

p p d iv u O,or

! B( 1 0h ) curl ( u X B ) O

, l B 0

to ge ther w i th the aux i l i ary r e l a t ions

E - u . x B

J curl B

q x div E .

! e us e the no tat i on d/dt to deno te the L agrang i an der ivat ive

d

5875+ u -V .

The sound spe e d ,a1

i s de f ine d by

32

p f ( p 9 6A

where p i s a g iven func t ion o f p and o f V1 ( the entropy p er

ma s s ) . For an incompre s s ibl e fluid,we repl a c e by

dd%' O

,d iv u 0

2Se e MH-VI

, Flui d Magne t i c Equat i on s ! ener al Prop er t i e s .

p c o n s t ant throughout r ather than cons t ant on e ach p ar t i cl e

path . S imil ar ly) i i

‘

a cons t ant through out t h e fluid we

c all t h e flow i s entrop i c ,wher e a s und er the cond i t ion the

flow i s c all e d ad i abat i c .

In the s e e q uat ion s we have a s sumed inf ini t e c onduc t ivi ty

have o m i t t e d_d i s p l a c em e n t current e l e c tro s t at i c for c e s .

Th i s report wa s prepa red as an account of ! overnment sponsored work. N e ithe rth e Un i ted S tates, nor the Comm i ss ion , nor a ny person acti ng on beha lf of theComm iss ion:

A . M akes any wa rranty o r representation , ex press or i m p lled, ! i th respect toth e a ccu racy, completeness , or u s efu ln es s of the i nformation conta i ned mthls report, or that the use of any ln fo rm a t lo n , appa ratu s , method , orprocess d is c losed m th i s report may not infring e private ly owned rlg hts , or

B Ass umes any Ila bllm e s wnth respect to the use of, or for damag es resu lt i ngfrom th e u s e of any informat ion, appa ratu s , method, or process d is closedin th is report .

A s used in the above,“person acti ng on beha lf of the Commiss ion " i n c l u des

any employee or contractor of the Co m m us s no n to the ex ten t that su ch emp loyeeor contracto r prepa res , hand l es or drs trrbu te s , or p ro vrde s access to , any m

form a tlo n pu rsu ant to h is employment or contract ! it h the Co m m i s s ro n

UNC L AS S IF IE!

AEC C omput ing and App l i e d Mathema t i c s C ent erIns t i t ut e o f Mathemat i c al S c i enc e s

by

A . A . Bl ank and Haro ld ! rad

1 5 , 19 58

C ontr ac t N0 0

The r e sul t s g iven in thi s r eport ar e,in large p art ,

i dent i c al t o tho s e pr e s ent ed in number I V o f the o r ig inal

s e r i e s o f mime ographe d l e c tur e no t e s on magne to -hydro dynami c s

i s sued in l9 5h . The c l a s s o f p ar alle l flow s cons i der e d in

S e c t ion 2 o f th i s r epor t ha s b e en ext ende d d S e c t ion 5

h a s be en add ed .

Pre fac e

1 . Introduc t ion‘

2 . P ar alle l Flows

3 . Two - ! imen s ional

C ons erv at ion o fL aw

One - ! imens i onal

Tr ansver s e Flow s

C ir cul at i on and Bernoull i ' s

Tr ans ver s e Flows

NY O -ou8 6

A . A . Bl ank and Harol d ! rad

1 . Intro duc t i on

Al though the comb ine d fluid and el e c tromagne t i c e quat ions

gener ally r epr e s ent a f ar more compl ex sy s t em than that o f

c la s s i c al fluid dynami c s,there ar e a number o f import ant

ins t anc e s for wh i ch the two sy s tems are mathemat i c ally ident i c al

upon appropr iat e ident if i c at ion o f symbol s . The exi s t enc e of

the s e exac t mathemat i c al analogi e s p erm i t s the immedi a t e appl i c a

t ion o f a we al th o f known re sul t s in fluid dynami c s to the

r e l at ive ly new and unexplor ed f ie ld o f fluid magne t i c s ° One

o f the s e analog i e s i s suff i c i ently ra n i f i e d to be tr e a t e d

s ep ar a t e ly .

1The re sul t s given in thi s repor t are mi s l e adingly

br i e f ; they allow the app l i c at ion o f many known t e chni q ue s and

r e sul t s to appropr iat e probl ems in fluid magne t i c s .

The fluid dynami c al a sp e c t o f one o f the s e analog ie s,i . e . two °

d imens ional non- s t e ady flow , inc lude s the ma ! or por t ion o f the

ent ire l i t er atur e o f fluid dynami c s . On the o ther hand,the

mathemat i c al ly ident i c al fluid magne t ic a sp e c t i s much more o f a

sp e c ial c a s e . Thi s r emark i llus tr at e s the far gr e at er compl exi ty

o f the to t al i ty o f flui d magne t i c theory a s comp ar e d to convent i onal

fluid dynami c al the o ry .

The general s e t o f e qu at ions whi ch i s under cons i derat ion

i s

d tp div u o

p Q% gr ad p fiv c u rl B x B

de dapdt

' + p div u O , ord?

0

6 E( l . h ) '

dtcurl ( u . x B ) O

,d iv B 0

to ge ther w i th the aux i l i ary r e l a t ions

( 1 0 5 ) E : -U. X B

J = .l curl Bp

q x div E .

! e us e the no tat i on d/dt to deno t e the L agrang i an der ivat ive

du -V .

The sound sp e ed,

al

i s de f ine d by

'g% r p

where p i s a g iv en func t ion o f p and o f Vi ( the entropy per u ni t

ma s s ) . For an incompre s s ibl e fluid,we repl a c e by

o,

d iv u 0

2Se e MH-VI

,Fluid Magne t i c Equat ion s ! ener al Proper t i e s .

p c on s t ant throughout r a t her than cons t ant on e ach p ar t i c l e

A

path . S imil a rly,i f V

! c ons t ant through out t h e flui d we

flow 1 8 c all e d ad i abat i c .

In the s e e q uat i on s we have a s sumed inf ini t e c o nduc t ivi ty

and have omi t t e d di sp l ac ement current e l e c tro s t at i c for c e s .

2 . P arall e l Flows

The momentum e quat ion c an be wr i t t en

Q J )

The form of thi s e quat i on sugge s t s that we lo ok for s t e ady flows

in whi ch B i s p ar all e l t o u,

B Au ;

in addi t i on , we cons i der only an incompr e s s ib l e flow , div u 0,

from whi ch it fo llow s that l i s con s t ant on e a ch s tre aml ine

( magne t i c l ine )

gg ( U n i 7 ll 0 .

S inc e e quat ion ( 1 . h ) i s s at i sf i e d ident i c ally,all that remains

i s to s at i s fy e quat i on whi ch t ake s the form

( 2 . h ) p ( u -l7 ) n gr ad p 0

wher e p i s a cons t ant on e ach s tr e aml ine,

p p iz/u

and p i s g iven by

p"

p

we re c all tha t ( a . h ) t o ge ther wi t h div u o,dp

*

/d t o

de f ine s an incompre s s ibl e s t e ady flow wi th d en s i ty p*

and

pre s sur e p%

.

3 To e very s uch flow,there ex i s t s a s t e ady

,

inc ompre s s ibl e,p ar all e l magne tohydro dynami c flow ; bo th p and

A c an b e arb i tr ar ily sp e c i f i e d on e a ch stre aml ine in t he magne t i c

analo gue . Bernoull i ' s l aw ho ld s,

v i z . the quant ity

i s con s t ant on e ach s tre aml ine .

The s t ab i l i ty o f the de gener at e c a s e p%

O, p

*c ons t ant

,

under the homogene i ty condi t i on p con s t ant ha s b e en tre at e d

by Chandr a s ekhar ( Pro c . Na t . Ac ad . S c i . , pg ,

I t should b e remar ked that no e quat ion o f s t at e i s re quir ed

0

b e twe en pw

and p*

.

Two - ! imens ional Tr an sver s e Flows

! e cons ider flow s i n whi ch all quant i t ie s are indep endent

of 2 but may dep end on x , y , and t . The magne t i c f i e ld i s

B, z

re s tr i c t e d to the z-dir e c t ion,

and the v elo c i ty i s two - d i m e n

s i o n a l in the p l ane I t

i s convenient to u se two —d i m e n

s i o n a l ve c tor no t at ion,int erpre t ing

the s ingl e component of B a s a

s c al ar . Equat i on ( l . n ) then t ake s

the form

dB0 1

F t 1( 3

3 !B d i v u 0.

In o ther words , c ons ervat ion of f l

t ake s the s ame form as c on s ervat ion o f mas s in thi s geome try .

The r e a son i s intui t ive ly c l e ar; in a plane flow,the e lement of

are a, dx dy ,

acro s s whi ch the flux i s conserve d i s the s ame as the

two -d imens ional "volu m e " e lement on whi ch ma s s i s c on s erved .

Equat i on c omb ined wi th yi e ld s thi s re sul t in the

form

dEE( B/p ) O ,

the r at io of B to p i s c on s t ant fo l lowing an e l eme nt o f fluid .

The e qu at i ons and may b e rewr i t t en in t erms of the

"to t al pre s sur e

p*

p

https://www.forgottenbooks.com/join

\0

B w ld q de p d ( -

p( 3 0 8 ) Td tz

+

3

The analogy c an be made exac t by impo s ing addi t i onal

re s tr i c t ions

a u p p o s e con s an oug ou c an e e m i na ed romS t t t hr h t B b l i t f

the e quat ion of s t at e t o ob t ain

N e w) (32

Thi s e quat ion o f s t at e s at i s fie s t he usual thermodynami c

convex i ty r e quir ement s ; in p ar t i cul ar we have

5 r

"7

de pTdn

and

( ii i

)2

l

lm

"!

I

!l

l

S! “

4

"0

FS

' Nno

an N

1: !

( b ) Suppo s e Q cons t ant throughout . ! e int erpre t Q*

a s the

thermodynami c entropy and obt a in the e quat ion o f s t at e

vii

) mm 51

;

From the re l at ion

2n 1

d e p (NE)

(1.(i i?

we s e e that , by t aking

T B/p ,

we obt a in a convent ional thermodyn ami c s truc ture . The

sound spe e d ,a*

,i s given by the s ame formula , a s

previously .

11

( 0 ) Suppo s e that q? f (n ) . Mor e pre c i s e ly,suppo s e t hat the

cu rve s Ygx ,y ) c ons t ant c o inc ide wi th the curve s

a the ini t i al ins t ant ; we de s cr ib e thi s by s aying that v5"i s a

( po s s ibly mul t i - valued ) func t ion o f Q ini t i al ly . S inc e the value s

o f nand V?ar e c arr i e d w i th the fluid el ement

,the r e l at ion

e7) f (Q) p er s i s t s for all t ime . Ei ther

2or lz

k

c an b e cho s en

a s the entropy ; l e t us t ake The pre s sur e e quat ion o f s t at e i s

l

7 ) p ( p , q )“

! ?

the sound sp e e d i s a s b e for e,and the thermodynami c ident i ty

3 7. 1( 3 Q

e p d (p)

fo llows wh en we t ake

x lt

T Tup

( d ) In a one - d imens i onal flow ( var iable s x and t ) , the re la t ion

!?

f (7)i s s at i sf i e d for arb i trary ini t i al dat a , and the form u a

l at i on ( 0 ) fo llow s .

All c onvent ional two - d imens ional ga s dynami c al the ory appl ie s

t o the abo ve magne t i c analogue s , pro v i de d that no sp e c ifi c a s s u mpo

t ions have b e en made r egard ing the e qu at ion o f s t at e . One of the

int er e s t ing fe atur e s o f the s e hydromagne t i c flows would s e em to

b e the po s s ib i l i ty of cre at ing arb i tr ary e qua t ions o f s t at e at

will by the proper cho i c e o f in i t i al confi gur at ion . Ano ther way

o f lo oking at thi s i s tha t one c an constru c t a fluid w i th a s ound

sp e e d whi ch i s an arb i trary func t ion of po s it ion much more e a s i ly

than in a convent ional gas .

- 12

S inc e the pr e v ious ly de s cr ibed analog i e s ar e exac t,the

convent ional r e sul t s on int e gr al s o f the mo t ion fo llow under the

s ame condi t ions a s quo te d for the c onvent ional fluid s . However,

it i s i l luminat ing to der iv e the s e re s ult s d ire c t ly .

The c ir cul at ion a s s igned t o a clo s e d curve rfl

i s de fined by

( LL . 1 ) C u o dx

where 0‘ i s a p ar ame tr i z at i on of r“ . For a curve

, ra ( t ) , whi ch

i s mov ing wi th t he flui d ( 0‘ i s f ixe d to a fluid element ) we have

Thi s w i ll no t b e z ero and yet admi t any s igni f i c ant

exc e p t in the two ~ d i men s i onal tr an s ver s e flow s ! us t

In that c a s e ( B o V )B O and

dc 3p

0

! t )

C ir culat i on will b e c ons erve d con s t ant ( homogeneous

incompre s s ible flow ) or if pi,c

func t ion o f p alone . The

lat t er wi ll be the c a s e when b o th l and n?are c on s t ant through

out the flow ( of . e quat ions and The

general c ond i t ion for c ons ervat ion o f c ir cul at ion ( o f . e qua t ion

i s that a po t ent ial, ( 2 ex i s t s at i s fying

(m i ) Tl53 dvf

‘

d ( e*

pei

/p ) -1-dp

*d ( 2

u !

I f the flow i s irro t at ional,

(M 5 ) u Vfl

and if the po t ent i al 2 exi s t s i f 7 and a?are cons t ant ) ,

then,u s ing ( u o V ) u curl u X u V (%—u

2), we f ind that

( L h é )

or ( Bernoull i ' s l aw )

I

t

co c

t

( it -7 ) gr+ 512

h*

-_ fl _

wher e the " to t al enthalpy" h*

i s

M

( u . 8 ) h*

ew

p%

/P ~

For a s t e ady irro t at i onal flow wi th a po t ent i al 2, we have

wéuz) %vp

*0.

from which,t ak ing the do t pro duc t w ith u and us ing 5% u

° ‘7 ,

we obt a in

gi

gdg u

zh""

Q ) o .

In th i s c as e, % u

?h*

2 i s c on s t ant on e ach s tre aml ine .

One -! imens ional Tran sver s e Flows

Thi s geome try i s,in a s ens e , the dual of the one tre at e d

Se c t i on 3 . a s sume that all quant i t i e s ar e ind ependent o f

x and y ( the r e l evant var iabl e s are z

and t ) , that B ha s two component s,Bx

and By!and u ha s only a z - component .

! e again have ( B o V )B O,and c an wr it e

the momentum e quat ion in the fo rm

Bu au 1 n 0

Fig . The e quat ion for B , int erpre t ing u a s

a s c al ar , i s

-a—B a

0.

a t

For the magni tude o f B we ob t a in

a 6

Thi s to ge ther wi th and the c ons ervat i on o f ma s s i s a

de t ermined sy s t em ( only the magni tude of B ent er s in More =

over,i t i s exac tly the s ame sys t em that wa s tre ated in Se c t ion 3 ,

spe c i al i z e d to one sp ac e d imens ion , 2 ,ins t e ad of two

,To

comp le t e the so lut ion o f the or iginal probl em,we no t e tha t

impl i e s that the dire c t ion o f the ve c tor B i s a cons t ant follow

ing a fluid e lement .