An introduction to the equations of magnetogasdynamics

FacUteitder1.uchIvaart.enRu' tevaartteclloielt Kluyverweg 1

2629 HS Delft

AN INTRODUCTION TO THE EQUATIONS

OF MAGNETOGASDYNAMICS

BY

..

•

...

APRIL, 1957

A1~ INTRODUCTION TO THE EQUATIONS OF MAGNETOGASDYNAMICS

BY

J. A. STEKETEE

"

ACKNOWLEDGEMENT

This review has its origin in a seminar lecture delivered at the UTIA in the autumn term of 1956.

It is a pleasure to thank Dr. G. N. Patterson for inviting me to write up the material of this lecture in this review and to express my gratitude to him and his staff for the encouragement and friendship I continually receive.

.

'

'

.

With the intensive study of strong shock waves and the frequent contacts with astronomy, aerodynamicists have recently been led into the field of magnetogasdynamics and magne tchydrodynam ics.

For the aerodynamic research worker entering this field several obstacles present themselves. In the first place he has to make himself familiar, for the first time or ag ain, with the concepts of

electromagnetism. In the second place he wiU find it difficult to estimate the physical significance of results obtained in research papers as he cannot properly appreciate the approximations made in the usuaUy postulated equations of m oti.on,

This review attempts to give some support in overcoming these obstacles.

In the first place a very shcrt discussion of MaxweU's equations is given where some of the classical analogies with familiar relations of hydrodynamics are pointed out.

In the second place it gives a systematic 11derivation" of the

equations of magnetohydrodynamics assuming that the equations of Navie r . Stokes are known, This review has its origin in a seminar

• NOTATION 1. INTRODUC TION 11. PART A lIl. PART B REFERENCES APPENDIX ._T ABI.EOF CONTENTS Page iL 1 2 15 31 32

E H D B P M

i-J

is

q s

e

j.J..

ex

X

d

f Tij. P!' c v ui

P

p

•

Xi NOTATION

electric intensity or electric field strength magnetic intensity or magnetic field strength electric displacement

magnetic induction polarization

magnetization

current density vector

current density vector of the conduction current surface current density vector

charge density

surface charge density dielectric constant permeability polarizability susceptibility conductivity ponderomotive force

component or Maxwell stress tensor Poynting vector

speed of light = 3 . /0 1.0 cm/sec velocity vector

component of velocity in xi direction density

pressure

!YJ

:)2

U Q T

~

cp' Cv

X

=

cP/cv À R viscosity

second viscosity (Stokes' assumption is 3'12+2'1 =0)

Internal energy / unit mass

Heat supplied / unit mass

temperature

dissipation function / unit mass

specific heats

ratio of specific heats

coefficien t ofhe a t conduction

1. INTRODUC TION

In magnetohydrodynamics or hy d r om a gnetics we study the

motion of flui ds wh ich are susceptible to elec tr i c- and magn e tic influ e n c e s.

In addition to th e ordinary visQ .: - and pressure - for ces the r e appear electromagne tic forces in the equations of motion, whic h are connec t ed in some way wit h the equations of Maxwell. The proble m s wh ic h ar i s e can the r e fo r e be indic ate d mos t correctly as mixed problems of hydr o

-dynam ics and ele ctrom a gnetism .

Wh ile magnetohydrodynam i c s experiments are difficult to

G"T:,eat.eina la boratory, there are sever al branches of physics and as t ro om y

wher e proble m s of this type are of importance, for example

(1) Astrophysics - Interstellar ion ized gas in a ma gnetic field, sta bility of spiral :neb ul a e, cosmic rays

(2) Geoph ys ics - The dynamics of flow in the core of the earth,

the ear t h's m agn e t ic fie l d

Fur ther one could pe rha ps think about fluid flo ws in electrolytes or di s

-charge s in gases.

As thi s review is written for aero- and hydr odynamicists we

shall sp end no time on the equations of motion of ordinary hydr odynamics. A short discus s ion of the Maxwell equation s however will be gi ven . More

details can be found in th e stand ard text bo oks (F or example

L

1, 2, 3J )

A diffic ulty one faces in star ting with electromagne tism is the

choice of units as se veral sy ste m s are in us e.

In this review we use th e non ration a lized Gaussian sys t em

with thr e e fu ndamen tal units . The fundam e n tal units are the units of m ass,

time and length, while no separate unit of el ectric cha r ge will be intr oduced .

(The unit of charge is de fine d by Coulomb's law). This has the advantage

tha t a proportionalit y fa c t o r appeari ng in se v era l equations has the magni

-tude and dimension of the velocity of light. In an app endix he important

equations sha ll al so be written down in r-a t iona l fz e d rn ,k. s , u "ts o

In most branches of electromagnet ism it is sufficient to con

-sider phenome na inbodies at rest and in sev e r a l book s the disc us s ion is

restric ted to thes e phenom e na. An essential aspec t ho wever of ma gnet o-hydrodynamic s is the motion of the fluid or th e ioniz ed gas and this in ro

-duces some fundame ntal differences.

S

tateá

roughly the situation is th e following; supp ose we ha v e an observe r at rest in an electric field. The only thing the observe r

no tic es is the electric field. As soon as the observer star t s mov ing

ho we v er it will seem to hi m that there is not only an ele ctr i c- but also a magnetic field.

In the usua l Eulerian description of hydrodynamics we select poi n ts an d sitting in the s e points we watch the fluid pass. In this way we fi n d all he sign ificant properties of the flow as function of position and

time, for example the vel ocity v in the form

v = v {x, y, z, t)

In a.ccord anc e with this description we shall also consider the

el ec tromagnetic propertTes as fu netion of position and time, for example '~ ..z

the magnetic induction ~ in the form

B ::: B (x, y, z,t)

If we wi sh to cons ider ho wev e r what the magnetic induction

will be for a fluid element which oecupies the position (x, y, z ) at time t

one has to rem e mber tha t the fluid element has a velocity v with respect to

he (x, y, z, ) fram e and it would th e r e for e be wrong in general to conclude

tha t the ma gnetic inductio fo r the fluid element is the B which we wr-öte

d.own ab ov e .

We sha ll se e tha t thi s situation creates the coupling between

the vel o city fiel d of hy d r odynami es and the electromagnetic field.

This rev iew is div i d e d in two parts. In_p~H~t A the equations of Maxwell for bodi e s at re s t are given and the modifications are indicated

which have to be ma de if the solids or fluids are in motion.

In part B the equ ations of electromagnetism are c ombin ed..

with the ordinary equations of hyd r o- and aerodynamics to obtain the

ma gneto hyd r odynamic equa tions.

u

,

PART A

1. The equatio s of Maxwell which describe the space-time-geometry

of the electrornagne tic field in bodies at rest are

di v B ::

0

1

div D =

_'

_{C, lf}

~ '0]) (1. 1) 4l'ï

.

I

J

curl 'B' =

_1-

+-

_at

c

c

I

.

'a ld.

eurl E ::

-C

vi;

wh er-e- B

-

magnetie induction

H

-

magne ie-int e ns ity or -field strength

D

-

electrie displacement

·

e

..current density vector

charge per unit volume

10

speed of light

=

3. Kl cm

I

sec

Inspection shows that the set (1. 1) represents eight scalar equations with sixteen unknown scalars. We have therefore to obtain another e ight equations. We shall obtain these r-elations which connect respectively

~ with Hl D with EI and

't

with E. I

The vectors Band Hare related to another vector ~ while D and E are related to another vector P by the following two relations:

where

B = H + 4 l f M D

=

E

+

4

T

P (1. 2)

M - magnetization or magnetic moment

I

unit volume P - polarization or dipole moment

I

unit volum e

The vectorsP and M are zero in vacuum as they describe a change in the medium caused by the electric- oît[llagnet1c~_fi~lds.

In an "i s otr"op i c medium the magnetization arid polarization are proportional to H respectively ~I that is

M='X.H P= r::Á E (1. 3)

The coefficients

X

and " 0<.. (respectively called susceptibility and polar -fzahil.rty) are properties of the material. It will be assume d throughout that

X

and 0<. are independent of position and time. Introducing the magnetic permeability

JA

and the dielectric constant

E.

by

p.

=

I

+

t(Tr

'X..

we can write instead of (1. 2) cE.

=

I

r u

tr

oc

D =

E.

E (I. 4) (1. 5)

where

fA-

and

é

are constants of the mater-Ial, which according to our former assumption are independent of position and time.

The equations (1. 5) supply six scalar equations. Another equation to complete the sys~em of equations is Ohm's Law; in many substances the current dens ity is proportiOfal to the electric intensity

and we have ,

·

i-

=

d

E (1. 6)

where

We assume also that

cS

is independent of time and position.

The equations (1. 1), (1. 5) and (1. 6) form seventeen scalar

equati ons with sixteen unknown s..and it seems that we have one e quation..

to o muc h. This is only apparent as we can discard the first equation of

[L, 1) provided we intr oduc e the initial condition that at some instant

div B

=

O. For if we ta k e the divergence of the fowr..ili.i..equát'.i0nJ.pf..{1.1}<:

we have Irrim edfa tely

(d i v B)

=

0 d

a-t

and as div B = 0 at som e instant it foUows that div B

=

0 throughout time. Taki ng al s o the divergence of the third equation we find when using the

sec o id equation

.

div ~

+

~

=O

ot

(1. 7)

2. Compar ing thë-e quat ions obtained so far with the weU known

equations-of hydrodynam ic s we notice first that the vector field B behaves as an inc om p r es sib l e fluid without sources and sinks.

-The vecto r fie l d D behaves in the same way at points where th e re is no free charge

cr

.

but if

CV

i=-

0 at a point. this charge makes th e div D equal to 4

ïlq;

at that point . This is similar to the liydro- .

dynamical case of an inc om p r e s s ib l e fluid in which a source or sink is :.:..:~

present. ',·,,,

Ij

If we omit the term

;f

in the third equatione the remaining

equation has a close analogy with the relations for vortex filaments in an

in c om pr e ssib l e fluid; inte g r ation of H along a closed contour gives then on

'using StokesI Theorem:

-If the current is restricted to a thin conductor-:of cross section

a

.

which int e r s e c ts the surface

,sI

we may write: '--'

(2. 2)

If we ha v e a thin vortex.filament of cross section Q,. and strength

ya...

in an incompressible non-v i s c ou s fluid which moves irrotationaUy out

-side the vortex filame nt. we have for the circulation along a circuit which

sur r ounds the filam ent

0

-It follows that the magnetic intensity and current density behave similar to respectively the velocity and vorticity in hydrodynamics. This analogy can of course also be notdeed on simply comparing the equations

curl v

=

+.

and curl H

=

~7T

.i:-The Eq. (1". 7) is the equation] ,of conservation of electric charge and is similar tothe equation of continuity which expresses the conservation of mass. .

For a more systematic and thorough discussion of these analogies, which may be useful to the hydrodynamicist one can consult ReL (4).

3. To complete the description of the electromagnetic field we need to know the boundary conditions which have to be satisfied at the surfaces of the media. These conditions are easily obtained from the Maxwell equations.

Consider first a small volum e element in the shape of a thin slice, such that its top and bottom surface Iie on

opposite sides of the interface. Apply

-.ing Gauss' theorem and the first equation of (1. 1) we have for this small volume

V

(3. 1)

where the second integral is taken over the complete surface of

V

while

.!1.

denotes outside normals. If we make the slice very thin the surface integral is practically equal to the integral over top and bottom surface. As the shape of the top and bottom surface is irrelevant and as c._

the outside normals point in opposite directions we find that

(3 . 2)

where the subscript h denotes here the direction of the normal to the interface in one direction.

Considering the same situation for the second equation of

(1. 1) we find

(3.3)

On the surface integral the same argument as before can he applied. If the slice is made very thin the volume integral of the charge becomes a surface integral of the surface charge S over the interface of the two

media, that is

(3. 4)

Considering then again that the top and bottom section (a n d therefore also the interface section) are completely arbitrary, and selecting a proper direction for the normal to the interface, we obtain:

::D

In ])J,h -

41T

5

(3. 5)

In parttcula r if there are no surface charges we have

.1)/11

-.D.2.

n

=

0 (3 . 6)

•

To obtain conditions for the tangential components we have to use the thitd and fourth equation of (1. 1). Therefore we consider a smal! circuit lying partly in each medium. Taking the integral of E along this contour

we have, em ployjng Stokes' Theorem .

If we make the contour very thi.n, the flux of B through this contour becomes negligible, and we obtà in in the limit

-.:

E..

+

=0

I .2. (3. 8)

In exactly the same way we obtain for ~

j

H

.

d

s

.

=

ij

ewJ..

1-1.

h

ei.

.~

=

~Tfh..n

diJ

~ ~

/1

ij

YJ

:d;3

1$

; 5 ; 3

(3. 9)

If there is an electric surface current

as

we obtain then

(3. 10)

For very thin contour and a proper direction we have then

and in particu lar fo r no surface current

(3. 11)

/-1,

i:

(3. 12)

4. It is well known that th e electromagnetic field exerts forces on electr ic charge s and on electric currents. If we consider a unit volume element of matter in which there is a charge

q".

and a current

-:L

th e force exerted on the vol u m e element (o r on the charges and currents irtit ) is

(4 . 1)

This force is usu all y ca lle d th e ponderomotive force or Lor'entz force. The first term is th e electros tatic force. The second term on the right hand side is only present if the r e is an electric current and is in a direct

-ion perpendicula r to both

i

and B . /

This second term shows a close analogy with the force which one cal.Is the Joukow s k y force in classical airfoil theory, although Kelvin and Rayle igh seem to have known about it before hirn.

If we have a cylinder in a two-d im e n s i o n a l potential.f1ow with con s tant ve l o c it y

ru

at infinity and there is a circulation

C

ar ound the cylinde r, then a force F is exerted on th e cylinder of magnitude

(4. 2)

L

in the direction perpe ndic ular to both the axis of the cylinder and

lIL.

To see the analogy most easily imagine the cross section of

the cylind e r very small so that it becomes a vortex filament, which is

in the direc tion ---perp endic ular to

ru.

-

:

From the discussion in 2 we have seen that the velocity and vorticity respectively correspond to the magnetic field intensity (o r by (1. 5) with the magnetic induction) and current

density

i

.

The force on the cylinder (vortex filament) is in the direction perpendlCular to the velocity

1)1

and the vorticity and with this corresponds in the case (3 . 1) the force perpendicular to the magnetic field and cur.rent

density vector .

'-We have written the ponderomotive force as a vector. His to-.._

rically great importance has been attached to the fact that the force can

stress-tens o r

T.

.

.

..

I.t~

Us ing the equations

H.

1} we can write (3. 1) in the symmetrie

fo r m:

(4.3)

+

H.

~

13

-I- _ I /

c

.wJ

E

4-

J..

'a

B)

x.

1)

L.j1T - -

41ï

l

-

c

ot-

-where the two te r m s in the second line are each zero. Combin i ng the terms we obtai n

The fir st te r m inside brackets is similar to the second term within brackets.

Wr i ting ou t th e first compone n t of the first term one finds:

(4.5)

we can write

f

in the

where

where we us e d (1. 5). It is then easily seen that

no ta ti on of Cartesian tensors in the form

1t-'0

P.

_...

ot

.

TL~

-

4~

L.Di.

Et

+

B~

H

_{j ) -}

~ ~'t

Lg·])

+

B .

.ti)

(4.6 ) and (4.7)

T

i

j

is the Maxwell s,.:tr.e.ss tensor while p* is. the Poynting vector

5. To find an equation for the energy we have to find the work

(5. 1)

The rate'at wh ic h this force performs work is

~

,~

=

}V

.E

+~ ~.(ix.

B)

wher e v is the velocity of the charges. But the charge multiplied wit h its vel ocity is the current density vector t an d so we obtain

(5. 2)

This expression can still be brought in another form if we us e the

equations (1. 1). We ha ve then .

(5. 3)

Maki ng us e of the relations (1. 5) and some vector identities we obtai n

,1.

E

= -

2. {-'- /

r=

,

D

~

B.H)} _

~

'P

~

"t"' -

'Ot

aT l - -

- -

-...

-

..-._-~

(5 . 4)

The expression between accolades is usua lly called the energy density of the electromagnetic field. One may think of this energy as located par tly

in the empty spac e and partly in the dielectric occ upyingtheapa c e,( C om

-pa re formulae

U

.

2

»)

.

6. Th e the o r y developed sa far was concer ned with the elec tro

-magne ti c field in media at rest. To obtain the changes which ha ve to be intr oduc e d if the me d ium is in motion one can fol.Iow.itwo method s . Th e

one method uses th e results of the special theory of relativity .. These

results will the r e fo r e hold up to the velocity of light, which is for many

hydr odynam i c cases unnecessary, while the results are st r i ctly speaking

only obtained for systems which move uniformly with respect to each

ot her . The sec ond method is more straight forward but will give wrong

answer s if ~becomes very large so as to approach the speed of ligh t . In

this.iaectton we shall state the results obtained from the relativity theory

and derive the s arne results with the second method in the ne x t section .

The ba sic as s umption of the special theory of relativity is the pri ncipl e

of aequ ival e nc e whic h states that all frames of reference in uniform

motion with respect to each other are completely identical for all mech

-anical and optica.l experiments. This means that physical Iaws, expr e s s e d

in coordi nat es of different fr a m e s of reference_must be of th e same form ,

if th e reference frames are only distinguished from each other by a

unif o r m motion with nespect to each other. Mathematically this state

-men t requires that physical laws must be invariant under a Lorentz

-Transformation. which is in its simplest form:

ti:

t

-V"X.

-c,

whe r e

Y

is the velocity of the (x ' , ti) frame with respect to the (x ,t) -frame, or _-I{ the velocity of the (x, t) .. frame with respect to the (x ', t

') -fr ame, c is the speed of light) x and x' Car-t e sfa n coordinates in the

direct-ion of

V

,

t and t' the local tim e s .

Sub ject ing the Maxwell's equations to such a transformation and r-equiring tha t in the ne w frame the equations have to be identical

wit h th e equa tion s in the ol d frame one ob t a i n s relations betwee n th e old

field vectors and the new on es ,

If th e quantities in our Cartesian frame are denoted without

dashes , and th e quantitte s'

în

another frame , having a un iform velocit y

v with re s pe c t to the fir st on e, wi th dash e a, the n we hav e for example for

t:re

longitudinal and transvers al com p on e nt s of E, B, E' and B' th e , .... .'"

expres sions

In particular of ~

<:.<

I and we put '(

=

I (which comes down to neglect -ing terms of order v2 and smaller) we obt a i n the complete set of relations

. ~ .

BI ::: B

v

X

E

H'

=

H _

~

x

1) c..

_c..

EI

=

,:-

₊

v

_x.

_B

.])1 _{]) 4-} v

_H

-=-x

-

c.

c

(6. 3)

pi

op

v

_M

'1 \/

-

x.

:f

-

+

~

Co

MI

=

M

+

_-

~

x.

-p

-

c,

7. To obtain the rela tions (6. 3) in a physically clearer way

we st art from the experimentaLfact behindthe fourth equa1i. on of (1. 1)

Consider first a closed circuit wh ich rnay move in some way. Let th e circuit be denoted by I at time t and by II at time t

+

~ t where At is smal! (aninfi nit e s irn a l strictly). Let the r e be further a surface which

R

has I as edge and one-which has II as edge, which we denote by

s,

and

SJ[:

respectively. Let the circuits .

surface" R such that

~

'

"slI:

and Renclose a volume V. If unit normals drawn inwards are denoted by n a surface element of R may be written as

II

d.~

.

= -

~

.

)<,

~

.

~

t

A volume elem ent of V may be denoted as

(7. 1)

(7. 2)

As we have

Faraday 's law of induction now states that the electromotive force generated in the circuit is proportional to the rate of change of magnetic flux which passes through the (moving) circuit. The magnetic flux at time t thr ough.

the circuit is

fl.B

(t).1l

d.

~

The magnetic flux at time t

+

6.

t

.

is

- ij

B

{t-1-oiJ.

J2

.

d.;S

~

JJJ

dW-

B

(+

+-6+)

d.1j-=

0

V

the last integral can be replaced by

!loB

(i:

+6.

-1:).

n

ei;5 +

.f!B

{++

t:.f)..!2.

ei

i3

The

flu~

passing through the circuit has therefore changed in the time interval 6. t with

IJ{

B

(f

+tl-t) -]

{ie

)}.n

ei

;5 -

f

IJ.

(i

+t>-/:)(rLs

X

':1.)

ó-t-I~Z

Passing to the limit l::. t ... 0 we have for the rate of change of B

i j

~:

.

12.

J.~

-

f(!!.

X

~)~

;3,7;

which may be written also as

Q'{

~f

-

CMA1{v

X

B)}

r2-

.

d. ;3

Faraday's law can then be written as

(7. 3)

(7. 4)

This can finally be put in the form

JJ

CtUt-.l

E.!1

d;j

= -

~

jJ

~

,!2

d

~

.

~.%:

; { ' •

and as this holds for any circuit we deduce that

§:.'

=

E

+

-f

L

Y.

x

~)

·

(7 . 6 )

(7 . 7)

(7.8)

In exactly the same way we obtain two other relations by considering the third equation of (1. 1). Considering the e le ctr-ic dis-placement current through the moving circuit we obtain first

-fJ

J.)

[++6-&)

!l

rl~

=

JJ])

(++ó-l:)

n

diS

+

~

~~

~

1f!d(/:

+6.t)!2

d;3

+

fjf

d#J..

Jj

(t

+ó-i:)

cLV.

Using (7. 1) and (7.2) we obtain for the rate of change of D

JJ{

~

-

Cw't-l

{~

l(

.D)

+.Y

~

dl }.

.!:1

d

~

cl.x

The third equation of U.1) becomes then for the moving circuit and written in integral form:

J

.!i

I,

r!á

=

~7r

fh',"

ti.

~

+

~.JIJ~~

-

cw.J.f':l

X

lJ) ...

l!

~ll

}'!2

rJ.;S,

As this

relatio~olds

for arbitFary circuits it must also hold in differential îor-m, which may be written

(7. 10)

Comparing this with the equation which holds in a body at rest we have

H':

J./

~ r..~x.D)

"

+=t-(7. 11)

The other relations of (6.3) can then be obtained with a certain plausible argument. although there is no need for these relations.

In vacuum we ha v e D = E are

B =Hand the equations (7 . 8) and (7 .11)

E.

1

=

E.

-4-

~

f..

~

lC.

t!.)

Hl _

~

~

l'!-.

X

F)

(7 .12)

In a medi um we have th at E' is related to 'B and E and it seems plaus ible

to assume tha t BIis then also related to B and E in such a way that the rela tion reducesto the second equation

oT

(7. 12) in vacuum , This argu

-men t supplies

B'

_B

I

L'!-x.

E.)

-

,

Co

In the same wa y we obtain

J)'

_-

1) _~

,

_L:!..

_xli)

_·

Co

(7. 13)

(7. 14 )

It is then very sim pl e to obtain the relations for Pand M when we use

(1. 2) and the re l ations first obtained.

-From the formulae obtained in (6 ) and (7 ) it is easily ch e cked th at an electric field E for an observer at rest is for an observer moving

with velocity v an. electromagnetic field with

v

EI _

E

_BI

₌

_x

_E.

Co

n

~ 15)

Expressi ng the vectors in th e frame at rest in terms of the dashed ve ctor s 1.

while ne g l e c ti ng terms of order ~2. and smaller. it is checked th a t th e resulting relations could be obtained immediately if one observed th at the frame at re st has a velocity -v with respect to the dashed frame.

8. The electric current observed by the stationary observe r is (8. 1)

x

B)

The_fi r_s t term on the right we eaU the conduction current and we shall assume as before that Dhm's Law shaU hold for this current. Further we shal.l uaual.ly write ~ Instead of

1

1 • The second term on the

R. H. S. is the;conv~ioncurrent àhd is simply due to the transport

of the medium with respect to the stationary observer. From Ohmts

Law we hav e the n

=

6

EI :.

G

.

/

E...

+

V

This is one point where we have a coupling between the Maxwell equations

and the hydrodynamic ve l oc ity field.

The second point is in the expression for the Lorentz force.

The force on a movi ng element is

(8.3)

Transforming the vectors to the stationary frame while neglecting terms

of order

.L

gives .

c.z.

(8 . 4 )

(8 . 6) (8. 5)

as bef'or-e, where j howeve r is no w given by (8 . 1). Further we shall

then wr ite f instead of f '.

It should be point ed out he r e that most of the relations obtained

in sections

i

to 6 are based on the equations (1. 1) and therefore will

contin ue to hold for the field ve c t o r s considered as funetions of (xy z t )

although the elem e nt s of fluid occupying the position (xyz) change from

moment to moment.

One has only to be car e f ul in applying Ohm's Law as we have

see n ab ove, while the r e is fur ther a point to be added to the relations in

5. The formulae (5 . 3 ) and (5.4) hold unchanged but there is a slight

.modification with respect to (5. 1) and (5. 2); the electromagnetic force

!.

performs work on the fluid elements given by the scalar product

!..

~

where v is now the veloc ity of the fluid which is only part of the velocity

of the charges.. Sa we have

t·

~

= [

~

E

+

~

{

(Cf

V~+

1)

X

.B}]-

V

.

=1}'f.

.1:.

-

~

f'{v,,- B)

Furthe r we have from (8 .2)

4

~LV~B)~

Sub s t itution in (8.5) gives us

~

.

.

.

1

~.

V

=

~~.

E..

-

'Jol

cr -

E.)

~

E

.

f

-

4}

For th e first te r m on the right we can write also (5 . 4 ). while the second

te r m on th e right is the Joule heat which is dissipated in the conducting

III PART B

9. We turn now to th e discussion of th e equa tions of magneto

-hydr odynamics. The difference between th e s e and th e ordi n ary equ a tions

of aerody n arn ics is the ap pea ran c e of th e ponderomotive force whic h is

expressed in v and the var iables descibing th e el e c t r o m a gn e tic field. We have in the notatien of Cartes i a n tens or s:

(9. 2) (9. 1)

'db

.

.

ft

t..

dX'

_<t

*

+

a:

.

(pu..~)

=

0

~

o~ lFlJ.

~)

+

*.

lP

~

~

Ltf)

=

f

X

L

i-t

where:

p -

density

u~

-

velocity component

XL

-

compon ent of grav ity force per un it mass

Pit

-

com p onert of -hydno dy namic st ress tens o r

4

t - com p onent of elec trom agne tic force pe r unit volume,

~

defined by (8 . 4)

The stress tens or P~L is defined by

p

~"

;-

'Y)

~"

()IJ.

~

+

1') (

~

r

L _J:J..

_&l-

_{d ?c. K' .J}

_d

_')C,i

+

d(A,

~)

O'lC.··

(

(9. 3)

wh ere - hyd r ostatic press ure

- viscosity coeffic ierrts, assumed consta nt.

It is clear th a 1; if we wish, the force fi can be replaced by th e tens o r exp r es sion (4 . 5 ).

In ordinary aerodynarnics (with fi'

=

0) th e equat ion s (9. 1)

and (9. 2) form a set of four equations with th e five ünkn o wns ui' pand

p

.

To complete th e system one ad ds the e quat ion of state relating p and

p

with th e te m pe rat u r e T (o r inte rnal energy U = cvT) and an equa tion fo r the en e r gy con taining amongs t ot her U = cvT which is

usually put in th e form of an equation for the.te m pe r atur e .

In th ï s .w a y we arrive at a system of six equations with th e six unknowns ui, P. p and T. In thec ase of an incompressible flu id th e fou r equati on s of motion (9. 1) and (9. 2) become independent of T while th e equ ation of state degenerates to

f

=

const,

In the most usual gasdynarn ic case th e equation of state for the ideal gas and th e temperature equation ar e r-eplace d by th e isentropic

rela t ion

p=

c ,'"

(9. 4)

-so th at the system (9. 1), (9. 2) and (9.4) gives five equations with the five unknowns p, p and ui'

In the ma gnetohydr o dynamic case fi

i=

0 and there enter in

the equati ons the additional unkno wns q, ..aÎ!. E and B whic h are conn e c t e d

with the ot h e r electromagnetic ve ctors by the equations

O.

1), (1. 5). (8. 1) and (8. 2). It has to be poin t ed out explicitly that (1. 6) does not give the relation between j and E anymo r e but that this equation is replaced by equ a tions (8 . 1) and (8.

2)

whic h rel ate the ele ctr om a gnetic veetor-s with

the ve loc ity field.

It was shown in section I that the electrom a gnetic field is

des c ribed by sixteen equ atieris with six te e n unknowns . In these equations

there app ea r s no w also the ve l o city v whic h is connected with the equations

(9. 1) and (9. 2) and the two equattonsmenti one d before which form six

equ a tion s with six unkno wns.

I follows that the com pl ete set of ma gnetohyd r o dy n a m i c

equations is a set of twenty-tw o equ a tions with twent y- t wo unknown s, It

wi ll be seen th a t these equations are us ually simp lifi e d so that the number

of equations and unknowns is considerab~y re duc e d. .,

It may be suitable to indic ate the cons e q u enc e s of the restrict- . .'

ions we have imposed on the system. The es sential rest r ic tion has been

that the material constants, di e lec tr ic con s t a nt , pe rmeab ility, con d u ctivity,

viscosity etc. ar e constants ind e pe nd e nt of pos ition and time and therefore

also of ten perature, pressure etc. .If this restriction were no t made, we shou ld ha v e to introdu c e equation s whi c h giv e th es e material constants as

functions of te m p e rat ur e pre s sur e, ma gnetic induc tion etc . Many more ph y s ical ph e nom e na could in this way be inc o r p o r ate d , but also the

clumsi ne s s and complic ations would increase considerably . The

com-promi se adopt e d se em s re a s o n a b l e when one considers th a t these constants

in general do not va ry ver y much with in limited regions of the flow para-mete r s . The only mate ria l pr o p e rty whi c h is not assumed constant is

the den s ity. Th e equation of stat e we sha ll use will eitherb~ the equation

p

::

con s tan t or the ide al gas la w

L

=

RT

.

F

(9. 5)

In the next se ction we proceed with the con s t r uc tion of the eq u a t ion for

the tem p eratu r e.

10. Wh e n we multi ply equation (9..2) w.it h Ui and sum over th e

subscript the resulting equation can be pu t eas ily in the form

I

!-

o u. (). ·)

+~

/

.!-

PWU, I U, .)

..:.

p

u.,

.J(:.

+~

lu

,

h..)

l

:2.. I l. l. t)x '

l

2. \ l. "

r

'

1

"

L 'O)C .

l (,

r ('

l.

_

f

b..

Ç)lA.~

1-

l). '

cd .

~

(la. 1)

1-(

l. C>')C. • '"

1

t.

. ~

electro-magnetic forces. This term can also be written in the form (8. 6 ) or in

ter-ms-of work-per-formed.by Maxwell stress'ès,',l Using (5.4) the.te r m ..'

can be m odif'ied still fur-the r; àrid it should he,no t i cdd-t h af cther-e-ts an.«

energylë'örtt a'c-t b e tweerf the ktnet äc-energy. ofthe.m a a s motroneand (the..~~

e lecte omagnettc..fieldAHier.gy..", -~ . ~_1:, . '

The following step is to write down tIE equation for the rate

of change of energy, both internal and kinetic. This energy we write

u

o,

2)

per unit volume, where

pfU.

1:

P

tA..~

l.Lt

- thermal energy per unit volume

- kinetic energy of th e mass motion per unit

volume

The first term gives us the kinetic energy of the molecular motions

and may be writte n as -\

u

o

.

3)

It has to be pointed out tha t th i s is permissible due to our assumptions

of cons t a n t material properties so that electromagnetic and thermal

phenomena are largely independent of each other. If the assumptions

were not made a number of other specific heats had to be introduced and

this section of the review would become considerably more complex.

A change of the energy (lD. 2) can be effected by several

means. They are:

(1) The transport of U and the kinetic energy by th e flow

(2) The work of the body forces.

(3) The work of the surface stresses.

(4) The work performed by the electromagnetic forces.

(5) Heat sup p l ied by conduction

(6 ) Jo ul e heat supplied by the conduction current

Writing down these te r m s we obtain the equation:

u

o

.

4)

Where

À

is the coefficient of heat conduction.

Using the equations (8. 6) and (5. 4 ) the equation can also be written in

the form

l.[p

i«,

1

Uw'

/,A.')

+...L

/E...J),..

B

.H)l

=

_.2

{ClÁ'

lfU.+

a

t i l .2. L. l.

a'1r

l - - . - -

f

à~• I ;( l.

L }

S"

(10. 5)

where the left hand side of the equation represents now the rate of change of total energy, internal, kinetic and electromagnetic . The electromagnetic en e rgy may again be thou ght of as situated partly in the medium and partly in the spac e occupied by the me d i u m. Several writers on electromagnetism prefer to omit th e part of this energy situated in the empty space when the int erna l energy of the medium is discussed .

It is no w easy to construc t an equation for the heat which is sup plie d to the vol u m e ele m ent . Substrac ting (lD.1) from (lD. 4) we obtain

1. /

o

ru.)

+

~

t

P

\t-1

ru.)

=

bj.

i.

~

+

t i

+

À

~

T

.

(10 . 6)

ot

l \

(,)'X. • v

r

Q

'Zl ')(. •

Ft

The left han lside of this equation

t

c an be written as

F

I

~

tU.

where

~

t

~=1.-+~

'-:J>t

ot

(

ê)

k.J

Th e firs t law of the rm o dynam i cs can no w be written

~

_

]JU

+

p

L

/ ,)

.:Dt -

J:)-t

..Dt

L

f

.

or in th e form

o

~

=

o

.

.::D

U. _

L

-!..e.

uo,

7)

\.:D

i

I

.:IJ

ot

f..J)

1:

whe re Q is the heat add ed per unit ma ss.

In t rod u cin g

~

, the dissipation function per unit mass, we have

ïf\

C>

U:

P

f

~

=

P

fL

~x.;

F

Combining (IO. 6), (Lü. 7) and (lD. 8) gives

l·l

cS

(10. 8)

UO. 9)

which indicates that the heat added to the Huid is due to three causes

U)

viscous dissipation (2) Jru l e heat from th e electric conduction currents and (3) Fourier heat due to th e conduction from neighbouring fluid elements.

The equation (10. 9) mayalso be described as the equation for the " e ntropy-production" if we write the left hand in the form

(10. 10)

where

~

entropy per unit mass.

If we consider (10. 3), where Cv is assumed constant, the

.-equation (lD. 6) can also be written in the form I

C

,DT

ih

p.Dr.>

+

4..1

~

>..

6.

T

From thi s last equation one obtains easily the is entropie relation (9 . 4 ) if the motion is adiabatie (o r is entropie) so that aU the heat supply terms vanish.

With this seetion we have eompleted the diseussion of the general equations and the next point wiU be to simplify them to their

usual form. _ . -...

---11. In th e equations obtained so far the eleetrie and magnetie fields are treated as c ompletely tequiva.lènt c. In magnetohydrodynamies it is usuaUy supposed that the e l ec tr-ic fields are only due to induetion and it wiU be seen that due to this faet several terms in our equations wiU be smaU eompared to others. In partieular there appear terms in whieh the ratio Va. appears. These terms wiU be negleeted as we

c ..2.1

have assumed throughout that \I !c.2. <:::<: I

To obtain these simplified equations we estimate the order of magnitude of the te r m s appearing in the equations. The magnetie field plays a dominating part in this seheme and we shaU denote its order of magnitude by

[HJ

:

H

In the same way we write [v ]

=

v, [x

J

=

L

The time seale so defined is t =

ok

and for simplicity's sake we assume tha t th i s is the only time seaYe. We obtain from the fourth equation of (1. 1)

[c~t

E ]

-

_c.

v

~H

_L

and

[E.]

v

c,

~H

v

c.

[q,J

=

The seeond equation of (1. 1) gives

~~H

L

In the thtr-d equation of (1. 1) we have forthe last te r m Vl

-

_c2.

tE.

~

_L

H

and this term ean be omitted as

é.

IJ-

wiU be near 1 usuaUy.

For the first term on the right hand side of this equation we have using

[~q,~J

-

_-

_c~v'L.

~

_L

[

4~

1

J

c1

\}

t-

H

-

_c2..

The first expression c an be omitted according to our assumption. The second term may be even smaller than the first one when

d

is smal.l, but in an ionized gas the conductivity is high (on e even assumes ri=.co)

and therefore it is usual to retain the c onduction.current.

In the equation (8. 4) for the ponderomotive force we have

[e"s]

[~ ~V x~J

[~

4

X

&J

-e.~1.H'2.

L

é: ~~

1-12-L

cS

V

b:'Z.

H'"

c'-The first two expressions can again be neglected and only the third one

is retained.

Further we ha ve still to estima te the terms in the Maxwell stress tensor. These are:

[

E

,~l

[.ê.

.

HJ

-[

-P~J

v2.

E.

~1.

H2..

C4.

~

Hl.

V

/.A.

H

2..

Considering th en the expression (4 . 5) for fi we obtain

This estimation shows that only the first term of the Maxwell stress tensor has to be retained.

12. If the simplifications of section 11 are introduced in the electromagnetic equations the equations assume the following forms:

div B

=

0 div D

₌

41tT q

.

curl H

₌

_-

4lr' J c

-curl E

=

-

1

'dB

c-

ot

B :: _{~H D'}

=

Eo

E J

=

c5

(

~

+

I

v

X

~)

e, {12 . 1) U2. 2) (12. 3) (12. 4) (12. 5) (12. 6) f

=

1 (J"x B ) C - -f_i

=

dd",

J(

T

&. )

"t

.

. ::

.1- B, H

·

(l. LjlT" t. ( wh ile fu rther I

fiT

s

1.( -

B. H

-(12. 7) (12. 8 ) (12. 9) v , f

=

E.J - ~.

I.

<r

d (

I

B H) _

c1

LA}

_?*

E .J = -

ot

8-rr _

r -(12. 10) (12. 11)

To thes e equations , the equations of motion (9. 1) and (9. 2) have to be

adde d, where _{f i} is given by (12 . 8), plus the equations of section 10. in

wh ic h the te r m ~.D ha s to be omitted while (10. 5) is replaced by (12. 8).

Due to the simplifications of the electromagnetic equations it is convenient to express all the elect romagnetic vectors in terms of B. The final system of equations of magnetohydrodynamics can there

-fOre be put in the form of the or-diriàny hydrodynamic equations (in which

~ appears) and an additional equation for B in which v appears.

To obtain the equation for ~ we notice first ~rorruX12..3) that

and

d -

"

(

GUfi-t

.§.

x.

t -

'-I

'Tr~

Furthe r we"h a v e from (12. 6 ) (12. 12) (12. 13)

E

_-

_-

4:-~ (12 . 14) and

With (12. 4): and (12. 12) the last equation becomes:

~

- C.M-t (V )(.

B) _

ot

-

-or on using (12 . 1) also.

(12.15)

Combining the last equation with the hydr odynam ic equation we have nine equations with nine unknowns ~,~, P.

p

,

T. Once this set is solved the other electromagnetic vectors are immediately obtained from (12. 12), (12.14) and (12.5).

Several authors have noticed the analogy of Eq. (12. 15) with the vorticity equation of hydrodynamics. Using the identity

remembering Eq. (12. 1) and assuming infinite conductivity the equation

(12 . 15 ) becomes

"0

B

+(V

.

\l)

B _

I

~,

'\1)

~

+

a /

'V .

v) '::

0

at

- - -

l. -

L -

-

(12. 16)

The equation is exactly similar to the vorticity equation in a non-visc ous, compressible medium which reads:

(12.17) Just as the last equation insures the conservation of the vorticity intensity and the ",t r a n s p or t a t i on of the vortex lines with the fluid (See for example

[5J ),

so equatron (12. 16) insures the conservation of the intensity of the

B - lines and their transportation with the flow. That the B - lines in a fluid with infinite conductivity are transported with the flow was shown in a very nice qualitative way by de Hoffmann and Teller [6

J .

If we consider in such a fluid a small B-filament we obtain another interesting relation. Let the cross section, length,density and B of the filam ent be

If thesequantities are at time t

+

II t

.l,

we have from the CDnservation of the strength of B, that

The conservation of mass gives

and so

--

_p,

_Ol,

u

a

.

18)

In particular if the length of the element of the filament has not changed we have

.:B

o

-Fo

-

-(12 . 1 9 )

and magnetic induction is proportional to the density.

13. It is now suitable to consider a few special cases of the

equations of motion which are of particular interest for shock tube WeTk.

We shal! therefore assume that the vectors appearing in the equations wil! depend only on the space-coordinate xl.

The equations then become:

(13. 1)

(I3 . 2)

L=

RT

F

0 3.4 ) where

P,~

L

.

11..

-pot _

I

ëb:::

_I

2-while further U

=

Cv T and e2 and e3 are unit ve c t o rs in respectively the x_2- and x

3- direc tion s .

If the situation, we cons ide r , is steady the above equations lead imm ediately to

0 3. 6)

f

(;.,u,:4 -

:J

OtA.~

B,

B4.- .:

C(J.)

-

_O)C.

_J

-411' /'"'"

.a-r

lAl tÁ.

a

-~

OU~

₈

1

B~

:.

C

tJ)

-

~

Cl

')CI 4"-~ (13. 8) . (13. 9)

0 3. 11 ) 0 3. 12) (13. 13)

(a)

=

.

C~

'0

.B.a

,,~ I (,)

The equation (12. 1) gives us immediately that in (1_{3. 11 ) C 4} = 0 and so BI

=

constan t .

If we omit in the equations (13. 6 ), (13. 7 ) and U3. 10 ) the magnetic terms and the derivatives the equations reduce to the Rankine -Hugoniot conditions for the plane norm al shock-wave.

If we omit the derivatives and assume u2

=

u3 = 0 the equations

(13.6),- (13 . 7) and (13. 10) become:

pu...

l ='

Cl

(13. 6)

0 3. 14 )

which are the relations for -t h e normal magnetohydrodynamic shock wave. As

B.

is constant under our assumptions it can be omitfed in (13. 14). The relations (13 . 8) etc. are the relations for the oblique shocks.

The second case we shall consider is the non-steady is e n t r o p i c flow. The condition of isentropy requires that no heat is supplied to th e :c

fluid elements and we have to negleet the heat conduction and viscosity while further assuming infinite éonductivity to make the Joule heat

vanrsh. Further we shall assurne u2

=

u3

=

0 which is reasonable enough in the shock tube. The equations (13. 1) - (13. 5) then becorne

(13. 1)

~

+

..l...

(plJ.

1)

=

0

ut

VX,

k(f~')

+

k,

(~u.~)

-.2.

~

+

---L

ê)

BI

-.J..-

1-

(B .

.B)

'0

'>Cl i.(

iJ"'fA

a

x-,

diT

f

0 ,)G, - -(13. 16)

ft:

IJ

UIL

+

i

u.~)

+

~,...

.a.

~

J

+

~JpU.,lfU.

+

1-

u.

f)]

=

= -

1.- /

u,

p) _

--L

~

f

u

I (

B

.

8 ) -

u.

1

BIJ.)

(13. 17)

()x:.,

l

~lr ~

ê)')(,.1

L

- -

J

.t.

=

RT

\

(13. 4) or BI

=

Constant

/lA,B.,2.)+

e.J

~

lu,B)=,0(13.18)

L

-

O~I

l

-.!. while further

';)

+-

e~

-

êh::,

êJ

BI

=

0

0')(:,

It is easily shown on using (13. 1), (13. 16) and (13. 18) that (13. 17) and (13.4) are equivalent to the isentropic relation

=

Constant

p

r

The remaining equations can be written

of)

»»

'dlJC..,

-=+-

+

u.. ,

...:.+-

+

p -

=

0 ê)

t

OX-,

ê):>c,

(13.19)

~

I

~

(2.

~)

P

-

n ; -

B.:a.

+

B-a

(13.20) I

e

x, , r.Jlr~p I.I~I

OER

₊

Vvl -~

BJ.

- -

+

.B2.

-

~t..t

I = ( )

Ot"

V~c., . 'ICI

ra

B~

+

C.... I

o.B

_d

+

~ .

OU,

0

ot

a

s:

.j _"ë)1C. 1 I (13. 21) (13. 2 2)

If _{the x2 and x 3 axes are selected such that B 3}

=

0 and if we denote B2

=

B the equations simplify stiU further.

It is noticed on inspection that the equation for B is similar to the equation for

p .

The easiest solution of B is therefore obtained . by assuming B proportional to

p .

This is indeed the case if we remember the discussion at th e end of 12. The variables in the problem depend only on xl and slices of fluid can therefore be compressed only in the xl -direction while their cross section area in the x2 x3 - plane remains

constant. A B -filament in the x2 x3 - plane wil! therefore preserve its

length and as -

ei

=

00 Eq. (12.19) wiU hold.

With th e s e simplifications the equations become

le-

+

u.. ,

!L

₊

p

'ê)

-

U I

-

-

0

'Ot

_{C>X/ O~I}

ê>u.

1

... u

'OU,

-

~

.B

98

-

_o"t

_{I ~')C.,}

-

-f

è)

x:.

I LiTr~

p

o:lC ,

(13 . 1 9 ) (13. 2 3 )

..B

=

C~

F

(13. 24) The right hand side of (13. 23 ) can also be written as

(13. 2 5)

where

= the speed of sound

,

As it is only the purpose of this review to obtain the equations we shall not discuss the Alfven waves further. For general discussions on the waves which appear in the linearized approximation the reader is referred to the investigations of Herlofson

(,7]

,

van de Hulst [8J and Ba'iios [9, 1~.

14. Finally we shall make a dimensional analysis of thedifferent terms in the equations. We shall therefore write down a representative term in the different equations of motion and write its order of magnitude under it.

The equation of momentum we represent by the terms

[

~

~ (~

f "

.

IJ.

~'j

)1 _

_{- -}

[0

"o~:J

P l

+

[~

TQ~.lo')C~

d

/

~

4- _Q'>Cj.

ê)14~)J.f-[

~

_0;('

() (I

4ir

"B.

~ a~

~ .)1

(

4"

ij

ij-

(14. 1)

PV2.

PV2.

~

\/

B~

- - - +

-

+

-L

L

L4

~L

In this way we introduce the charaeteris tic ratio's

.B~

L

The first one is the well-known Reynolds number Re. The second one .

gives the ratio of the magnetic energy to th e kinetic energy. In particular if we wis,h to obtain strong magnetohydrodynamic effects this ratio should be or order one.

The last one gives the ratio of th e magnetic to the viscous force. This ratio is however not much used. It should be remembered that the electromagnetic force can also be written in the form of the Lorentz force (12.7). If we apply Ohmts Law (12, 6 ) to this term we find tha t the order of magnitude is

d

\J

'BoL/Cl. .

If we now write down the ratio of the magnetic force to the viscous force we find

d

Boa

I:/'!}

c.a..

This parameter has found a certain use and its square root has been called the Hartmann-number

[11]

which we denote by

Ha. _

BL

V

~

Co

~

(14. 2)

The second ratio of the Lor entz force to the inertia force can then be written as

v

C~

-

-( H

0..

)2-Re-

(14.3)

One may think that for infinite conductivity the Har-tmanrr-number will

become infinite and that the Lorentz force would dominate Eq. (14 . 1).

This however is not correct as Ohmts Law (12.6) cannot be used to

compute the current if (;;' ~ 00 . For infinite conductivity the current

is formed from (12. 3) while to prevent the current from becoming

infinite we must have E

+

..!.

( v x B)

=

0 in that case, so that (12 . 6)

- c

gives an indeterminate answer.

The next equation to be considered is the equation for B

(12 . 15). We have

02.

15)

~J

-+

Bv

L

[~J-

(ewt1

L~~ ~)J "L;~~

à.

B

v

ca.

B

L

~

T

G

iJ'-

Lj.

c.'"

The ratio introduced here is

Lj1r

~ ~

v

L

We have already seen that it is neces sar-y for strong magnetohydrodynamic effects that the magnetic and kinetic energy are of the same order so

that

v

(14. 4)

.Further it will be necessary that the ~'d:Lffusion"of the magnetic fields,

which is represented by~!le right hand side in (12. 15J is small, which

requires

<:.<

I

_{(14. 5)}

Combining (14 . 4 ) and (14 . 5) we obtain the well-known Lundquist nurnber

Lu..

=

6

..B

L

V

JA:

'

»

I

.

P

(14. 6 )

This expression shows that it will be difficult in general to produce

rnagnetohydrodynarnic experiments in a laboratory where L wiU be small,

which wiU require very high values of B to compensate and obtain

Lu» 1.

(10. 1 1 )

pv~

L

The ratio's introduced here are well known; the ratio of the second term on the right hand side to the term on the left is the well known but

name-less parameter

"%

e

.

The ratio of the dissipation term to the inertia

term on the left hand"side can be written as (R e )- l

vl.j,

ti.

The ratio

pC"

G

V

L:L.

is the Plclet number which is the prod:uct of Prandtl

.

L

o ·

.,

>.:

é

and Reynolds numberJ that is

pé

=

'P~.

Re.

(14.7)

If Ohrn's Law is used to express the current we have the ratio

:2-=

(Ha.)

(14 . 8 )

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. .1 1. Abraham, M. Beeker, R. Panofsky, W. K. H. Phillips, M. Frenkel, J. Bjerknes, V. Bjerknes, J. Solberg, H. Bergeron, T. Kotschin, N. J. Kibel, 1. A. Rose, N. W. de Hoffmann, F. Teller, E. Herlofson, N. van de Hulst, H. C.

-Banos , A., Jr. Ba'ii'os, A., Jr. Hartmann, J. Lazarus, F. REFERENCES \'

"The Classical Theory of Electricity and Magnettsrn ", Blackie and Son Ltd, 1950 "CIas stcal Electricity and Magnetism",

Addison -Wesley Pub.

co

.,

1955

"Lehrbuch der Elektrodynamik", Julius Springer, Berlin, Band I, 1926, Band II, 1928

"Physikalische Hydrodynamik", Julius Springer, Berlin, 1933

"Theoretische Hydromechanik", Akademie Verlag, Berlin, 1954

"Magnetohydrodynamic Shocks", The

Physical Review, Vol. 80, 2nd Ser. No. 4,

Nov. 15, 1950, pp. 692-703

"Magnetohydrodynamic Waves in a Com -pressible Fluid Conductor", Nature, Vol. CLXV, p. 1020, 1950

"Problems of Cosmical Aerodynamics", Proc. of Symposium, Paris, 1949 Central Air Doe. Office, Dayton 2, Ohio The Physical Review, Vol. 97, No. 6, p . .1435

Symposium, Proc. Roy. Soc. A 233, Dec. 1955

"Kgl Danske Videnskab'Selskab" , Math. -fys. Medd. 15, No. 6-7, 1937

APPENDIX

Electromagnetic Relations in

Non-rationalized Gaussian system Rationalized Giorgi system

div B = 0 div B = 0 div D = 4

-rr

q div D = q curl H

=~

+

+

I

àD

curl H

~

+

o~

-

= c

_ot

ot

curl E =

-

-

I d~ curl E =

oB

ot

c,

_-at

B = H + 41rM =

_~~

B

=

yto

(~ + M)

=

fJ-

_~. D = E + 41r P =

e:

E D =

c=.o

E + P = ê, E

L

=

d

E

_t

=

cS

E

div

_L

+

oq"

= 0 div

i

+

~

=

0

ot

ai:

BIn

-

B2n

₌

0 _BIn

-

_{B 2n}

₌

0 HH

-

H2t

=

Lj"Tr_c,

1~

Hlt

-

H _2t

=

js Elt

-

E2t

=

0 E lt

-

E 2t

=

0 DIn D2n

=

41rs DIn

-

D2n

=

s Ei

₌

E +v x B BI _{= B} .1 (v x E) ~

c

2 Hl = H

-

(v x D) II -

-

--

.

f v = E i =

- .l.

f

..!.. /

E.

.

D

-1- B.

H)I -

div p*

ot

L2 l - -

- -

f

- \ .) c - I = BI = B Hl = H f = q E

+

.!

(j x B)

-

-

- *"

=

01:'

~

L _c

.E.

fA

'0 'P(" fi

d1C.~

c-

ot

..,- = 1 ( Dl' E J.

+

B· H.) -~ ij ~1r I J

- -L

~

..

(E D

+

B H)

&ïr

t.~ -p*

=

c (E x H) 41r f ~

=

E

i

=

1.

_

(

_

I

(F

.

J:)

~'BJI)~-

div p*

- öt

lelT'

= -

--J

EI = E + 1 (v x B) c 1 (v x E) c 1 (v x D)

DI

=

D

+

.!

(v x H) c DI = D

+

1 (v x H) c . J

=

t

I

=

i.

f.v

=

E

t

- qv

t i

rS

J

=

tI

=

i

f . v = E · t

-Tables of Units

.Quantity Symbol Dimensions m. k. s. unit

Length m Time T sec Mass Force .kg Newton Charge density q Magnetic induction B Magne tic intens ity H Electric intensity E Amp

I

.

·m

Volt

I

m Coulomb Amp

I

m 2 ··Mh o

I

·m

Coulomb

I

m 3 Coulomb.Z m 2 We.ber

_.

_.

I

m 2 Q Q L-3 L -2 T- 1 Q M L T- 2 Q-l L-l T- 1 Q L- 2 Q . M T- 1 Q-l M-l L-3 T Q2 D Displacement Conductivity

a'

Charge Current density

i,

J Permeability Henry

I

m M - l L-3 T 2 Q2

Quantity Sym bol Charge density q Electric intensity E Magnetic induction B cm .dyne gram sec gauss oerstedt state oulom b .s.ta t m h o / cm statcoul / cm 2 Gaussian unit statamp / cm 2 .statvolt / cm statcoulomb / cm 3 Dimension L T M M L T- 2 1 L 3/ 2 T-l M"2 1 L-3/2 T-l M"2 .!. 1 T- 2 M2 L-"2 1 1 T- 1 M2 L -"2 1 1 T- 1 M2 L-"2 Mi L-"21 T- 1 1 1 T- 1 M~ L -"2 T- 1 D Conductivity Magnetic intensity H Force Displacernent Mass Current density

i.

J Charge Time Length Permeability dimensionless

Institute of Aerophysics, Universityof Toronto lnstituteof Aerophysics, University of Toronto

An Intr oductio n to the Equ ati on s ofMa gne to ga s d yn amics An Intr oduction tothe Equations of Magnetogasdynamics

1. Magnetohydrodyna m icsan d M~~·hetv ~a öG.y:uunics 4. Flu id Dynamics I St ekete e , J. A. UT IA Re v i e w No. 9 Shock Wa v e s Equa tio n s of Motio n J. A. Ste k ete e , April , 1957, 2. 3. II 35pp., 2ta b les _{J. A. Stekete e, .Ap ril, 1957,} 1. Mag n e t o hy d r od yn am ics and Ma gneto gasdyn amics 4. Fl uidDynam ic s I Ste k ete e , J. A. 35pp. , 2tabl e s 2. Shock Wa v e s 3. Eq ua tio n s of Motion II UT IA Revie wNo. 9

The purposeof thi s reviewis to faci lit a t e the st udyofma g ne t o hy d rod y nam ic s and

magnetogasdynam icsforae r o dyna m icsctenttsts . The reviewis divided in two parts.

In pa rt Aa short discussionof Maxwell'seq u a ti ons in isotropicmedia atre s t is given

while themodific ations are indicatedwhichhave tobe mad e if themedia are in motion.

In part B the equations of Ma x we ll and thehydr-odynam ic equations ar eco m bined to

ob tain theusual eq uations of magnetohydrodynamics . No specialsolution s of the equat

-ions are discussed;on theco nt r a ry the discussionfinishes where the ordina ry research

reportbegins .

The purpos eof thi s reviewis to facilitatethe studyofma g n e t o h ydrod y n a mics and

magnetogasdynamics for aerodynam ic scientis ts . The review is divided in two parts.

Inpart A a sh o rt discussionof Ma xwell' s equationsin isotropic media at rest is given

whilethe modifications areindicated wh ichhave to be made if the media are in motion .

In partB the equations of Ma x we ll and th e hydr od yn a m icequationsare co m b i ne d to obtain

the usual equa tions of magnetohydro dynamics. No special sol utions of the equa tionsare

discus sed;on the co ntr a ry thedi scus s i on fin i s hes wheretheor-dinar-y research report

begins .

Copies obtainablefrom:lnstituteof Aerophysics,Universit yof Toronto,Toronto5, Ontario Copiesobtainable from:lnstitute of Aerophysics, Univer sity of Toro nto,Toronto5, Ontario

UTIA REVIE WNO. 9

Institute of Aerophysics, University of Toronto

UTIA RE V IE WNO. 9

lnstitute of Aerophysics, University of Toronto

An Introductionto the Eq ua tio ns ofMa gn eto gasd yn a m ics An Introducti on to the Equa ti on s of Ma g n e t o ga sdynam ics

2ta ble s Shock Wa v e s Eq uations ofMo tion 35 pp., 2. 3. April, 1957 ,

1. Ma g netoh yd r od ynam i csand

Magne to gasdynamics

4. Flui d Dy na m ics

I Steketee, J. A. II UTIA Review No. :1

The pu rpose ofth i s review is to facilitate thestudy of magn etohydrodynamicsand

magn etogasdynamicsforaerodynamic sc ieritists . The re vi ewis divided in two pa r t s.

In part Aashortdiscussionof Maxwell's equationsin isotropic mediaat restis gi v e n while the modificat i ons are indicat e d whichhavetobe made if the media are in motion.

Inpart B the equa tion s of Ma xw e lland the hydrodynamic equationsare com b i n e d to

obtain the usualeq ua t i o ns of magnetohydrodynamics. No specialsolutionsof the eq ua t

-ionsare discussed;on the contrary the discussionfin ishes where the ordinary res earch

re por t begins.

J. A. Steketee,

The purpo s e of this reviewisto facilitate the study ofmagnetohydr od ynami cs and magnetogasdynamics fo r aerody na m ic scientists . The revie w is divided in two pa r ts. In pa rtA a short dis cussionof Maxwell's equation s in isotr o pic mediaat restis given while themodificationsare indicatedwhichhave to be madeif the mediaare in motion.

In partB the equations of Maxwell and the hydrodynamicequationsare co m b i n e d to

obtain the'usual equations of magnetohydr od ynamics. No special solutionsof the eqüat

-ions ar e discuss ed; on thecon t r a ry the discussionfinishe s where theor di na ry resear ch

reportbegins .

J. A. Ste k ete e, Apri l, 1957, 35 pp ,, 2tables

1. Ma gn e t oh y d r od y na m ics and 2. Sho c k Waves

Ma gn e t o ga s d yna mics 3. Eq uations of Motion

4. Fluid Dynamics

I Steketee, J. A. II UTIA Review No. 9