SIMILARITY RULES EN MAGNETOHYDRODYNAMl-CS
BASED ON MULTl-FLUID THEORY

Thesis far the Degree of Ph. D.
MICHlGAN STATE UNlVERSlTY
Joab Jacob Blech
T964

[HESIS

This is to certify that the
thesis entitled

SIMILARITY RULES IN MAGNETOHYDRODYNAMICS

BASED ON MULTI-FLUID THEORY.

presented b9

Joab J. Blech

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in M. E.

Major @rolessor

Date Mg 9 /d: /ffi7/4

0-169

% Z 1». %W04eé‘

 

 

 

 

     
 

HERA R Y
Michigan State
- University

 
   

ABSTRACT
SIMILARITY RULES IN MAGNETOHYDRODYNAMICS
BASED ON MULTI-FLUID THEORY

by Joab Jacob Blech

In the present work similarity rules are derived
in magnetohydrodynamics, in the physical Space, based
on a multi-fluid theory. The basic hypothesis of the
multi-fluid theory is that the fluid consists of a
number of fluid components, each with its own intrinsic
properties (such as molecular mass, charge, etc.). Each
of the Species is assumed to be inviscid and non-heat
conducting. The postulated fundamental equations for
each fluid component are: equation of state, first law
of thermodynamics, conservation of mass and conservation
of momentum. In addition two Maxwell's vector equations,
describing the electromagnetic field, are inserted into
the system of equations. The flow is assumed to be
multi-diabatic, i.e., there is injection of momentum as
well as energy (heat) by means of sources from outside
into the various fluid components.

We assume a steady flow which depends only on two
spatial coordinates; but in the present multi-fluid
theory such a flow is not a two-dimensional flow in the
usual sense. The velocity, the electric and the magnetic
fields have three components. After a reduction of the

number of fundamental equations is made, the system of

Joab Jacob Blech

equations is linearized, i.e., it is assumed that a
first order small perturbation theory describes adequately
the flow field.

In the present case of MHD a mere linearization of the
system of equations seems to be insufficient for obtaining
similarity rules. An additional procedure, i.e., some
sort of a smoothing process, is applied in which a crite-
rion for neglecting very small terms is introduced, thus
leading to a simplified system of equations which governs
the flow. A correlation between this simplified system
of equations of the compressible flow and correSponding
systems of equations of the incompressible flow is estab-
lished for the case of aligned fields, in which the velocity
and the magnetic fields in the undisturbed stream are
parallel, and for the ease of crossed fields, where the
velocity and the magnetic fields in the undisturbed stream
are perpendicular. Pressure coefficients for the indivi-
dual Species and for the gross fluid are calculated and
correlated. AS a Special numerical case, a fully ionized
plasma is considered and the ion pressure coefficient
is plotted vs. the ion free stream Mach number for

various orientations of the magnetic field.

SIMILARITY RULES IN MAGNETOHYDRODYNAMICS
BASED ON MULTI-FLUID THEORY

BY

Joab Jacob Blech

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1964

1
F} 7 III- L“. "

’I‘O

NEOMI

ACKNOWLEDGEMENTS

The author is greatly indebted to Dr. M. Z. v.
Krzywoblocki and wishes to express his appreciation for
the guidance he has received throughout the writing of
this work. He also wishes to thank Dr. G. H. Martin,
Dr. G. E. Mass and Dr. C. P. Wells for serving on his
guidance committee.

ii

CONTENTS

LIST OF TABLES O O O O O O O O O O O O 0

LI ST OF FIGURES 0 o o o o o o o e o o e 0
LI ST OF APPENDICES O O 0 O O O O O O O 0

INTRODUCTION . . . . . . . . . . . . . .

CHAPTER I.

1.

OF FLOWS O O O O O O O O O O

Fundamental Concepts .

FUNDAMENTAL EQUATIONS AND CLASSIFICATION

2. Fundamental Equations of the Gasdynamic
subsyStem o o o o o o o o o o o
3. Charge and Current Equations, Ohm's Law . . .
4. Fundamental Electromagnetic Equations . . . .
5. The Final System of Fundamental Equations . .
CHAPTER II. QUASI-THREE-DIMENSIONAL STEADY FLOW . .
1. The Governing System of Equations . . .
2. Quasi-Stream -Function and Potential Equations
3. Generalized Bernoulli and Crocco Equations .
4. The Final Quasi-Three-Dimensional System . .
5. Incompressible Flow . . . . . . . . . . . . .
o Non-DimenSIOnaliza‘bion o o o o o o o o o o o
7. Linearization . . . . . . . . . . . . . . . .
CHAPTER III. SIMILARITI CF FLOWS . . . . . . . . .
1 Approximate Governing System of Equations . .
2. Correlation of Flows . . . . . . . . . . . .
3. Pressure Coefficients . . . . . . . . . . . .
40 Application 0 O o o o o o o o o o o o o o o o

REFMENOES O O O O O O O O O O 0 O O O 0

iii

Page
iv

vi

LIST OF TABLES
Page

Table 1. Correlation for the s-th fluid component
quantities . . . . . . . . . . . . . . . 83

Table 2. Correlation for the w-th fluid component
quantities O O O O O O O 0 O O O O O O 0 87

iv

Fig.

Fig.

Fig.

1.

2.

3.

Fig. 4.

Fig.

5.

LIST OF FIGURES

Streamline geometry . . . . . .

Ion pressure coefficient for H2

2,—‘o7500000000000

Ion pressure coefficient for H3

H:=-.50...o.......

Ign pressure coefficient for R:

oo=-’25"”"""°

Ion pressure coefficient for H;
E: = 0.00 and H; = 1.00 . . . .

.75,

.50,

.25,

-1000,

Page
50

99

100

101

102

LIST OF APPENDICES
Page
APPENDIX.A. Some Remarks on the Functions Bs . . . 10}

vi

INTRODUCTION

In the past, many formulas for the similarity relations
between incompressible and compressible fluids in an
isentr0pic gas flow, were derived. A collection of such
rules in the physical Space is given in (8). The Kérmén-
Tsien technique which employs the hodograph method can be
found, for example, in (7, pp. 336-340). Similarly, some
attempts to derive similarity rules in diabatic flow, i.e.,
a flow with heat addition by means of sources, were made (4).
In recent years there has been a tendency to derive various
similarity rules in magnetohydrodynamics.

From various attempts in the past we may quote and
discuss briefly the following:

(i) v. Krzywoblocki and Nutant (5) derive a similarity

rule for an inviscid, non-heat conducting, diabatic flow
which takes place in an electromagnetic field, with excess
electric charge equal to zero, following the technique of
Kérmén-Tsien in the hodograph plane and assuming a simpli-
fied pressure-density-entrOpy relation. Although a similarity
rule is derived, the disadvantage of this procedure is

that equations of the electromagnetic nature are not trans-
formed into the hodograph plane. The electric and magnetic
fields are treated as known functions. Moreover, the

1

2

correlation of correSponding coefficients in both stream-
function equations of the compressible and incompressible
flow requires that there be a certain relation between
the vorticity distributions of both flows. A Special
relation is also imposed on the Jacobians of the trans-
formations in the physical Spaces.
(ii) McCune and Ressler (6) treat the two-dimensional
steady case of a highly electrically conducting, inviscid,
non-heat conducting, isentr0pic flow passing over a thin
body. A single partial differential equation for the
current field is used to study the flow. This differential
equation is derived from the linearized fundamental system
of equations of the hydrodynamic and electromagnetic nature.
The discussion is separated into three main cases, depending
on the orientation of the externally applied magnetic field:
The case of aligned fields, where the magnetic field is
parallel to the velocity field of the uniform undisturbed
flow, the case of the crossed fields where the magnetic
field is perpendicular to the free stream velocity, and
the case of an arbitrary field angle. In each case,
procedures are deve10ped for the solutions of the
magnetoaerodynamic problems involved and the compressibility
effects can be studied through the solutions.

In the works mentioned previously the ionized gas
is treated as a single fluid. It was pointed out in (2)
that if one seeks a single fluid magnetohydrodynamic

formulation in which the current density is considered

3
as an unknown, a counting of variables and equations shows
the necessity of an additional vector equation which is
usually taken as the generalized Ohm's law. The influence
of gasdynamic effects on the electric current density has
been completely neglected. One way to improve the descrip-
tion of the mechanism which governs the electric current
density is to use multi-fluid theory. In this formulation
the fluid is assumed to consist of several fluid components,
each with its own intrinsic pr0perties (such as molecular
mass, charge, etc.) and with its own thermodynamic state
variables. Conservation equations from the macroscopic
point of view are then postulated for each fluid component.
There is no necessity for Ohm's law Since the electric
current density is defined by means of the velocities and
charge densities of the various Species, and the former
are governed by the conservation equations of the indivi-
dual fluid components. Thus the effect of the various
forces on the electric current density through the velocity
vectors and momentum equations can now be treated exactly
from the macroscopic point of view.

The main purpose of the present work is to derive Simi-
larity rules, avoiding the pitfalls appearing in the works
of previous authors. A multi-fluid theory was employed.
Each fluid component was assumed to be inviscid and non-
heat conducting. In Chapter I, section 1, the dependent
variables and a list of the fundamental equations are

introduced. In section 2, equations of state, first law

of thermodynamics, conservation of mass and of momentum

for each fluid component were postulated and the passage
from the Species equations to a gross fluid formulation

is discussed. In order to obtain correct momentum and
energy equations for the gross fluid, it was assumed that
the fluid component flows may be multi-diabatic, i.e.,

there may be injection of momentum, as well as energy (heat)
by means of sources from outside into the various Species.
Equations of state and for the internal energy of each
component are allowed to deviate from perfect gas equations,
so that it is possible to derive perfect gas equations

of state and for the internal energy of the gross fluid.
Various results pertaining to the charge and current
equations and to Ohm's law, which were obtained in (1, 2)
are summarized in section 3. The fundamental electromagnetic
equations, i.e., Maxwell's equations, are introduced in
section 4, and following (9) two possible formulations of
the final system of equations are given in section 5.

In Chapter II, section 1, a steady three-dimensional
flow which depends only on two Spatial coordinates is
assumed. In the multi-fluid theory, one cannot assume
that such a flow is two~dimensional in the usual sense.

The velocity and the magnetic field have three components
which have to be calculated from the fundamental equations.
The electric field component in the third direction is shown,
from the fundamental equations, to be a constant. The

electromagnetic system of equations is further reduced

into two differential Equations for prOperly chosen functions,
the spatial derivatives of which give us the remaining two
components of the electric and magnetic fields. In section 2,
the single quasi-stream function* and potential equations
for each Species are derived. Those functions are associated
only with two velocity components. The third velocity
component is governed by the momentum equation in the
correSponding direction. Generalized Crocco and Bernoulli
equations are derived in section 3. In section 4, a
summary of the governing system of equations is given.
The quasi-stream function and potential equations for the
incompressible flow are given in section 5, and the final
system for the incompressible flow is discussed. Section 6
merely introduces a nonmdimensionalization of the various
quantities and equations of the flow, after which a lineari-
zation procedure is carried out in section 7, i.e., it is
assumed that a first order small perturbation theory
describes the flow field.

In Chapter III an analogy between the compressible
and incompressible flow is obtained. For this analogy it
is necessary to simplify further the linearized system of
governing equations. Thus in section 1, following a criterion
similar to the one given in (11), very small terms are
neglected in the linearized system of equations. Certain

pairs of coefficients in the linearized quasi-stream function

 

* A three-dimensional steady flow has actually two stream
functions (3).

6

equations are approximated by their weighted mean. In
section 2, a correlation between the simplified linear
system of equations of the compressible flow and correSponding
systems of equations of the incompressible flow is established
by means of linear transformations of the coordinates and
of pr0perly chosen relations between corresponding
quantities of both flows. It is necessary here to distinguish
between the two separate cases of aligned fields and of
crossed fields. Finally, in section 3, pressure coefficients
for the individual Species and for the gross fluid are
calculated in both flows and the relation between them is
given. A Special case is chosen in section 4, and the
cOmpressible pressure coefficient is plotted vs. the free
stream Mach number with the externally applied magnetic
field components as parameters.

In ordinary isentr0pic irrotational flow it is
sufficient to linearize the equations in the physical
Space in order to obtain similarity rules. In a rotational
flow of such a character, some additional assumptions
must be made to take care of the vorticity effects. It
seems that in the present case of MHD the procedure known
from the classical gas dynamics is absolutely insufficient
for obtaining reasonable similarity rules. The additional
procedure which is applied in the present work is some sort
of a smoothing process after linearization which is actually
equivalent to neglecting some terms of smaller order, and

taking mean values. It seems as if this procedure can be

7
considered as a first approximation in a chain of a successive
approximation process as applied to the similarity rules.
Higher approximations are then obtained from a more accurate
system of equations of the flow and the results of the first
approximation. In the present work only the first approxi-

mation is considered.

CHAPTER I

FUNDAMENTAL EQUATIONS AND CLASSIFICATION OF FLOWS

I.1. Fundamental Concepts

We consider the plasma, on the base of a continuous
medium, as a mixture of n fluid components, each with its
own intrinsic properties (such as molecular mass, charge,
etc.) and each with its own thermodynamic state variables.
From a macroscoPic point of view, the following quantities

are sought:

T5 = temperature of the s-th species;

p5 2 pressure of the S-th Species;

P, : density of the S-th Species;

u: = i-th component of the velocity field of the sath Species;
T : temperature of the gross fluid;

p : pressure of the gross fluid;

P = density of the gross fluid;

u = i-th component of the velocity field of the gross fluid;
= i-th component of the electric field;

H‘ = i-th component of the magnetic field;

where s=1,2,...,n and i:1,2, or 3. Counting the unknowns

8

9

we get 6n+6 quantities to be determined*.

The following gas—dynamics equations are postulated
for each fluid component: Equation of state, first law
of thermodynamics, equation of continuity and equation
of momentum.

In addition, two Maxwell's vector equations, describing
the electromagnetic field, are inserted into the system
of equations.

Counting the equations we get 6n+6 relations.

I.2. Fundamental equations of the Gasdynamic Subsystem

The following relations are postulated for each of
the fluid components, assuming that each component behaves
like an inviscid, non-heat conducting fluid:

Equation of state:

TasrRSPS—E}, (1.201)

where R5 is the gas coefficient of the s-th component.
The density, fl;, is assumed to be given by:

Patna), . (1.2.2)

 

* The quantities T, p, p, u; depend on T5, p5, P,, u;
and, therefore, will not be counted.

10
where m6 is the molecular mass of the S-th component and
1; is its number density.

First law of thermodynamics in a form:
dQ5= dU5+ 1osd((o;‘), (1.2.3)

where dQ5 is the energy addition per unit mass into the
s-th component and U6 is its internal energy per unit mass.
With the aid of Eq. (I.2.1), Eq. (1.2.3) can also be

written in the well-known form:
()1 Q.= dI, - 0:11.10, , (1.2.4)
where I5 is given by:
[5: Us +R5Ts. (1.2.5)
Equation of continuity*:
%%*afi'fi((0.u’§)’ 6., (1.2.6)

where Cg is the mass source, per unit volume, of the s-th

component. From the conservation of mass it follows that:

:5; 0, (1.2.7)

9'1

 

* Summation convention is used for repeated tensorial
indices but not for subscripts distinguishing between the
fluid components.

11
which implies that there is no addition of mass from out»
side.

Equation of momentum is postulated (1, p. 8, Eq. (1)):

- Xi}

6 ° 8 ' ‘ _ 9.) 4" / 4’ 8 s
fi(@su::)+ SEARS“; Uzi) "'fi€+ Fe + OSZS- ax} , (I.2.8)

where i=1,2, or 3 and F; is the i-th component of the
body force acting on the s-th Species. This force may

be written as:
a z a :
F5 : F s + Fee + F09 , (1.2.9)

where Ffi;is the i-th component of the non-electromagnetic
body force such as gravitation force, etc., F; is the i-th
component of the electromagnetic force and Fé is the i-th
component of the interaction force, i.e., the force on

the s-th component due to all the other kinds of Species
in the fluid. By Newton's third law of motion, we have:

Z F0: “‘2 0; (121,2.3). (1.2.10)

S=i

i
6525 is a momentum source associated with the mass source

6%. Following (1, p. 9) we require that:

n L
;6925=O; (i=1,2.3). (1.2.11)

if
X5 is a momentum transfer tensor, associated with the s-th

12
fluid component, the significance of which will be explained
later.
Next, we derive an energy equation analogous to the
one given in (1, p. 8, Eq. (1)). Assuming that QS is a
differentiable function, then from Eq. (1.2.4):

 

 

 

 

8 g a 5 —i , .

835’ aid: —[05 %% 9 (3319293)» (I.2.12)
3 31-5 ‘1- ()5

985 : at ~ (05 $4? (I.2.13)

Multiplying Eq. (1.2.8) by u: and using Eqs. (1.2.1),
(1.2.5), (I.2.6), (1.2.12), (1.2.13) we get after a few

manipulations:
a - x“
Edge-[0505) + g—fl(ésuas .(95{ SJ“: 'FFSUé) = 111;}T5t-(L; fi'f
i ' A.
+55( I5 QS-Z u": LL: U’SZ»), (I.2.14)
where 85 is given by:
g. = é—patéui ”110.11.. (1.2.15)

Equation (1.2.14) is the energy equation, analogous to the
one obtained in (1).

Define the following gross fluid variables:

11.
P: :1 (a, , (1.2.16)

13
(0,11: , (1.2.17)

11

(“A '

F‘=

Mg SM:

FT; 1 (1.2.18)

1

(D
11

. ' _ .. ’11) ~' - . --

XL3= 10131.1( -P8L3 +;(X? will; 11: +1056”), (1.2.19)
, . 11 . 1 m

S = Ii- Pw' LL” +i0U = ; (“é“(OSLL: Us + P5115): 5:? SS (1.2.20)

PQFFEMPQLLUPMFA 37,(eu5+p111)+11 7171' +

91

+;[11§F;— 9—— ———(e, 113+ +1.11) +§TWPQ +3—;(P.Q.ui)—
143+ (I o z:

1153—)“ 6, ,- ,——11:;_:11 +;11 )] (1221)

where 6&6 is the Kronecker delta. The diffusion velocities,
U1

5!

are defined by:
1 1 1
if, = 11,5 - 11 . (1.2.22)
It is evident from Eqs. (1.2.16), (1.2.17), (1.2.22) that:
I", .
2P5”; =0 . (1.2.23)

The pressure, p, is defined in (1, p. 9) to be* :

 

* The pressure dealt with in this work is the gas-
dynamic ressure. The radiation pressure is neglected (see
9’ p. 12 Q

1,4

FE::PS MP u‘ , (1.2.24)
Another possible definition is (9, p. 10):

@2105. (1.2.25)
We will ad0pt the latter proposition for the pressure.
Using Eqs. (1.2.11), (1.2.16), (1.2.17). (1.2.18),

(1.2.22), (1.2.23), (1.2.25) in Eqs. (1.2.19) to (1.2.21)
it can be shown that:

- 7" , _
X)A =:1(X:1+P,1IJUJ), (1.2.26)

[DU = 2(F5US + 71(051}: U2): (1'2'27)

+17; 2:) ‘37, (55,0 5U, 5,55% (1.2.28)
Adding each of the individual Eqs. (1.2.6), (1.2.8),

(1.2.14) over s, and making use of Eqs. (1.2.7), (1.2.11),
(I.2.16) to (1.2.21), (1.2.25) we obtain:

fiflpui) = O , (1.2.29)

- 1~ *fi
397(3011‘) 1- ”1311“” 111) 3311.1 ‘1— 3:, , (1.2.30)
a—at—(é -PQ) wefifia 118 -P()115‘ +1111?) =~

7%
=H1 F “Lia—F". (1.2.31)

Equations (1.2.29), (1.2.30), (1.2.31) are the continuity,
momentum and energy equations for the gross fluid.

In the equations of the single plasma components we
introduced terms representing momentum and energy flux
into the s-th.component from outside ( LX721 -@—(FSQS)+

3X3 1 81
+38X3()DS OSLL 1135)) . The inclusion of these terms made it

 

possible to obtain correct momentum and energy equations
for the gross fluid. A correct continuity equation for the
gross fluid was obtained by the requirement that the density
of the gross fluid be equal to the sum of the densities of
the component fluids, and that the mass flux in the gross
fluid be equal to the sum of the mass fluxes of the component
fluids. We will define a "perfect gross fluid flow" as
one for which Eulerian formulation is obeyed, i.e., as
in our case,.XTLEO , due to nonmexistence of'a momentum
transfer into the gross fluid. A perfect gross fluid flow
will be called ”adiabatic" or "diabatic", depending on
whether Q does or does not vanish.

In a diabatic fluid flow we assumed that the energy
is added into the particle with no viscosity and heat

conductivity present; similarly, in our case of perfect

 

 

.‘

16

gross fluid flow, we observe, by inSpection of Eqs.
(1.2.26), (1.2.28), that in general there exists an
injection or subtraction of momentum and energy into a
particle (X: 9&0, 054:0 ), hence the Species flow is
diabatic not only in an energy sense, but also in a
momentum sense. We, therefore, propose the name
"multi-diabatio flow" for this model which is more general
than the diabatic flow.

From Eq. (1.2.26) we get in the case of a perfect
gross fluid flow:

’11
X(>< +1.11, U1) =0 . (1.2.32)

5:1

Q
One way to satisfy Eq. (1.2.32) is by choosing )(5 in

the form:
x‘1—-- 1
s _ P6115113 - (IE-2.33)

In the case of an adiabatic perfect gross fluid flow
the left-hand side of Eq. (1.2.28) vanishes, and therefore:

:[UJ F.f——,1—,—,(e.ug +10: 111) +3t (1.0.)+9fi((D.Q.11§)—
vii—X—s+ds(l Q.- uéué+U§Zl)+

9111

”111—111(1)”; 11.3)] = O . (1.2.34)

17
One way to satisfy Eq. (1.2.34) is by choosing QS such
that:

%
fiat-(PSQS)+ 330 (P Q LL ) _U5 151F1383A86 U1 P2527511) U51 99%?

L (r '\

" 9(Ie_Q5 — ”El-us 1’15 +1}: 2:1 -._ ucégfl<PSUJU3)) (I‘2°35)

Q
and if 2X; is given by Eq. (1.2.33) we get, with the aid
of Eq. (1.2.22):

—(FSQS)+ _8_3_( (SP 550 LL21) ' 7 451?: + 5%??(11 Us} +P5U51) ‘
—6,(I,—Q,—Ziu;u§ +U§Z:)-— uéfifipsvévé). (1.2.36)

Eqs. (1.2.33). (I.2.36), if taken into account, enable
{

one to eliminate )(51 , QS from the momentum and energy

equations of each fluid component.

Defining the Operator %§ by:

= 2R1+U} 5X, (1.2.37)

2

it can be shown that the equation of energy (1.2.31) can
be remodelled with use of Eqs. (1.2.29), (1.2.30) into a
form which may eXpress a gross fluid first law of thermo-

dynamics:

ZFF'= TIT'+'P"&RT(P¢)- (1.2.38)

18
It is sometimes convenient to consider another form
of the momentum equation for each fluid component which is

obtained from Eqs. (1.2.6), (1.2.8):

 

 

 

l g 8 s _1' - 4'. 9X3,
516(3)“ + 111 19):) = 73.1. 165(Zé-u.) a). , (1.2.39)

where 121,2, or 3, and for the gross fluid, using Eqs.
(1.2.29), (1.2.30):

' i . -Q
m1 M 9 1 951 .
()(9’0 + 3W) :_‘03%+F '33)? , (i=1.2.3). (1.2.40)

 

Dividing Eq. (1.2.39) by p, , Eq. (1.2.40) by P’ sub-
tracting the second equation from the first, and using
Eq. (1.2.22) leads to the soecalled difference momentum

equations:

 

 

sp;1(Zf.-u:) , (1.2.41)

For the derivation of an equation of state for the
gross fluid, we define the gross fluid temperature as

(9. p. 11):

T = V1: 115T. , (1.2.42)

where 9 is given by:

9:}: )5 , (1.2.43)
5:1

The gas coefficient for the smth fluid component, Rs ,

is assumed to be given by:
R,=R,m;1 (141,) =R..(1+W.) , (1.2.44)

where RA is a universal constant and W5 is a function

representing the deviation of the state of the s-th Species
from the state of a perfect gas. We observe that if W5 : 0
then the s-th component has a perfect gas equation of state

since equation (1.2.1) becomes, using Eqs. (1.2.2), (1.2.44):
P5=RA95T5 , (1.2.45)
which is a perfect gas equation of state. R#,is given by:
111.512.7113. (1.2.46)

Inserting Eq. (1.2.44) into Eq. (1.2.1) and using Eq.
(1.2.2) furnishes:

70.=R..(1+w.)(o.1; = RA(1+W,)YSTS, (1.2.47)

Summing each of the individual Eqs. (1.2.47) over s and
making use of Eqs. (1.2.25), (1.2.45) we get:

2O

(FRANK/”T , (1.2.48)
where W is defined by:

W=()’T)—15Z__1WSY,T, , (1.2.49)

Equation (1.2.49) can also be written in the form:
10=R(0T, (1.2.50)

where R is defined by:

R= R m (1+ W)- -RP(1+ +;)\/) Rpm/m“, (1.2.51)

13:50))“ =(5::_!Ds)(:ys) (2:213) ,)(:Ve> . (1.2.52)

5:1

The symbol m is referred to as the mean mass of a particle
in the gross fluid.

In order that Eq. (1.2.50) be a perfect gas equation
of state it is necessary that R be a constant. We observe,
from Eqs. (1.2.51), (1.2.52) that the condition W20 is not
sufficient to make R a constant since the mean mass of a
particle in the gross fluid, m, given by Eq. (1.2.52) is,
in general, not a constant. If one assumes however that
m = constant (and W;O), then the gross fluid has a perfect gas
equation of state.

Next, a specifying equation for each fluid component

21
is derived. Introduce the entrOpy per unit mass of the

s-th Species, 55, by means of the following relation:
T5615, = 310, . (1.2.53)

It is assumed that the Specific internal energy of the

s-th component is given by:
U,=C,,,(1+1\,)T5 , (1.2.54)

where A5 is a function which represents the deviation of
the Specific internal energy of the S-th component from
a perfect gas Specific internal energy, i.e., the imper-
fections of the gas. 0,5 is the heat capacity of the s-th
fluid component at constant volume if it would have been a
perfect gas, and will be assumed to be a constant.

Inserting Eqs. (1.2.53). (1.2.54) into Eq. (1.2.3),
deviding the resulting equation by TS and using Eq. (1.2.47)
furnishes:

(1.35: C15“. 1A5) .1—5-1 de + Cvs dAS-RPS(1+WS)P:deO (I.2.55)

Integrating Eq. (1.2.55) from a zero subscript initial
state to some and state gives after the elimination of

the logarithms by means of eXponential functions:

T5 T5: = 65(51553; Ks-i WHSJSSJCJH , (1.2.56)

22

where Gs , K5 are given by:

G: 2x1019(A.-/\..) - (0A.? 012T. + (112110114): 4111.1, (1.2.57)
K.=1+R..C;i. (1.2.58)

Using Eq. (1.2.47) in Eq. (1.2.56) we get:

1156.))? 54315.1.) (1.2.59)

where the function 05 is given by:
C5 = 13.053355(1+1315)“11/1/21.)—1 Gs 9X10(“SSOC;:) . (1.2.60)

Eq. (1.2.59) is the Specifying equation (generalized pre-
ssure-density-entrOpy relation) for the S-th fluid
component.

Similarly, we introduce the specific entrapy, S ,
and internal energy, [1, for the gross fluid by the

relations:

T115 = (10, (1.2.61)
U=C.(1+A)T, (1.2.62)

where A, c1, have a similar meaning for the gross fluid as

A5 , c,s have for a fluid component.

23
It can be shown, by similar operations which were
performed to obtain Eq. (1.2.59) and with the additional

assumption m = constant, that:

1P= CPKW181111) (1.2.63)
where:
C: PoPJK “HA/X11110)—1 G W(—So CV1) , (1.2.64)
6 = WRMVAO) ‘LAT—1 dT + {19910141504115} ’ (1.2.55)
K= 111121.03. (1.2.66)
Equation (1.2.63) is the Specifying equation for the gross
fluid*.
It may be worthwhile to notice that the n+1 functions

A, A1 ’ A2 , coo , A“. are related. by Eq. (I02027),
remodelled with the use of Eqs. (1.2.54), (I.2.62):

PCvU (1):ij— 5:1: PMCv51H1—1'A “Mg—ENJUSI']. (I.2.67)
One cannot assume, in general, that:

AEA,EA,E...‘=‘AREO, (1.2.68)

 

* 1n the case where the mass m is assumed to be a

24

Since we would get one more relation between.P, T, PS, TS,
v; , thus overSpecifying our system of equations. If,
however, one assumes that Eq. (1.2.68) is true, then the
gross fluid as well as all the component fluids have perfect
gas specific internal energies.

Next, we calculate the value of (93%,)55 . Using Eq.
(1.2.47) in differential form and Eq. (1.2.58) in Eq.

(1.2.55) we get, after rearranging:
(3.6}: (11A,)d55: dp.—K.psf.i(’11(\1,)()?gt 111(1 W) (iv/gr +1111) 1).)1111. (1.2. 69)
where N5 is given by:

N. = (Ks-1M1;1 (W.-A,)(1+A,)-i. (1.2.70)

Assuming ()5 and 5. independent, also assuming that 10., As,

Ws , are, possibly, functions of R5and some other variables
independent of P5, we get, equating coefficients of dPS

in Eq. (1.2.69) and rearranging:

s O

(065)2 = (11%:35) = K5131” (1 N.+) p.13—P;103(1+w.)(14A,)-1]S (1.2.71)

Similarly, assuming m = constant,f)and S as independent

variables and p, A, W as functions off1and, possibly, of

 

variable, we can decompose it into two parts: m = mean
constant + variable perturbation. This will give us in the

final Eqs. (1.2. 63) to (1.2.65) an additional term which will
represent a deviation from their present forms.

25

some more variables, independent of P , we have:

2 9 _ 1 _
(06) 43%)5-7Kpp‘(i+N)+yloflf(13(1*W)(1+/)HS, (1.2.72)
where N is given by:

N =(K—1)K‘1(W—A)(1+A). (1.2.73)

I.3. Charge and Current Equations, Ohm's Law

Following (1, pp. 11, 12) we derive a charge and a

current equation.

Let P%,be the charge density of the smth fluid

component, given by:
PBS: 6595 , (I0301)

where es is the charge of a particle of the smth Species.

From Eqs. (1.2.2), (1.3.1) we have:
11.51.11. ; 21:263st (1.3.2)

Assuming 55: consonant, we have, multiplying Eq.
(I.2.6) by'#;and making use of Eq. (1.3.2):

$51,711:): 6515. (”'3’

26
Equation (1.3.3) is the equation of conservation of
electrical charge for the s-th fluid component.
Requiring that there be no input of electric charge

sources from outside into the fluid, we have:
71.
Z 6.). =0. (1.3.4)
S=i

The gross fluid excess charge density, Fe, is given

’71.

foe=ZPes , (1113.5)
5:1
and the electrical current density, J), is given by:
i. 'n. J. 1'. 1‘.
J =;pe.u, = J, +1111 , (1.3.6)
where J; is given by, using Eqs. (1.2.22), (1.3.5):
‘ _ 1
Ju — [0957}, . (10307)

J‘ is the current observed in a fixed system of coordinates
while J: is the current observed as ”moving with the gross
fluid". The term (mu‘ is called the convection current.
Adding the individual Eqs. (1.3.3) over s and using
Eqs. (1.3.4), (1.3.5). (1.3.6) leads to the equation of

conservation of electrical charge of the gross fluid:

27

_%%,h§E(Jg)g(). (1.3.8)

It may be worthwhile to notice that in the case of a
fully ionized plasma (n = 2), Eqs. (1.2.7), (1.3.4) lead to

the immediate conclusion:
@E@EO. (LEW

which means that there is no mass interchange between the
fluids.
Multiplying Eq. (1.2.8) by 55 and using Eq. (1.3.2),

we get the individual current equations:

.99T(Pesuvs)+3x1(resusug+5e,:x8):"J53—§1+65E+556,,:.Z (I.3.10)

The total current equation is obtained by summing the
individual Eqs. (1.3.8) over s and making use of Eq.
(1.3.6):

J‘ + 251(11uiuth?‘216105813);HF : “523- ”'3'”)

Multiplying Eq. (1.2.40) by Eksgji, summing the resulting

 

equations over s, subtracting the resulting equations from
Eq. {1.3-11). and using Eqs. (I.3.22). (1.3.5) to (1.3.8).

gives, after some Operations:

234+ +:[§éfi([°.,v:vf)) + ((3%;- + E95304} JJ) +

5:1

28

+2 [16%(135813 + X5] ' p.10": gag-(1081 + X”) =
{17.11: 22:11-11:11". 7......

S=i

Either Eq. (1.3.11) or Eq. (1.3.12) are the gross fluid
current equations (1, p. 12).

Next we state some results pertaining to Ohm's law.
For a more complete discussion, the reader is referred
to (2).

It was shown in (2) that a generalized Ohm's law is
contained in the difference momentum equation, (1.2.41),
as a certain limiting case. If we assume that all terms
in the difference momentum equatiOn can be neglected
compared to electromagnetic and interaction forces, then

Eq. (1.2.41), reduces to*:
(a: S 15‘): , (1.3.13)

where the force, 1; , is given by, according to Eq.
(1.2.9):

F; =-):és~+ F15 . (1.3.14)

The electromagnetic force, figs , is given by:

 

* Vector notation is used in the following
derivation.

29

F652P25( +Jsxg), (103-15)

where § is the magnetic flux density. The interaction

force, fim , is assumed to have the form:

F05 = Z 0651(E+__U’5) ) (I03016)'

+4

where kg are assumed to be constants. Substituting Eq.
(1.3.14) into Eq. (1.2.18) and making use of Eqs. (1.2.10),
(1.3.5), (1.3.6) and (1.3.15) we get:

—’D _.

F 1091:, ijfi+§oeuxfi, (1.3.17)

Inserting Eqs. (1.3.14) to (1.3.17) into Eq. (1.3.13),
eliminating'fis by means of Eq. (1.2.22), we have:

(PPes-Pspe) Eu, +(Pfe9175 *stu)x_B’ "
_:[oms+(175-fi) =0 ; s =1,2,...,n, (1.3.18)

where E, is given by:

—.§

Eu. = E + leB . (1.3.19)

If we consider P5, P85, 1),, , E, 0&3, as given, Eq.
(1.3.18) is a system of n algebraic linear equations
for the velocities is (also contained implicitly in

30
TL
3., = :Pesfis ). It is shown in (2) that I, can be written
5:1

in the form:
Ju=aEw+ (1E.XB+C(E.,XB)XE, (1.3.20)

where a,b and c, obtainable from the solution of the
system (1.3.18), are scalar functions of 95, (k5, my and
B. Equation (1.3.20), if inverted, yields:

---I- —.—§-

Eu = R11 Ju. +3 waB + (R11-Ri)B—Z(iIXE)X—é, (1.3.21)

where R" , R, and 5 can be written explicitly in terms

of a, b and c, as follows:

12.. = w ; 5 = 11114131212821",-

R, = 1.. .Bl)[(.-.13+)2+ 1:132)“ ; 1:11.: ;

1 "511114152131“ ; c11111—111311]. 1......

For the case of the fully ionized plasma (n : 2), Eq.
(1.3.21) becomes (See 2):

Eu = 641]& + gJux—E , (L123)

where §,<5 are given by:

31

5 =’( 1111” 1121312111212 3 $011 (1.1232111? +302‘)'2 , (1.3.24)

and Nzxazdu . The inversion of Eq. (1.3.23) can be

put in the form:

* *

J. = 6E... 1‘ 6‘16” + ngzfi-E’i -§(6*2 72139-1;th , (1.3.25)

where E:, , E; are given by:

‘IF -—.—0-

E1=( ~E.)B" ;E1=B'ZE1(E.1B); 15.51113. (1.3.26)

By inspection of Eq. (1.3.23) one is tempted to call 15
the electrical conductivity and EnixB , the "Hall effect".
When the Hall effect is negligible, Eq. (1.3.23) reduces.
to the generalized Ohm's law:

_. —. —+

Ju=6Ew =6<E+EXE). (1.3.27)

In a similar manner one may decide to keep more
terms in Eq. (1.2.41), in addition to the electro-
magnetic and interaction terms. Ohm's law will take then
different forms than the one stated above (2, p. 17, Eqs.
(22), (23)). The reader is also referred to (13) where
the derivation of the generalized Ohm's law in a three

component plasma is dealt with.

32

1.4. Fundamental Electromagnetic Equations

It is assumed that the gross fluid, as a whole, is
subjected to one electric field, E, and one magnetic field,
E. The equations governing the electric and magnetic fields
are the Maxwell equations (12, p. 2):

VXE + ‘3‘?" Z 0, (1.4.1)
VXFI “ g? = 3, (1.4.2)

where §,'5 are given by:

= eE: dielectric diSplacement vector, (1.4.3)

UH U11

= [13% magnetic flux density vector, ' (1.4.4)
6 is the inductive capacity and fie, the magnetic
permeability.

Following (12) we take the divergence of Eq. (1.4.1).
The first term is identically zero. Assuming that the

Operators V1 and g%'are commutable for E we get:
—9-— v-E —— 0
at _ , (I0405)

‘3 is, therefore, a function of position only. Assuming
that ever in its past history the field has vanished,

one concludes that:

33
V-‘B=o. (1.4.6)

Simdlarly, taking the divergence Of Eq. (1.4.2), using
Eq. (1.3.8), commuting the Operations V andgfi'on i, we
get:

Kat—(17» 11.4.7)

t4
1
”ts
(D
\J
H
c:

which becomes:
V'sze , (1.4.8)

by a similar reasoning which led to Eq. (1.4.6).

1.5. The Final System Of Fundamental Equations

Following (9, p. 14) we state two ways to describe
the final system of equations:
(a) Equations Of state, continuity, momentum and energy
for each fluid component and two Maxwell's vector
equations, i.e., Eqs. (1.2.1), (1.2.6), (1.2.8), (1.2.14),
(1.4.1), (1.4.2) where s runs over all n fluid components.
These are 6n+6 equations for the 6n+6 unknowns (T,, p,,'
(1,, ui, E‘, 11‘).
(b) Equations Of state, continuity, momentum and energy

for n-1 fluid components and for the gross fluid, and two

34

Maxwell's vector equations, i.e., Eqs. (1.2.1), (1.2.6),
(1.2.8), (1.2.14), (1.2.29), (1.2.30), (1.2.31), (1.2.50),
(1.4.1), (1.4.2) where s runs over n~1 fluid components.
These are 6n+6 equations for the 6n+6 unknowns (T5,p”%” u: ,
1, p, P, u‘ , E‘, H‘).

Additionally let us consider a system of equations
for a single fluid. Most Of the magnetohydrodynamical
problems which were dealt with up till now consider the
plasma as a single fluid. The electromagnetic phenomena
due to the existence Of differently charged particles and
a relative velocity between them in the plasma, was taken
into account through the introduction Of additional
variables, like the excess charge density,[%, and the
electric current density, 3} The final system Of equations
is supplemented by the equation of conservation Of charge,
(1.3.8), and by Ohm's law, (1.3.27). In this formulation
no injection Of momentum and no mass sources are present.

The final system is:
p=RpT, (LSJ)
+ 3,117.11)- 0, 45.21
g11111..§_,,11111111= —%§, F (1.5.3)

‘gt—(é- MP0) +3—)(éu’+ +1ou‘—— PQU)- (131:3, (1.5.4)

(1.5.5)

(I.5.6)

(1.5.7)

(1.5.8)

These are 16 equations for the 16 unknowns (T, (9’ (D, u‘ ,

E“, H

4'.

9 (De, J‘ ), with 6 being known.

CHAPTER II

QUASI-THREE-DIMENSIONAL STEADY FLOW

11.1. The Governing System of Equations

From this chapter and on we deal with the quasi-three-
dimensional steady flow of each component fluid. The basic
hypothesis for such a flow is that all quantities are
independent Of x3 and t. The equations valid for this
type of motion can be Obtained from the governing system
of equations given in Chapter I.

Eqs. (1.2.6), (1.2.8), (1.2.14), (1.4.1), (1.4.2)

become:

"3%;(F5U94r 1731135162“) 6., (11.1.1)
%(p,u:u;)+§%(psd;u§)= 5154+ F. +6.2: X: (11.1.2)
fi(p,u§u§)+5§y(Psu§ui)=--W 9X: +511? 65Z: X:,(II-‘-3)
5%(P.u:u:)+—937.—(t.ui 113.)= F: + Cali-Xi, (11.1.4)

37

9X‘( 5‘5?ng 11.: ”atrial/(1). ”111‘“ 11i-(D.O.11§+1p,11§o) :-

=uéF,‘+u:F: +u2F2- 11:? ‘ 11:31:: ~111X§+

 

 

 

 

 

 

+a.[I.—Q.—-:-1u.>’-+ 1122M: 2.11:2:1, 1......)
2E3 , 9E3 . :E2 at“
aleO, 9x1: , 6?-ERT:O, (II0106)
3H3 __ 1. :H3 _ 2. 8111 8H1 _. 3
3x2..J’ , ax1""J , 81?"”Y""J. ’ (11.1.7)
1
where XS, (i:1,2,3), is defined by the equation:
1‘. 31-11 3X12
X5 : “9‘15 *' -a—:X;ZE . (110108)

From the first two Eqs. (11.1.6) it follows that:

E3 2 constant (11.1.9)

TO determine H3, the first two Eqs. (11.1.7) are

used. From Eq. (1.3.8) we have:
_.....---.2. (II01010)
which, together with the first two Eqs. (11.1.7) is a

necessary and sufficient condition that -J2dx‘+J‘dxz be
the total differential of the function H3:

38
dH3= JWLW-—J2dx'. (11d.11)

Eq. (11.1.11), when integrated from a 0 subscripted
initial point to some end point along an arbitrary path

gives:
H3=HZ +(O(J4dx2—J20(X‘). (11.1.12)

It should be noticed that the first two Eqs. (11.1.6)
determine E3 only, and the first two Eqs. (11.1.7)
determine H3 alone. By having four equations determining
only two unknowns our system of equations becomes
underspecified, i.e., the number of unknowns exceeds the
number Of equations, unless two additional equations can
be supplied to the system. We note that the two electro-
magnetic equations (1.4.6), (1.4.8) which were derived from
Maxwell's equations (1.4.1), (1.4.2) cannot be derived
from those equations after one putsiéz 0 there. In our
case we will take them into account, thus making our total

system Of equations Specified by having the two additional
equations, using Eqs. (1.4.3). (1.4.4):

59F(/1EH1) + 99x2</“6H2) = 0 , (11.1.13)
$4,151) . 534.132) = 21.. (11.1.1111

The system Of governing equations consists, therefore,

39
Of Eqs. (1.2.1), (11.1.1) to (11.1.5), the third Eqs.
(11.1.6), (11.1.7) and Eqs. (11.1.12) to (11.1.14), which
are 6n+5 equations for the 6n+5 unknowns (T5, pg, 1%, u; ,
ug, ug, E‘, E‘, H‘, H2, H3).
We note that the third Eq. (11.1.6) is automatically
satisfied by the choice of a function wfixi, x2) such that:

9:933: ; E1=37:)(5. (11.1.15)

Inserting Eq. (11.1.15) into Eq. (11.1.14) and assuming

€z= constant, we get:

-é_;§g+(§%z= Pee“. (11.1.16)

Similarly, assume we: constant and introduce a function

§(x‘, x‘) such that:

H1==-%§5 ; H2=:%g% (11.1.17)

,

then Eq. (11.1.13) is automatically satisfied. The third
Eq. (11.1.7) becomes, using Eq. (11.1.17):

gip+ (973-172 : J3 . (II.1.18)

Eqs. (11.1.16), (11.1.18) will be used tO determine the
functions n(x*, x2), §(x‘, x“) reSpectively.

In the case where e is not assumed to be a constant

4O
(11.1.16) is replaced by:

.:.1e::1+:.<ea1=1e. 1......)

When IQ is not assumed to be a constant, the function f

 

is assumed to be such that:

feHi=‘%§1; few-5% , (11.1.20)

and Eq. (11.1.18) is replaced by:

t0? 38531185311133. ”“2"

We will assume in the present work that air, are constants.

11.2. Quasi-Stream-Function and Potential Equations

In Eq. (11.1.1) the functions 6% are known and given.
Suppose that there exist functions BS, by means Of which
the functions cg may be expressed in conjunction with (k

and ué, by the formula:

(11.2.1)

6,=-37(P, LL 13,)

         

with the condition that when there is no mass source into

the s-th fluid component, i.e., 13520 , the trivial solution

41
B8 = 0 should be taken. Inserting Eq. (11.2.1) into Eq.
(11.1.1), we have:

7371(1)}.43 QT,((D,11,=) 0, (11.2.2)

where P, is given by:
Ff 3,043,). (11.2.3)

We introduce the quasiwstream function %gxi, x‘) such

that:
(11.2.4)
then Eq. (11.2.2) is automatically satisfied by this

choice Of 5.

Eqs. (11.1.2), (11.1.3), remodelled, using Eq. (11.1.1),

 

 

yield:
5 3 -
11; 2:): + 11: 333‘; =-[o_;,‘—§—)%i+ {1“F§+Y , ; 312) (11.2.5)
where )3 is given by:
1311.16.12:- 111.) - X3]. (11....)

Setting 3 = 1 in Eq. (1.2.5) and multiplying the equation
by -u§, then setting 3 = 2, and multiplying by ug, adding

42

the resulting equations, furnishes, after a few manipulations:
— 1 aft. 2 an _ I{L1 9 a
“50"- 9x* L15 9715) ‘ (a 52:)“ + (D;("’L 5 9x: “ii—£2 +§u F:

142$)" Lt: Y: - LL: Y5}, (11.2.7)

 

where as , (A): are given by:

 

 

(if = (L12)Z + (11:)2 , (11.2.8)
3 9425 9H:
(1)., = 911‘ — 9111 . (11.2.9)

The term A): is the 1:3— component of the vorticity of the

s-th species, [5,, given by:
ws=Vle5 . (11.2.10)
Next, we represent the term (1); 95%, (1:3, 2), in a
different form. The enthalpy of the s-th fluid component,

15, can be written, using Eqs. (1.2.5), (1.2.44), (1.2.54),
in the form:

IS = crs(i+D5)T5 , (11.2.11)
where 0,15 , D5 are given by, using Eq. (1.2.58):

CTN}: C35 + RPS = Kscvs , (IIO2012)

43
D5: K;‘[A.+(KS=1)W.]. (11.2.13)

Substituting an eXpression for T5 from Eq. (1.2.47) into
Eq. (11.2.11), and inserting the resulting equation into
Eq. (1.2.12), differentiating the product, using Eqs.
(1.2.70), (11.2.12), (11.2.13) and solving for 32%fi3,

furnishes:

93WM1wX1/1) 3:49 w.:,—-3,[Xo)(1w.)(+.)"1A)
«41.30: (P N59331: . (11.2.14)

In Chapter I it was already assumed that As, W5 are,

 

possibly, functions of Psand some other variables
independent of 3k. Let us denote those variables by the
symbol.Ԥ: (i=1,2,...,k$), then we have:

35(3[113(1+W)()(;1+A)‘=]W35U§13(1+W.)(1+A9193%+
)1.

+11 535—[103(1%)1]§§;- (11.2.15)

Substituting Eq. (11.2.15) into Eq. (11.2.14) and using

 

me-

Eq. (1.2.71), we get:

323 =(K.-1)(1W.)(1A) 90“ 331‘ 06.9333+ +Z,L (11.2.16)

3X3

where Li is-given by:

3.

L}: 1
. 11911 .31

  

44 .
[(1111 W)(1+A.)"] 351

(11.2.17)

 

Eq. (11.2.16), remodelled with the use of Eq. (11.2.3)

gives:

3.333% (-1)+((1<. WW)“ As”) a). Pé‘(ws)2§3§-
«0692311939191 + L3, (11.2.18)

 

From Eqs. (11.2.4), (11.2.8) we have:

(3.51.916 i)2 + @3392. (11.2.19)

Differentiating Eq. (11.2.19) with respect to x1 and
multiplying the result by -u:, then with reSpeot to x2
and multiplying the result by ué, adding the resulting
equations and using Eqs. (11.2.4), (11.2.8), we get:

113133;). - 2.1111: .33— .1191 39—3. = (1119:3911 - 11:33)
+3.115(u.§%%-‘ _ (123%) (11.2.20)

Solving Eqs. (11.2.4) for u;, u; and inserting the
results into Eq. (11.2.9) it can be shown that:

“92.3% =psw:+(—$' (33.) +3335: (11.2.21)

45
Inserting Eq. (1I.2.18) into Eq. (11.2.7), substitutu
ing the resulting equation into Eq. (11.2.20) by elimination
of the term u:3%§-:Ué§%3, and using Eq. (11.2.21) to
substitute for the term LQ“%%~1 253% , we get, after

rearranging and using Eqs. (11.2.4), (11.2.6):
[(0692 ’01:)? (93%; ‘ 211.311: 33:32:33; + I: (065)2' (”92] T333237 =
= -34..)1... + 3: (399311.411111/.)(11A.)“--3—~.%2 — 1.: +

+ (065 Z 9_33X3_[1031(1”Bs)] + P:[F3 + 6521: _ X: J} 3 (II.2.22)

Eq. (11.2.22) is one possible form of the quasi-stream
function equation.

Next, we derive a quasimpotential equation by
introducing the two functions(&(x1, x2),3%$x‘, x2) such
that

lib—333:3. ; (j=i,2). (11.2.23)

It is readily seen from Eqs. (11.2.9)9 (II.2.23) that

gs must satisfy the equation:
.9. ‘1; - 32:1. -.- a): (11.2.24)

We add the condition that if ageao, the trivial solution
gs: 0 should be taken.

46
Setting 3:1 in Eq. (11.2.5) and multiplying the
equation by -u;, then setting 3:2 and multiplying by -u§,
adding the resulting equations and using Eqs. (11.1.1),
(II.2.2), (11.2.3). (II.2.18), (11.2.23), we get, after

rearranging:
[(06924 (us)]['3%z+§ ‘31] “Mafia—121133317 %&+%)+
1(1):.)- (11313393. 3.3133151 (1121*

z —3o;1F$—Y.]3 (11.2.25)

 

1(1):)()<.-[MW)(1A)

{-(1

Eq. (11.2.25) is one possible form of the quasi-potential
equation.

We assume that the body force, 1;, consists only of
the electromagnetic force, 1.. , which is given by:

—.

Ff2.33:5:391.115“? +/‘e(asxfl)]3 (11.2.26)

Inserting the correSponding components of Eq.
(11.2.26) into Eq. (11.2.22), using Eqs. (1.3.2), (11.1.15),
(11.1.17), (11.2.4) and dividing the resulting equation by
(06,): leads to the final form of the quasi-stream function

equation:

47

[1 (“6955“)ng -éZLL u§((oc)Z—9335—+[1-(uéwg‘)]izfi’$—

2W 3111(2))(1)2 :

" 1.1: + )Z (3 3) 1111111111 + (11.111211111111111 > ‘ 331‘

-1 3 5 3
+105 (652. "Xs) - 1.5)} (11.2.27)
Inserting the correSponding components of Eq.
(11.2.26) into Eq. (11.2.25), using Eqs. (1.3.2),
(11.1.15). (11.1.17), (11.2.23) and dividing the result-

ing equation by (agfi leads to the final form of the
quasi-potential equation:

[1-(uiocg‘)][m%z .312] (1:11: (11. (25%,— 2,51%)
111+ 1111313 11% 11‘
1123—11: 1111—15 1 (1 (1)1 11:1: 1:11-11:11)
+p;‘[e.Zi -Xi-6s(%¥?+ 32)] ii}. (11.2.28)

11.3. Generalized Bernoulli and Crocco Equations

Eqs. (11.1.1) to (11.1.4) combined with Eq. (1.2.12),

and written in vector form, give:

48
:VQS_VIS+(D:FS+YS’ (11:03.1)

 

9—5:—

—9

where the components 1: ,(j: 1, 2 ,3), of the vector Ys are
given by Eq. (11.2.6), and the operator j%'operating on

an s-subscripted quantity denotes:

d._u13 I429:“

af—-' us‘ir + u.3yi us ‘V . (11.3.2)
Let the position vector of a particle of the s-th

fluid component be denoted by 5;, then the velocity of this

particle, fig, is given by:

.1 i1
u5= 11$ . (11.5.3)

 

Taking the vector dot product of Eq. (11.3.1) by 3;, using
Eqs. (1.3.2), (11.2.26), (11.3.2), (11.5.3), multiplying
the resulting equation by dt and integrating along a
streamline of the s-th fluid component from a zero sub-

scripted state to some end state, we have:
1411.) 1 £111) “311: 11. 11.11. H, (11.3.1)

where Hso is given by:

1450= %'(u.Jz-*J.O (11.5.5)

(11.3.4) is the generalized Bernoulli equation.

49
From Eqs. (1.2.70), (I.2.71), (II.2.11) to (11.2.13)

we have:
15(065)—2 = (i’rP_.)(1<.—i)'i , (11.3.6)
where P5 is given by:
4
P. ={ (1 + W.)(1+A.)“ + (1.11.111): 11111111111} -1 . (11.3.7)
Inserting Eq. (11.3.6) into Eq. (11.3.4) we have:
21111.):+(11.)2(1+P.)(K.-1)11011-1101311 101.11% 11..., (11.3.8)
where H50 is given by:
_ 1 _

H50- 7(uvso)z +(MSO)Z(1+PSO)(KS-1)1e (11.3.9)
Eq. (11.3.8) is another form of the generalized Bernoulli
equation.

Let.A,be a streamline of the s-th fluid component
which intersects the x1 -x‘ plane at the point A,.(x;.,
xf. , 0). this streamline satisfies the equation of
differential type:

dxé:dx::dx2=u::u;:u: , (11.3.10)

Let drs be an arc element of A. and let dr; be an element

50
of arc length taken along the curve C which is the pro-
Jection of A? on the x’-x‘ plane. We note, from Eqs.

(11.2.4), (11.3.10) that Q is a qg = constant curve.

3

X

( 35(1; ,X: 91: )

/

 

a:xz

 

 

 

 

 

 

 

 

 

Val
2WJ A.(x,‘ .x; .0)
'(h= constant

 

Fig. 1. Streamline Geometry

Let 35(x;, xi, 1:) be some end point on flu, and let

A5(x;, xi, 0) be the projection of B6 on the x*-x‘plane.
Let F, be any vector function defined in the three-dimen-
sional space and depending on the coordinates (1‘, 1‘)
only, i.e., for a pair (2‘, x2), )1, is the same for any x”.
Denote the two unit vectors, tangential and normal to the
F; curve, in the x‘-x2plane, by §,, h. respectively,

and let E be a unit vector in the x’direction. The{

components of jg in the above mentioned directions are

51
denoted by (fig, [Q3 /Q), and the components of the velocity
vector 1, in those directions are (3,, 0, ug). Using
Eq. (11.3.3), the relation as = £3: and the fact that all

functions depend only on the (x‘-x2) coordinates, we have:

8. _. 1 B. 1 g __1 g _
(A... /1.,-d16 =)Aso(/"$'u5)us M's: 5:;(fle Us)“ 0111, (11.3.11)

and
V( )1: f5 ”9) ‘ V( 5:11. "1191110119411. 1191131.»

A,

+3—31T.[):‘:(fu) ujaLFJfiS =:(/1 [1: uifl;‘)§s.§%—S[A3fg 1141111111.. (11. 3. 12)

Subtracting VHWJ] from both sides of Eq. (11.3.1),
using Eqs. (1.3.2), (11.2.26), (11.3.4) and the identity
for steady flow: % - V [%(LL5)2] = 5341:, we get:

1135111”. = —VH..+ VQ..-).v( (05011;) — v( (1.11;) +

As As

+YS+5$[E+/18(115Xg)] -. (II-3.13)

Substituting for E in Eq. (11.3.13) the expression quEéfi
which can be obtained when using Eq. (11.1.15), noting
that (DV'VI'dfiWl-vlo, using Eqs. (11.1.9). (11.3.11) where

-> AC
1”; is taken to be equal to k, and using Eq. (11.3.12)
where 15 is substituted for }C, we get in Eq. (11.3.13):

52
wsxu5= -VH.,.,+VQ.. +3571). 1.133 V“ 1121:1112) —
P50
1311:1131 — 3%:[10115-111'1y1q1. + 1:15 + 131+

Q

+3.E3’2 +),/141151111). (11.3.14)

It may be worthwhile to notice that the zero subscripted
functions are constant along streamlines and, in addition,
are not functions of x3, thus, zero subscripted functions
are constant on the cylindrical surfaces qg:= constant.

The component of Eq. (11.3.14) in the direction he

is:

3" _ 3 fl; = __9____Hso+ 9050 _ 3 3-4 ’
(.05 LL LLS 971:5 9714+ 9715 ((55—1971: 65 5——E 597—1;(gou's “’6 dr‘
Q

- .3972“ (25.115) a: 1111].. Y:+36/118(E5x-H)n , ”(11.3.15)

where (fi‘x flit is the component of 15x E in the fis

direction. Solving Eq. (11.3.15) for a@ we have:

GT, 9% 59h

1.34:1 3—H» 10—» 5—1-). 111111-11)

1 Gus
_94—71.[),(Yfla;)a: (1,1411 +Y:+ mix/fl 5,111+) ”12111—3. (11.3.16)
Q
Transforming from ( R, n.) variables to ( R, Q,) variables,

we have, using Eqs. (11.2.4), (11.2.8) and the relations:

’J_X_ __ ‘4. 9X:_ 1- -1.
911, 11:11,, 11..

 

9 ‘ ._ 9 SE: 9 5 3X1 _ — —
9%5— 9 9n. + 9x2 9115‘Psuvs- (II.3.17)
Using EQ- (11.3.17) we have:

_1_=

3 ,9 s =- — _3__
3715 atysfi P9u53(+)5 0 (II03018)

 

Eq. (II.3.16) becomes, using Eq. (11.3.18):
(1):: Psi-fig};— +d'Q:o ”3%-fiE393—q1Uu2-Hdr)
[W° 112:1] 1115-11“ s 1s/4e1fisx9)“]*

2112
+LL: 94);}. (II03019)

 

 

Eqs. (11.3.16), (11.3.19) are two possible forms of the
generalized Grocco equation. Transformong from (F;, n,)
variables to (x‘, 1‘) variables, we have, using Eqs.

(II.2.4), (II.2.8):

 

z 5
_9_=’3X£_’3_ + _9_'X =P5'1E‘1(§-(£—L+ 33%???) (II.3.20)

3%: 9n. QX‘ 971,371 QX‘ 31111111

and for the component ”2 of an arbitrary vector,/'l',:

= "riflfit/«i * $33), ”I'M”

where f1,‘, f: are the components of /15, in the directions
1 , x reSpectively. Using Eqs. (11.1.17), (11.2. 4),

54
(11.3.20), (11.3.21) in Eq. (11.3.19) we get:

2

a»: = 12(— 33‘ 353: + 21-33?) ‘31313‘2 333133-31?"

 

 

 

fa
33x3“ uiujdf‘s) - 5%33WY5'L335'1dF53+ Y1-
‘35fe3FQIHB%+ ”2%; W) 3x31}. (11.3.22)

Eq. (11.3.22) is the form of the generalized Crocco

equation which will be used in the present work.

11.4. The Final Quasi-Three-Dimensional System

There are two ways in which the governing system
of equations can be formulated, depending on whether one
chooses the quasi-stream-function or the quasi-potential
function as the unknown to be solved for. In the present
work, the.first formulation is given and dealt with.

The quasi-stream function equation, (11.2.7), becomes,

using Eq. (11.3.22) and rearranging terms:

—(u:oc; H633?- Zu: 31:04; 973%}: [1.3 (ugwgflgafigfi
[.Psa;1(.g_31:° + 2%) + :;3%3%)3£7330331-B5)3 +
+1131 >111 >31 1 3—3: ~31- M 33 331»

55
335

1313-333 1211:3331 13:31-33“ 3
3 L"1"‘1333113113311311313+“ H333-
-3o;‘6523133;‘Xi)333 3 (13.4.1)

 

Multiplying Eq. (11.1.4) by 1-Bs and using Eqs.
(1.3.2), (II.1.1), (II.1.17), (II.2.3), (II.2.4), (II.2.26),

we get:

13 33— 33 13: 33113 31313 1313

+33’Bs)[6s3Z:‘LL:)‘X:] . 1 (11.4.2)

 

Eq. (11.1.12) becomes, using Eqs. (1.3.2), (1.3.6),
. -6 l 3 6 lo
(11.2.3), (11.2.4) and the relation dqg-fik—dx +34%]; .

=H:+;313o33—B53-1033" (11.4.3)

It may be worthwhile to notice that in the case of a
fully ionized plasma (n22, Egao) Eq. (11.4.3) can be

integrated and one gets:
2
3
+33 1 ; 353333.333). (13.4.4)

Using Eqs. (1.3.2), (1.3.5) in Eq. (11.1.16) we get:

 

Gzn + 32" - _
(9x82 (9x11): __ E 12 x5 P5 . (11:04.5)

Eq. (11.1.18) becomes, using Eqs. (1.3.2), (1.3.6):
31 31 _ W
#41 563?“ ggsfguz. (11.4.6)

Eqs. (1.2.1), (1.2.59), (11.3.4), (11.4.1), (11.4.2),
(11.4.4), (11.4.5), (11.4.6), are considered as a system

of 5n+3 equations for the 5n+3 unknowns 1,, p., p.,(%, u: ,

H3, 3, f.

11.5. Incompressible Flow

We distinguish between the following two cases of
incompressible flows:
(a) Gross fluid is incompressible (P = constant). In
this case the component fluids' densities, P5, need not
be constant, only their sum is a constant, since P=Z;Ps.
(b) Each fluid component is incompressible ( Rs: constant;
6 = 1,2,...,n). In this case the gross fluid is incomp-
ressible. In the present work only case (b) is considered.

To derive a quasi-stream function equation for this

type of flow we start from Eq. (II.2.21)*, and using Eqs.

 

* Note that .1 is not a constant in the incompressible
flow due to existence of mass sources or sinks, de, in
this type of flow.

57
(11.2.3), (11.2.4) and the conditioneqzconstant, we have:

2

z
T337337 3. $33, = 13.3»: + ;3‘§333‘33133‘333'B133 (11.5.1)
Equation (11.5.1) is the quasi-stream function equation
for the incompressible flow. w: will be calculated from
the generalized Crocco equation which was derived in
Section 3.

The quasi-potential equation is obtained from Eq.
(11.1.1), using the condition P5:constant and Eq.
(11.2.23):

925 l 35 s "‘1
33333113331 1131 » 111-1"

where gs is calculated through Eq. (11.2.24) and one of
the forms of the generalized Crocco equation which was
given before.

In the single fluid formulation it is possible to
obtain the stream function equation and the potential
equation for the incompressible flow, from the stream
function equation and the potential equation for the
compressible flow, by setting the velocity of sound in
each of those equations to becw. this result carries
on to the present type of flow, for if we set czs=°°
(s=1,2,...n) in Eqs. (11.2.27), (11.2.28) we obtain Eqs.
(11.5.1), (11.5.2) reSpectively.

When a formulation with the quasi-stream function

58
as an unknown is chosen, in addition to the quasi-stream
function equation, (11.5.1), one has to consider in the
incompressible flow Eqs. (11.4.2), (11.4.3), (11.4.5),
(11.4.6), and n Bernoulli's equations, with the accompany-
ing condition. R5: constant. In order that the system
of equations for the incompressible flow be Specified,
there is a need for n additional equations. Those must
be supplied by means of n given relations between the
pressures, p5, and the temperatures, T5, for each fluid
component, given, for example, in the form of the

equations:
f5(p5,T9) = o; s:1,2,...,n. (11.5.3)

Note that Eq. (11.4.5) in the present case contains
a known constant on the right hand side, thus, if prOper
boundary conditions are given for n, this equation
"decouples" from the system of equations and one may solve
Eq. (11.4.5) for q first, and use the result in the quasi-

stream function equation where n appears.

11.6. Non-dimensionalization

We introduce the following non-dimensional quantities,

denoted by the symbol N:

59

x‘=x"L; ns=%$L; F.=F.L, (11.6.1)
i NC - 2
u5=u5 u‘fo"o ;u'5=u'$u’$¢, (II0602)

5.31.1; H‘=1"1‘H., (11....)

$115.1. ; Ei=EiEoo , (11.6.4)

_ 1’!

P9: FSPS‘”; Ps=fisP5~(usw>2 ; T5:LT_$T500; P5=P5P5m , (11.6.5)

cps-1).)..u..l_; «1.3.11.4. ; 3.4321»... (11....)
Q5: 65 (wavy. ; HS = HSoo(|/LSoo)2 , . (II°6‘7)
a): = C3: IJ.,.,.L.1 , (11.6.8)
65 ' gs Paco use» L1 3 X: =X: £35000;is [—4 ;
i. "’ ' “’4'. -
Zs =Z: usao; Y:= Y5 )Dsm(u’$m)2 Li, (11.6.9)
~ 1 w L z 4
.=o¢.w.. ; L. = L, (u...) L , (II.6.10)

where Ts"° , F5” p5,, , u... , H, , Em , 065,, L are some
standard temperatures, densities, pressures, velocities,
magnetic and electric fields, velocities of sound and
length reapectively.

Using Eqs. (11.6.1) to (11.6.10) in Eqs. (11.4.1),

60
(11.4.2), (11.4.3). (11.4.5), (11.4.6) we get:

[1111:3211 .)1—,1g.- 21:1: (52; 11..)23—31‘13. +

 

*[1 (113533 ”Mtg-)7 = 1.11.)“(—%:—flj 93215:) .
+E:(2%)(23’15H13U- 9))"(55:Maw)2((K.-1)(1+1+..'W)(1+A)909x;

1‘

 

 

~

 

 

 

g3 11: — 3. 1:1” 1-11.1113 1.131111% 3,. -
"-55%- :;2) + (1‘B5)[55( 3—332) ’32.] , (11.6.12)
(‘13 =11: + Z R... 1011-13.)“ 111. , (II.6.13)

2N ‘ " ~ 2 31 a
V )1 = 4: RMR; RC: (0.; ‘7 =W+ % , (11.6.14)
70
V2? = :Rmsfifii , (11.6.15)
$1

where M5”, Rms, RHS, RC5, RE., are given by:

61

VLm= u.~AQ:, (Mach number) (11.6.16)
R7115: 6.»...11...L HZ, (1312311133: 33:16:30 (11.6.17)
Raf 112(1):»)2 (>2: L125. , (33151116230 pressure (11.6.18)
R1) ,..1....)2, 1:111:13) 1......)

- _ -1 -1 -1 (electrical field
R55 ‘ [ac/(e LLsoa oo , parameter) (II.6.20)

REng'EsE‘: E:, (11.6.21)

where c is the velocity of light, and E: are the components
of the standard electric field vector, 1;. Using Eq.
(11.1.9) we choose E3 = E: .

The correSponding incompressible equations to Eqs.
(11.6.11) to (11.6.15) are obtained by setting H5, = 0,
Fs=:1 in those equations.

11.7. Linearization

We shall treat the problem of flow under uniformly
applied electric and magnetic fields such that a first
order small perturbation theory is sufficient to describe
the flow field in the neighbourhood of the origin of the

disturbances in question.

62
The velocity vector of each one of the fluid compo-

nents is assumed to have the following components:

I 1 ,
11. =u.‘.+ 11., ; ué=u:. . 112w; , (11.7.1)

where uéw is the velocity of the undisturbed uniform
flow of the s-th fluid component and ug+ (121,2,3) are
the perturbed x‘ components of the velocity reSpectively.
4
We assume u; »u., .
The externally applied magnetic field has the

following components:

.1

H as H:

= constant ; (121,2,3). (11.7.2)
Hence the resultant magnetic field has the components:

H‘ = H: + H; ; H;:» H; . (11.7.3)
Similarly we have for the electric field*:

E = E, + E? ; slaw E; . (11.7.4)

We also assume:

(as = P5w+ Pep ; P5» >> Pep. (II-705)

 

* From Eq. (11.1.9) we have E; = 0.

63
Eqs. (11.7.1), (11.7.3), (11.7.4), (11.7.5), written

in non-dimensional form, become, using Eqs. (11.6.2) to

(11.6.5):

8; = 1 + 61+ ; 8; = 8;? ; a; = as, , (11.7.6)
8‘ = E; + fig ; EL 2 E; + E; , (11.7.7)
(‘0'. = 1+ (5.1., (11.7.8)

Ni Ni “’4'.

where 3;? , (LT’ fig, EP' H”, Em, are given by:

fig: 11111911; ; ' (11.7.9)

8; = Hgnj ; 8; = Egnj , (11.7.10)

E: = HLHj ; E; = 8:8: . (11.7.11)
The function j can be Split into the form:

fl = Nii‘ + fiiiz + fl, , (11.7.12)

where 9!", represents the perturbation part for the electric
field components in the (x‘, x2) plane. Using Eqs.
(11.1.15). (11.6.1), (11.6.4), (11.7.4), (11.7.7),
(11.7.8), (11.7.11), (11.7.12) we get:

E; : 34L; (431,2), (1107013)
In a similar manner we split g :
§= “H:§‘+H:5(‘ “g1. , (11.7.14)

and get, using Eqs. (11.1.17), (11.6.1), (11.6.3),
(II.7.3), (II.7.7), (II.7.10), (II.7.11), (II.7.14):

H1 “—39 ; Hz: 1:?— (11.7.15)

1 9x2 1“ ax‘ .

We will assume conservation of mass of each fluid
component in the undisturbed stream, i.e., no mass
sources or sinks ( 6;.2 O and, therefore, BS, = o). If
we choose the non-dimensional quasi-stream function,(Ps.

in the form:
(PS: §z+q35P , (1107016)

we get, using Eqs. (11.2.3), (11.2.4), (11.6.1), (11.6.2),
(11.6.5). (11.6.6), (11.7.1), (11.7.6), (11.7.8), (II.7.16):

%% : -(1'B$)a:1 3 %%p = (1‘ B.)(a:p*(5.,)'Bs, (11.7.17)

It may be assumed, since we are dealing with a first
order small perturbation system, that the non-dimensional

mass source of the s-th fluid component, 3}, is small at

65
least to the first order. It can be shown* that Bs is
of the same order as 3;. Thus, neglecting second order

terms in Eq. (11.7.17). we have:

N N

5 ~ Gs "’ ~
111,. =-u:,; _‘Lr -1151? *Psr'B$° (11.7.18)

9X‘ 9X‘

In order that the electromagnetic forces, fin, , be
small at least to the first order we assume that those
forces vanish in the undisturbed stream. Using Eq.
(11.2.26) we have:

-.

Em+/)e(EsmxHao)= O . (11.7.19)

The components of the velocity vectors in the undisturbed

stream are (us; , O, O). The three component equations

of Eq. (11.7.19) are, therefore:
i_ , 2_ 1 a. 3__ 1 a
Eco—0) E00-fleu'$¢o GD, Ew‘ f'eu'fw‘l‘L) S:1,2,...,n. (1107.20)

It follows from the last two equations given in Eq.
(11.7.20) that either:

ESEHSEEEEHEO, (11.7.21)

01‘:

 

* Using Eq. (A.4) which is derived in Appendix A.

66
uioo : ul‘” 2 °°° :: undo: 1100 9 (II.7.22)
i.e., the velocities of the n fluid components in the

undisturbed stream are equal.
Summarizing, from Eqs. (11.7.20) to (11.7.22) we

have one of the following two possible cases:

.0

(a) 11 = 112,, = 0.0 : um): um ; E: = O ; E: : renal-13
E: : -/uumH: . (II-7.23)

(1107024)

m
8m”
m
’11:
s w
m
0

O

(b) 1:15 13.21:,

In the present work the case (a) is dealt with.
Using Eqs. (11.7.11), (11.7.23) in Eqs. (11.6.19),
(11.6.20) we get:

RE HUM) (H32) FH‘Miffi; Rgf‘Hi.‘ (11.7.25)

We assume that the equations of state and of the
internal energy of each one of the n fluid components
follow perfect gas laws in the undisturbed stream, i.e.,
Wags A955 0. We also assume, since we are dealing with
a first order small perturbation system, that W5, A5 ,
are small quantities, at least to the first order,
throughout the disturbed flow field. Similarly, it is

85...... that X.‘.. -_-. o, and that X,‘, 2.2’.‘ (1=1.2.3). L93;

67

(3:1,2), are small quantities, at least to the first order.

Inserting Eqs. (11.2.6), (11.3.9), (11.7.1) into
Eq. (11.3.8) and using Eqs. (11.6.1), (11.6.2), (11.6.4),
(11.6.7), (11.6.9), (11.6.10) we get:

é—(asfy + 11:? + (325 Mew)2(i+P5)(Ks‘ 9-1 _ 55 =
=(Msw)TZ(Ks-i)fii, (11.7.26)

where T15? , 5, are given by:

(11.1.1 = ZWQV , (11.7.27)

3:61
i=(m1i5fiRwfigsL: 1811162211356”.+:.(Z:-11.)11.1. (11.7.28)

Note that P5 , which is given in Eq. (11.3.7). vanishes
in the undisturbed stream, since W. , is do. Solving

Eq. (11.7.26) for ((32.12 we have:
—i
(52:)-2=(11P5)(1+(Ks-1)(M5..)2 [59.11; —:-(\ 1169?} . (11.7.29)

We assume that (Ks-1)(M,.,)2Ti;1, , %(K_., -1)(M5,,)2(u51,)<<1,
thus neglecting them in Eq. (11.7.29) we get:

(-5) -2 (1 +P5()[1+(Ks 1)(Me..2.)5]1. (11.7.30)

Furthermore, we will approximate 13.5 , 55 by chosen

68

constant mean values 155 , 55 reSpectively, thus having

in Eq. (H.730):
(1.): +- (1111+11.111111211“ = 10151111- 1:17.11 ’

Using Eqs. (1.3.2), (1.3.6). (11.1.17). (11.2.3).
(11.2.4), (11.6.5). (II.6.6), (11.6.16) to (11.6.18),
(11.6.20), (11.6.21), (11.7.1), (11.7.5) to (11.7.8).
(11.7.11), (11.7.12), (11.7.14), (11.7.16), (11.7.17),
(11.7.23), (11.7.25) in Eqs. (11.6.11) to (11.6.15),
noting that Q5“, H5“ are universally constants, i.e.,

‘%%? E €§ETEE , neglecting second and higher order terms,

assuming ii? is small at least to the first order, 33}: fig,
and making the following changes in notation: E‘ = x,

£1: y,LP ZP‘jsr q);,u:1,=ws,H;:-.Hz, fir'n , ff‘f,
(5.15 0. , 5,: a.,Zézzsx,z§=z%,23=z,,,x,=x,,,

"L

x; = 11%,, 11;: x,,, ’0’, = 0,, 1.5 = 1.5,, '11; = 1.3, we get*:

F;%%+%;% FeeRm-iRHS‘g‘P HR... H.111 “msR ngflifih}

+(1.211.111H.+R..R,$R.a(1-F;)%§r+11., ' 1......)

3—1" R..R,g1é+,,11-11 1.11.135, +R..R,,;B, +

 

* In order that the equations which were derived
above comply with the linearized case, the 0 subscripted
state is taken at the undisturbed stream, 1. e., the
subscripts 0 and1n are assumed to be equivalent.

69

 

wig—X”, (11.7.33)
H; 2: RMQV, +£(1-BJ1113]; 3?} = :Rm+%%, (11.7.34)
qu= : RngfiR: 11...), (11.7.35)
V’f =:me,, (II.7.36)

where F5”, RH: , G6 are given by:

Pem‘j- (~6 HMsoo) , (13.7.37)
RH ‘ 5=RH:H.: ='/(. ”H... ~ .103. (LL... , (II.7.38)
G.=-35§-{13.R+..R+.-B {1.4.5.1.— X.,) 1. S [a.(z.- 1)-

-X.x] dx} (1'P$°°)[(Kd)(1*ws)(1*As)-igas L53} (”'7'”)

In the incompressible case thC). (r:1,2,...,n).
Taking the derivative of Eq. (II.7.35) once with respect
to x and then with reSpect to y, and commuting differen-

tiation signs we have:

V2(%:H =0 ; Vz(%‘) =0 . (11.7.40)

Assuming that the perturbed electric field, E;, vanishes

at large distances from the origin of disturbances, we

7o
obtain, using Eqs. (11.7.10), (11.7.13). 35%}: 0 at
large distances. Using this result and Eq. (11.7.40)
we have:

9 _ 9
9% ‘ a} " 0 everywhere, (11.7.41)

i.e., in the incompressible case, the perturbed electric
field vanishes identically everywhere. We assume in the
compressible case, that the right hand side of Eq.
(11.7.35) 18 approximately a constant and, therefore,

the same result is obtained for this case.

CHAPTER III

SIMILKRITY OF FLOWS

III.1. Approximate Governing System of Equations

To obtain similarity rules we apply, additionally
to the linearization, a procedure which could be called
an equalization of the order of terms, i.e., we will
further simplify the governing system of equations

neglecting very small terms.

Let m“ = min.(m¢) and assume furthermore that*:

4
mdm,<<1; r:1,2,...,n; r #66; e,6 #0 . (III.1.1)
In addition we assume:

RQZRE; 029.12le << 1, (111.1.2)

where 63, is obtained by using Eqs. (11.6.17), (11.6.18)
in Eq. (III.1.2):

 

* Thecx-th fluid component may be the electron
fluid. ,

71

72
Qg=e..(p..,.m5: e“)? . (111.1.3)

$L,is called the plasma frequency of the w-th Species*.
Assuming that the number density of each fluid
component at w is the same, we have, using Eqs. (11.6.17).

(11.6.18), (11.7.38):
1Kmfl¥<;:‘= 3+6: ; ‘{H*¥{::=qn47fl: §

‘

RHiRH:=%¢m:H; :5 (“1.2.35 +=1.2.....%). (111.1.11)

Taking the derivative of Eq. (11.7.36) with reSpect
to x, substituting Eq. (11.7.33) into the resulting
2
equation, dividing by RMRH‘, using Eq. (111.1.4),

commuting the Operators V2 and £1 and rearranging, we

get:
.1311}; V‘fiH ‘ (251‘ )2 (6f 63 mm?) =

= Z { e: e;zm,m:[H;H3—e% " (1:11:19? * H: H:B+] +
+ 61. e: meH‘ (61.21} ‘X+2)} . (III.1.5)

Neglecting small terms in Eq. (111.1.5). following the
assumptions made in Eqs. (111.1.1), (111.1.2), we have:

 

* Similar assumptions to the ones above can be
found in (10, 11).

L—H: H: 7.1 — H: H1111— - H: H.111.-
—Z 9.9.: meH‘(61Z12-Xn). (III.1.6)

Inserting Eq. (111.1.6) into Eq. (11.7.33). using

Eq. (11.7.38) and rearranging furnishes:

LU. =RmeRH:(%§(Di-%§i()—)+RWSRH:(%%-%)+RMSR1fi-(9BTB)

Tb

_ e5 ejmwm: Z(:e+(6+Z+z‘Xn)] + 65Z52 —X52 . (III.1.7)

Taking the derivative of Eq. (11.7.32) with respect
to x, substituting Eq. (111.1.7) into the resulting
equation and dividing by 11.3.3.1 , we have, using Eq.

(111.1.4):

RniRK-‘(Ps «ab—qu+ ”(33352): szmeseéH: H: 7n.m:Rmi Hug—3%
[3.2. esemeHZ'mimf {Hm H:( (3x‘1%%)+
+H1H.1(%§(1-%1) H.:-H.1.(R —R.)

-— .111. :1. e 1.(.z. 1.)) R1.R.:(.1z.- x.))—

 

- _ .. 61
~e.e;: .1. mgH1H1 :.R.: ...(S,w.11)+
P9
TD
- _ - - 693
+ 5100 6.18.: ma‘mSile-w H: 2 (e1. 3);) +

tq

74
-3 R“ LG:
1m. H. ex . (111.1.8)

Neglecting small terms in Eq. (111.1.8), following the

assumption made in Eq. (111.1.2) we have:

.1. R1. 1: e: H; H: 111: .121) H:H.:1(%?(1 - 3.131%
«111111111118 +H.:H;1(R.-R.) -

_ Rx“ R11: ;[E—r €;1(6+Zu-X12)]+ R711:R1):(6-121‘.tmx:11+z)}9

'11
+ :90 8.8’imxm?R.1.R.:;(e.%%) = O . (III.1.9)

For sfiw, Eq. (111.1.9) becomes, neglecting small terms
in view of Eq. (111.1.1):

'7).
291%:01 (111.1.10)

1+1

or, using Eqs. (11.6.17), (11.7.34), (111.1.10) we get:

1x (111.1.11)

LHl‘O,

and since 32:0 in the undisturbed stream we conclude

that:

H2 2 O everywhere. (111.1.12)

75
For Szw, using Eq. (111.1.10) in Eq. (111.1.9) we get:

; e+(6+Z1-2 _X‘hi‘) = ew(6fi6Zw2.-Xw2). (III.1.13)

Inserting Eq. (111.1.13) into Eqs. (III.1.6),
(III.1.7) we have:

3 H211)- H: H: 131- H; H.113. —
R;R..:(6.. Z... Xi), (111.1.14)
33—1 =RR. R(—311.-—3—3+ MM (131 331) +R..R.(B. -.+B)

+6eZsz“Xs. for s #06 + w. = o . (111.1.15)

,

(11.7.32) becomes, using Eqs. (11.7.41), (111.1.12):

2 924’s + 82w5_

Fees 9x3— 38,” FstmeRméQ— 93.5 F:me9RH: we
”RmsRH: $6" 111. (ix) + 6.. (111.1.16)

Eqs. (111.1.14) to (111.1.16) will be considered as a
system of equations governing approximately the n-
component fluid flow. The equations of this system can
be solved separately by solving first Eq. (111.1.16) for
Q3, after which Eq. (111.1.14) is solved for .§, and Eqs.
(111.1.15). (111.1.16) can be solved for ig,(#. (six).

76
111.2. Correlation of Flows

The system of equations governing approximately
the compressible flow of an n-component fluid will be
correlated to n systems of equations governing
approximately the incompressible flow of a correSponding
n-component fluid. By correlation it is meant that there
will be derived simple relations between correSponding
functions of the compressible and those of the incom-
pressible flow° This procedure can serve, then, as a
first approximation in the correlation of the more general
system of Eqs. (11.7.32) to (11.7.36). Having this first
approximation one can try to derive, by an itterative
procedure, a higher order correlation, using the system
of Eqs. (11.7.32) to (11.7.36).

The equations for the x-th fluid component and each
of the s-th fluid components separately are coupled
together and will be taken as one system when compared
to the incompressible flow. We will denote functions
and coordinates in the incompressible flow by primes.

The equations for the s-th and w~th fluid components in
the incompressible flow are, from Eqs. (11.7.39):
(111.7.39). (III.1.15), (111.1.16), (here is {35: = 1,

(an: :1):

RT

91 , 91; ' 2 5’ 7 '
$5 + 3—39;— =Rm9RHi 13%— - R7115 Rt): 7H,; -

wgdx)+G§, (11:.2.1)
1.11m .431 133,912-11 Mg— 1.31) +.:R;.R (B B2)
1622;. *X... , (111.2.2)
17.1—1thme WM)
0 = 11: R..R,;.R;- 6:2.» X5;
1ti5.[6;(z.;—1)-X;H+}; ww- n+1-2+)

1;

 

The equations for Hg, y; have both Laplacian
Operators on the left hand side. If we want to correlate
the equations for 4k,(%, of the compressible flow to the
corresponding equations in the incompressible flow by
means of a linear transformation between the (x, y) and
the (x’, y') coordinates we have to introduce one further

approximation by introducing a weighted mean value Ffi.:

2.

F9m=ESP52mf(i-ES)P¢ZOO; 05 Eséi, (111.2.5)

where e; is a weight factor, the value of which must be
known. We approximate in Eqs. (11.7.39). (III.1.15),
(111.1.16) for the s-th and the w-th fluid components

1

the ’85: and the a. by F9: , and get:

78
(113111—913 W162R.R,.1.1g (5.:R..R.,.;6 —

—R,.R,,.1g-(L6.Bx) + 6., (III.2.6)
1—1—1=R-R (111— 111) RR .(21 2—‘(1)+R,.R.;(B.-B.)+

+6.Z..-X.., (111.2.7)
(1.2%(w 1.3117: R.,R,,;11— ,3 6., (In...)
6.=— —1——B1 1.. :R..R, R (a: (:.Z.,—X,,)+-.1g( (1.12.1)-
-X..] d1}1(1-pf.)[(R.-1)(11W.)(Mlgm L 11:]; (=1 (5,1) (111.2.9)

We correlate the system of Eqs. (111.2.6) to
(111.2.9) with the system of Eqs. (111.2.1) to (III.2.4).

Introduce the following linear transformation:
X = x; 3: {55003, (III.2.10)

and assume that the dependent variables in the two

flows are related by:
Lys=a.(y;; ngéws’; (p.=a.((1.’, (111.2.11)

where as, b5, a06 are constants which will be determined.

Using Eqs. (111.2.10), (111.2.11) in Eqs. (111.2.6)

79
to (111.2.9) we have:

—)g)(7- gig: stR’MSRH 33% HartlfiRmRH :21.

m6 '1 asfimeRH; gig-(Lu; dx’) ’1 01':ng G. , (111.2.12)

'1.

9w; - a z 3 :6 _ _ 3 :5 3 0,6
37'— : I351 RmRH; (as—3%: _ 0... 33") + (36" flsiRmRHg(053% - ad???) +

 

(.1R,.R,;(B.-B.)+B:(s.z.—X.), (In...)
34:11: .11: = .-.R,..R 16+ afi”i.(111.2.11)

a: F; = " 0,; {35: 95T§f+a+ m-RH2B1- — a: (6+Zt3-X113) +
1a: F111‘3H7H~[11(Z11” 1) ~X..] dx’} 1
+ .1 . 2 . 11.1.9.1-
0.,F...(i-P..)[(K.-1)(1+W+)(1 A.) 33 L13]; ($5.06). (111.2.15)
In order to correlate Eqs. (111.2.12) to (111.2.15)

with Eqs. (111.2.1) to (111.2.4) we require the following

relations to exist:

(.1: R..RH;=R.;.R;; ; a.(}.“R..RH;=RQ.RQ;, (III.2.16)
(1.0: :RnsRHfRésRfi; ; a.(r;‘RmsRH;‘-'R(5Rfi; , (111.2.17)

FmRmsRH: =R:«+R;: ; P; 0:: 6,»: 6;, (111.2.18)

80

RMRH;(B;B»)=RJSRJ;(BQ‘BL)3€‘(6SZSIX,.)=6§Z;-X;2, (mm)

where r = s,w . In addition we require that the curve

r; be transformed into the curve r2, i.e., projected
streamlines in the compressible flow transform into
projected streamlines in the incompressible flow. This
leads to the so-called "streamline analogy" (8, pp. 180,
181), which requires that Qg be transformed in the same
way as y, thus, using Eqs. (IIIo2.10), (III.2.11) we have:

_-1 I
a5= Pan. (111.2.20)

From either the second Eq. (III.2.16) or the second

Eq. (III.2.17) we get:

a¢=a5. (III.2.21)
From Eq. (III.2.16) we have:
35a? '1 unless RmsRH:E RJSRfii'é-O. (III.2.22)
From Eq. (III.2.17) it can be seen that;
’f’s 0: = Fem unless R'msRH: E szsRég 50. (III.2.23)

Since the first Eqs. (III.2.22), (III.2.23) are conflict-
ing each other we will limit our correlation to the

81

following two Special cases:

1 H130.(R R E 2 2 SQ) 3 4__
Case (a): Hana a” , ms H: RmRH: , sas ‘Fem- (III-234)
Case (b): H:5H:’EO$ (RmsRHfRn:3R;leD),”Quiet-‘1. (111.2.25)

Using Eq. (111.2.20) in Eqs. (111.2.24), (III.2.25) we

have:
bs = 1 in case (a) ; bs :‘E1 in case (b). (III.2.26)
Q
We assume the following relations:
rmfimL; e,= e; (haw); u, = LLL;
_ ’
Ye,“ 9:. ; L‘ L . (111.2.27)
Thus, in order that the relations in the first Eqs.
(III.2.18) be established, we assume, using Eqs.
(II.6.17), (II.6.18), (III.2.27):

Fecal—i: = 030’, (III.2.28)

and, in order that Eqs. (III.2.16), (111.2.17) be satis-
fied we choose, using Eqs. (II.6.17), (11.6.18), (III.2.27):

Hi=H:’ ; Hi= Hi, . (III.2.29)

82
To satisfy the second Eq. (III.2.18) we assume, using

Eqs. (111.2.4), (111.2.15), the first Eq. (111.2.18), and
Eq. (III.2.20):

B1,:B; ; 61.: 6a;- ; 6rZ+x= 4:21;; Fsmng'fx= &;Z;3;

X-Hs'X-Lx , stX+3=X;3;

Féfli—p&)[(K+-1)U*W+)U*A5i'§§i‘ bdrm (1'=6,o¢). (III.2.30)

The eXpression on the left hand side of the last Eq.
(III.2.30) is assumed to be negligibly small since
1 - [5,: = 1 - 1 + Wham)" : (3'64‘."11I¢.,,)2 is assumed to be a
small quantity.

The first Eq. (III.2.19) is automatically satisfied
when using Eqs. (III.2.17), (III.2.24) and the first Eq.
(III.2.30). In order satisfy the second Eq. (111.2.19),

we assume, using Eq. (III.2.26)3

65252: 6,, 25,2; Xsfx; in case (a), (III.2.31)

Feuéazsfézge; stX53=X;z in case (b). (111.2.32)
Denoting the nonndimensional perturbation velocities

in the y direction by v;, we have, using Eqs. (III.7.18),
(III.2.1o), (III.2.11), (111.2.20), (111.2.21):

83
Emu: W 3 (FM). (111.2.33)

A summary of the relations between correSponding
quantities of the compressible and the incompressible
flow, when the s-th fluid component is correlated, is

given below in a table form:

Table 1. Correlation for the s-th fluid component

 

 

 

quantities
Compressible Incompressible
1n+,e+,u,,ys,H:,H:,H: mksémhyéfiffifififi 3'
L 2. 3 U __ ’ 31 )
Ego=0)Eoo,E°o’L ”:0,P6p:, on,L
9», 15,13“ «5+, eZu, {231113, psi: vi, 3,1,5; 42,1,

6,2,, X“, X? {3; 6,32,; , Xi, , (5;: Xé,
Case (a): 1175,6152”,st 715;, 65’ Z532L ,Xs;

Case (b): 2J5, 532551,st F’s: w; , P526;25;3;Xs;

 

 

 

 

where r:s,x .

For the correlation of the weth fluid component we
consider Eq. (111.1016) for s:x combined with the second
Eq. (III.1.15). We also consider Eq. (III.1.14).

Introducing the linear transformation:

X =.1 ; ‘fr-1m‘1, (111.2.34)
and assuming;

(For: 0.x 05%); a = (III.2.35)
where a“, b are constants which will be determined, we

get, using Eqs. (11.7.41), (III.1.15), (III.2.34),
(111.2.35), in Eqs. (11.7.39), (III.1.14), (III.1.16)

for s:%:
32;
jig—Jr 9.312%: waRmeH: jx-gfi— — a: Patina (III.2.36)

 

Cari; Goo = " a: Fa;oo QB +0.0, RW‘RH :ocB " 0.1%ng Xx3)+
+ 1:,” fijgexzw- 11 - xxx] 11+} +

+aZP+i[(-1(i 1111111131?“ 1411, arm-371

%§+—=‘a PH: Higg— 3x7 a.(3“H:H;‘ €35?-

 

-1 11:11: 13,- 1111,: .311 (Hz... 11,) (mass)

Requiring a streamline analogy we have:

a,=-P;, (111.2.39)

85
In order to correlate Eqs. (III.2.36) to (III.2.39)
twith the correSponding equations for the incompressible
:flow we will require the following relations to exist,
‘using Eq. (III.2.39):

)

PmRmRH; = meR;: 7. F2126“: G; (111.2.40)

£4ng H: H: = HZ’H’: ;1}’11,113.23Hm‘Hi’=HI;1 , (111.2.41 )

Yi‘Rfii incl“ m3 Hfi6Z,,;5‘RmRH “X; ,, (111.2.42
We assume the following relations:

41mm; e,=e;; 11m=u;; »m=11;; 15L);

meH2- H5; Hot=H.:,; H:=H:, (111,2,43)

From the last three Eqs. (III.2.43) we have:

= 11:01 1+ F;Z(1+p§m)( H:’ 11:12] $- , (mg...)

In order that Eq. (III.2.41) be completely satisfied
we use Eqs. (III.2.43), (III.2.44) and choose:

’6: PM.“ [3;oo(1-P “X: H?) J 2:. (III.2.45)

86

The first Eq. (III.2.40) is satisfied by the assump-

tions made in the first six Eqs.(III.2.43). In order

that the second Eq. (III.2.40) be satisfied we assume,
using Eq. (III.2.37) and the first Eq. (111.2.40):

8.553.; 6,261; daze“: 6.1ZL.;

meéwagz 6::Zia. 3 Xxx: XLx; Pume3r-Xo:3,;

-i 2 ‘1_Q£
Paco(i-wa)[(Kw’i)(1_+w¢)(1+/A\oo) 33% +L“8]%D, (III/2.46)

From Eq. (II.6.17), (II.6.18), (III.2.44), (III.2.45)
we have:

11mm; 11;: ,1: =9...

(111.2.47)
Using Eq. (III.2.47) in Eq. (III.2.42) we have:

606206} = (3;; 60:20:} ; X062 = Pecfa-oxcc; . (11102-48)

Summarizing the correlations of the w-th fluid

component in a table form:

87

Table 2. Correlation for the math fluid component
quantities

 

Compressible Incompressible

 

H1 2 }+3 , 1 1 1 U H2, 4. 3’
mw’ew,uw’yw) 00, w, m, mw)ew,u/ijw, m) m ’ “U U,

E:=O,E:, 3,1. E;’=o,{1;:E:,E:',1_'

 

z 3’ ’— J
1+: 1:1:11111112111111» 1111+
Bead“ 6¢Zxx,6oon3 Bl)é.:,6.:Zo:x ,F;: 612023,
6.42012 ,XoLX7Xp¢8’Xo(3: Fiodzaiz, Xéx,F>;toXo2%, [371: a;

 

 

 

 

III.3. Pressure Coefficients

In this section we invalidate the changes in
notation which were introduced for the perturbed
quantities in Chapter II, Section 7.

The s-th Species' pressure coefficient is defined by:
_ -i —2
Cfs- 2(T35-10500) boo H1500 . (III0301)

Bernoulli's Eq. (11.3.4) becomes, using Eqs.
(11.3.5), (11.6.2), (11.6.4). (11.6.5), (11.6.7),
(II.6.9), (II.6.17), (11.6.18), (11.6.20), (11.7.1)

88
(11.7.4) to (11.7.11). (11.7.23), (11.7.25). (11.7.28),
neglecting second order terms and using the result 3-:

constant which was obtained in Chapter II, Section 7:
CLJP+IsLL7§-5s= Imuif , (111.3.2)
where:

15”.: = Cf5(1+Ds)Ts LL: ; mu: :CPSTS LL: , (IIIo3o3)

:1.=1.+1.+1..+i.; 1.5100111; ; :1.» 11.11.: (”11+ ;

N

a d
_. N3 ”L . . ~ Ali-D N 5. Ni
fies—RWSRng Herdsx , 354’s [64215 1>+Xs]d«x . (III.3.4)
1*.” 11
Note that D50.55 0 since each of the fluid components
follows perfect gas laws in the undisturbed stream.

From Eqs. (1.2.70), (1.2.71) and the relations A...,w..§o

we also have:

(055.)2" K510500119; . (111.3.5)

-1
Using the relations 13500505: =R1osTSoo , RPs=C1os"Cve, Ks: Cps Cite ,

we have:

(04,.)2 = c,.(1<.—1)T.. (111.3.6)

Using Eqs. (II.6.16), (111.3.5). (III.3.6) in Eq. (111.3.3)

 

89
we get:

[.1113 = (1+D.)(1<.—1)‘1(M..)’2 T. T: ;

Law: = (Ks-1)-1(M.a)-2 . (111.3.7)

Inserting Eq. (III.3.7) into Eq. (III.3.2) and solving
for T,T,;,1 furni she 5:

T5 T5: =(1+Ds)-i[1+ (Ks‘i )(Msw)2 (56 -,u+:f>] (III.3.8)
We introduce the following approximations:
LRJ: 0LT.“R.LT;‘0)T. ; )mst.‘dP.“Ws)w(5:dP., (111.3.9)

where K6, W5 are some mean values representing A9, W,
in those integrals. Using Eq. (111.3.9) in Eq. (1.2.57)

we have:

6. =(T. Tsj-A) (p.p;-)"‘"”ws W¢P(_As). (111.3.10)

Inserting Eq. (III.3.10) into Eq. (I.2.56), making use
- _ ‘ A

of the relation Pepei=PsTj5a§ TSooTSt1+w5) which is obtained

from the equation of state, and solving for P5 3ywe

get:

p.319 =(T. T..‘)K‘HK‘Q (H) )W‘) “R 1W.) mp(‘A.+c?.AS.), (111.3.11)

90

where 55,A5, are given by:

— ~ _-

De= K:[/A\5+ ‘1)Ws] ; A85: 85— 8500 . (III.3.12)

Inserting Eq. (III.3.8) into Eq. (III.3.11) gives:

10510;: = { (1+D)-1[1+ (Ks‘1)(M5w)2(js - Eli-9]] Ke(K,-i)'1(1+fis)(1+WS)-i .

111/1W111...s11m1+11+111+1 . 1......)

We make the following approximations by eXpanding the
power and eXponential expressions in Eq. (III.3.13),
neglecting higher than first order terms and using Eqs.
(1.2.53). (11.6.71. (11.6.16), (111.3.4), (III.3.6) and

the relation K.=c..c;¢. :

WW: (“D-+111" ~1- . .. 1+1, 1......)
[111511115115 11.11““5 MMMI 1.1115111. 31...), (III 315)
(”Meme defied-1K.111+WJ1111+(K.—i)“A.-

"c..(1<.—1))T:0)0.” 110151;)11‘1 K.(M..o)3e.(111.3.16)

Inserting Eqs. (III.3.14) to (III.6.16) into Eq.
(III.3.13). neglecting higher than first order terms and
using Eq. (II.2.13) and the first Eq. (III.3.4) we get:

91
13510;: =1+ K51Msm1213esz. + 5. ’11:..1. (111.3.17)

Dividing Eq. (III.3.1) by f5, and using Eqs. (II.6.16),
(III.3.5), (III.3.17) leads to:

61.5,: 21111910523111 K: Mgi=2135f iflriiQ. (111.3.18)

Eliminating Tgmtin Eq. (III.3.11) by means of the
equation of state, using the relation £05930: 1 +156? and
Eqs. (III.3.12), (III.3.16), neglecting higher than

first order terms gives:
136 Pa: x 1+ 1453551” +WS—AS + K5(K5‘i)11v1$oo)2 354. . (III.3. 19)

Using Eqs. (111.3.4), (111.3.17) in Eq. (111.3.19) and

solving for fi$ we get:

152111501 (6— m. .. —u:.1- K21w.—/\.1. (111.3...)

”‘61

Inserting Eq. (III.3.20) into Eq. (11.7.18) we get:
2 :- =1715m)2 “1):?+(M500>2($5-K565i>-K:(W5’A$>— B5. (III-3021)

where 9‘50.» is given by:

(25m)z=1-11\’15m1a, (111.3.22)

92

In the incompressible case, the corresponding

Bernoulli equation to Eq. (III.3.2) is:
i ’ - - _ 9 ‘4. :
us,» +10; pg; uéf 35’: “pm $5“. us? (III.3.23)

where 5; is given by:

3;: 552+ 5525+ 5’»; j52=-RmsRH {Lo d1~fl§
G
553- RMRH juggdx 35’4=J[2’g(2§’—1)+X:’]d§‘. (1113.24)
at”

The pressure coefficients in the incompressible

flow are given by the following relation, using Eq.

(111.3.23):

C’- Him-105;) Lal‘ulf =ZHQ-flif’). (mm-5.25)

P5—

In the incompressible case the Eq. (11.7.18)

becomes:
gals? _ N1, ,
$2403“ LL51D “B5 . (III.3.26)

From Eqs. (111.2.10), (III.2.11), the first Eq.
(III.2.30), (III.3.21), (III.3.26) we have:

B15110 = 9:5: I115}, _ (7‘2: Macy-(35‘ K5351)+ K_i25i(w5flA5)° (III ° 3'27)

93
From Eqs. (III.2.1o), (111.2.11), (111.2.17),
(III.2.18), (III.2.20), (111.2.24), (111.2.30), (III.3.4),
(III.3.24) we have:

5’2=(F5w>2j52; 15;:j53 ; js,q:j54 . (111.3.28)

Combining Eqs. (III.3.4), (111.3.18), (111.3.24),
(111.3.25), (III.3.27), (III.3.28) we get, after a few

manipulations:

Ct“ (fiswszr’s +2 (ASH-[1- (Poi RmRHgfldez +

Q

+(Ks’1)[i'( (7mm) JL d0:- K;L(W5’As)}, (III .3.29)

The pressure coefficient for the gross fluid is
defined by:

-i -2

[fab-10mm“, LL00. (III.3.30)

It can be shown, using Eqs. (1.2.25), the first Eq.
(II.7.23), (111.3.1), (III.3.3o), that:

Cf: Z?” «:93? fZ’m-J (W “C105 (111.3.31)

521 6=L

where nm is given by:

7!.
nm=zfm+, (III.3.32)
'f‘hi

94

Similarly, in the incompressible case, we have:
3 1" i
— v y )1
Cr- gush: m)Crs, (IIIJJS)
where nn' is given by:
TL
‘n’m’= Z’M. (1115.34)

From Eqs. (111.3.29), (111.3.31), (III.3.33) and the
first Eq. (III.2.27), we have:

'3‘:

C; Z m;(n"m’){ 2;: Cf; + z sin-(pm RngJou +
5‘1 F‘
”MKS 1)—[1 (25.0) JfionLQm dJSW mAJJJ (III.3.35)

Adding and subtracting aij', to Eq. (III.3.35) and making
use of Eq. (111.3.33) we get:

C1: : CJJHW‘CJJ :m;JC,:5[1-h:x)2]+
+—2[[1(ps)‘gJRmRHfide +( (-—Ksi)[i (AssHJQols
" K;1(W5‘Ae)JJJJ (111.3.36)

,

where (’Ag)z is given by:

(%:)2=i-(Mm)z , (111.3.37)

 

 

1' II {III 1.1. III in.

95
and (mg)2 is the gross fluid free stream Mach number.
In the case s:w, we replace J2“ by “w in all the

formulas.

III.4. Application

Eq. (III.3.29) is applied to the calculation of the
pressure coefficients, Cfs, in a fully ionized plasma
consisting of electrons and singly charged ions (hydro-
gen ions). The subscript 1 refers to the values for ions
and the subscript e refers to the values for electrons.
We assume W;EA;EW¢EAQ§ O . (and, therefore, using
Eq. (11.3.7). P: =1“ R 5 Pei“ fi. 5 O ). Furthermore, we

make the assumption:
PdQ‘V’ PidOfiJoedQe, (III.4.1)
which becomes, approximating 9gp, by their values in

the undisturbed stream, using Eqs. (1.2.2), (I.2.16)

and neglecting me in comparison to m; :

dQ=th (III.4.2)

Following (5) we split dQ into the form:

otQ=oLQ + (quoéufid’r. (III-“'3’

 

 

Il.ul.|ll Ill-II" 1- III. I|.1:r I

96
vniere d5 is all the heat injected from outside into the
ggross fluid except the Joule heat, (J)‘°’(J)<Su)‘i is the rate
of Joule heat,<5 is the electrical conductivity of the

gross fluid and dr is an element of arc of the gross

.fluid streamline. Assuming d5 5 O, approximating (J')"'(Joc3u)‘i

by its value in the undisturbed stream and using Eqs.

(I.2.2), (I.2.16), (1.2.17). (1.3.2), (I.3.6), (III.4.2),
(III.4.3) we have:

th= (9.92)“, um cmfs‘i (111.4.4)

Assuming E; 2 1, Ge 2 O and using EQSo (I9309)!

(11.7.28), (II.7.31), (II.7.37): (III.2.5)9 (111.3.4),
(III.3.22) in Eq. (III.4.4) we get:

cps cg.[1——(Mm)]‘ +2[1— (ng Jim—W Ms) J OLE-—

N

‘UW W :JMJi at. m WJJ WP
’RMRHE'J"? NR RH _;.°Jutfdx JJJiJ . (III.4.5)

pl.

Eq. (III.4.5) becomes, using Eqs. (II.6.3). (11.6.7).
(II.6.16) to (II.6.18), (11.7.38), (III.4.4):

C? W- (HM + 2MLin'JMLw)Z]'1J(Ki-1)AgJJroLfi.-
”[AHzHZJ:dY’JJJ1 (K..- 1M) MUM JoLF+ A (H: 1121.de-

7‘.

97

-H’: Judi )JJJi . (111.4.6)

‘where Aé,"Am are given by:
Adi: (6i)2wam;iM£-:6-4; Aeri/‘eHeoLm‘z‘Mzt. (11104.7)

We calculate C; using the following values*:

 

Cé‘i; Kr: ; v..=1o‘8m‘3; [51%; 6=88-5W*""s

U.

OCLm=Zx104 m-sec"; Jail-“’1 JOLXZ=5MO-2;

PL P1

N3 ’Vi - ~49 -
mfdx = 5.102; e.= Lexie CPr. ; mg: Lumovfigo;

w

i.

[m imio" &3_.,n-a;2; Hm=102 3m. (III.4.8)

Figs. 2 to 5 show the results compared with the
isentr0pic curve Cf CH1 - (M...)2]-1 for various orienta-
tions of the magnetic field**° The relation between M“,
and Mn» is assumed to be (11, p. 10, Eq. (31)):

 

* The value of 0Q” is based on a temperature of
30,000°K approximately. <§ is calculated from the formula:

-4.
6 = 1.56x1o“xm‘"5x[1og(1.23x104xm"‘xn;°'5)] i 15%, where T av
30,000°K = plasma temperature, neew 102‘cm4'z electron
number density. This formula is given in (Spitzer, L.,
Jr., and R. Harm: TranSport Phenomena in a Completely

Ionized Gas. Phys. Rev., 89, 997, 1953.).

** The calculations were made on a CDC 3600 computer
at the Michigan State University computing center.

(III.4.9)

99

 

 

 

 

 

 

 

 

 

 

-1

 

-2 —

 

 

 

 

 

 

 

 

 

Fig. 2. ion pressure coefficient for
-- 0759 H: = ' '75‘

100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-1

-2

r
0
.1
+9
n
e
i o
00
.15
f o
f.
e
o:
C
3a
man
80
85
e o
r
P
=
n
03a
T:
o
3
g
.1.
F

101

 

 

 

0134',
4
3
r’ ".v (if;
2 \w

 

 

 

 

‘ \\\

 

01 02 03

/

 

 

 

 

 

 

 

 

 

 

 

Fig. 4. Ion pressure coefficient for
E: = .25, H: = -.25

 

102

 

 

 

 

 

 

 

 

 

 

OP;
4
3 ////////
,w
WCK'OW
/ $0"
2 ///r
/’/////A ?,B;LQELM/”’///
1 ' isenu:0pic
curve
0 . 1 K . 3 . 4 . 5 . 6
-1
-2
-3 ,

 

 

 

 

 

 

 

 

 

Fig. 5.

ion pressure~coefficient for
g: = -1.00, Hi 2 0.00 and
H: =

1.00.

APPENDIX A

Some Remarkas on the Functions Bs

Carrying out the differentiation in Eq. (11.2.1)
and using Eqs. (11.1.1), (11.3.2) we get, after a few

manipulations:
iii—I: X53 (1_Bs)]= — P2165 . (A.1)

Integrating Eq. (A.2) along a streamline of the s-th
fluid component, starting from an m subscripted state

it,

to some end state, and using the relation ug= HE_ ,

we have:

1_Bs = (1-Bsoo) MG J00 Peiuvjdsd’rs) . (A.2)
A.

In the linearized, steady quasi-three dimensional flow
we assume B500: 0, @fi P5W+P5f, u5=[(u5m+us‘f)2+ ([15sz +(Ltif)z]%,
it,” til-'5 , where dl"5 is an element of arc taken along
the curve r; which is the projection of the streamline,
13, on the (x‘, x2) plane. Using Eqs. (11.6.1),
(II.6.2), (II.6.5). (II.6.9) and the relations above,
we get , neglecting higher than first order terms:

103

 

(L3)

(AA)

 

 

10.

11.

REFERENCES

Grad, H.: General Fluid Equations. Notes on Magneto-
hydrodynamics, No. I, Inst. of Math. Sciences, New
York Univ., NYO-6486, 1956.

Grad, H.: Ohm's Law. Notes on Magnetohydrodynamics,
No. IV. Inst. of Math. Sciences, New York Univ. NYC-
6486, 1956.

Krzywoblocki v., M. 2.: 0n the Stream Functions in
Non-Steady Three Dimensional Flow. Journal of the
Aero/Space Sciences, 25, 1, Jan. 1958, p. 67.

Krzywoblocki v., M. Z., F. Horiuchi and J. Voda: 0n
the Similarity Rules in Diabatic Flow. Proceedings,
Third U. S. National Congress of Applied Mechanics,
Brown University, Sept. 1958, pp. 771-775.

Krzywoblocki v., M. Z. and J. Nutant: 0n the Simi-
larity Rule in Magneto-Gas-Dynamics. Acta Physica
Austriaca, Vol. 13, No. 1, 1960, pp. 1-18.

McCune, J. E. and E. L. Resler Jr.: Compressible
Effects in Magneto-aerodynamic Flows Past Thin
Bodies. Journal of the Aero/Space Sciences, 27,

19609 PP- 493'5030

Oswgtitsch, K.: Gas Dynamics. Academic Press Inc.,
195 .

Oswatitsch, K.: Similarity and Equivalence in Com-
pressible Flow. Advances in Applied Mechanics, VI,
1960, Academic Press, pp. 153-271.

Pai, S. I.: Magnetogasdynamics and Plasma Dynamics.
Springer-Verlag, Vienna, and Prentice-Hall, Inc.,
Englewood Cliffs, N. J., 1962.

Pai, S. I.: Gasdynamic Effects on Electric Current
Density in Magnetogasdynamics. Journal of the Aero/
Space Sciences, 29, 1962, pp. 483-484.

Pai, S. I. and S. K. 0h: Linearized Magnetogasdynamics
Based on Two Fluid Theory. (Unpublished).

105

106

12. Stratton, J. A.: Electromagnetic Theory. McGraw-Hill
Book Co., Inc., N. Y., 1941.

13. Zhdanov, V. M.: TranSport Phenomena in a Partly Ionized
GaSO PM, v01. 26, N0. 2, 1962’ ppo 280-2880