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AFOSR-70-0527TR MF-61

Â»EW yORK UNIVERSJTY

â€¢50UÂ«AMT )NSTITUTE-liRP*RV

Courant Institute of

Mathematical Sciences

Magneto-Fluid Dynamics Division

Hyperliptic Magnetohydrodynamic

Steady Flow Past a Point Source

Stanley Friedlander

Air Force Office of Scientific Research Report

March 1970

New York University

New York University

Courant Institute of Mathematical Sciences

Magneto -Fluid Dynamics Division

MF-61 AFOSR-7O-O527

HYPERLIPTIC MAGNETOHYDRODYNAMIC STEADY FLOW

PAST A POINT SOURCE

Stanley Frledlander

March I97O

Research sponsored by the Air Force Office

of Scientific Research, Office of Aerospace

Research, United States Air Force under

Grant No. AF-AFOSR-815-67. ,

Table of Contents

Page

1. Introduction 1

2. Properties of the Lundquist Equations

A, The Lundquist Equations 7

B. The Characteristic Cone of the Lundquist Equations 8

3. The Steady Flow Equations

A. Derivation l4

B. The Compressive Component l6

C. The Transverse Component l8

D. Inverting the Solution 19

E. Construction of the Steady Flow Characteristic

Cone 20

F. A Note on Uniqueness of the Steady Flow 24

4. The Characteristic Polynomial 26

5. The Derivation of the Solution

A. Statement of the Problem 35

B. The Method of Fourier Transforms 37

C. The Method of Solution and an Example 43

D. Derivation of the Solution 47

E. Verification of th. Solution 62

P. The Singularity S':!.'^faces 68

6. The Solution Near tie c laracteristic Cone

A. The Solution Near ^he Planar Segment 76

B. Steady Flow Across the Planar Segment 80

C. Solution Near Remainder of Characteristic Cone 82

-111-

ABSTRACT

The disturbance field generated by a point source is

obtained for steady, three dimensional, isentropic, magneto-

hydrodynamic flow. The flow is governed by the Lundquist

equations linearized about constant velocity, matter density,

and magnetic field. The flow is assumed to be hyperliptic,

i.e., the negative of the free stream velocity lies within

the fast wave front of the time-dependent characteristic

surface but outside the two cusped slow wave fronts of this

surface. There are both real and complex characteristics.

New dependent variables are introduced which separate

the flow into transverse and compressive components. The

transverse component is propagated one-dimensionally with

the Alfven velocity. The determination of the compressive

component is reduced to finding the fundamental solution of a

fourth order homogeneous differential equation with constant

coefficients. The solution is required to satisfy smoothness

and causality conditions and a condition on the growth at

infinity.

The fundamental solution is found by extending the

plane wave representation for the fundamental solution of

hyperbolic equations to the hyperliptic case. The fundamental

solution for a hyperbolic equation is extended to include plane

-V-

waves with complex wave speeds. This function will not be

continuous across the plane z = 0. A solution for z :/ of

the homogeneous equation is then added so that the resulting

2 2

function is smooth across z = 0, x + y / 0. This function

is shown to he the fundamental solution for the compressive

component of the steady flow.

Since there are two directions for which the real wave

speeds coincide^ it is possible that the solution is singular

across the part of the plane ruled by the tangent half lines

to the line segment joining the slow wave fronts of the

time-dependent characteristic surface. The steady flow is

shown to be smooth across this surface. The singularities

of the steady flow across the forward-facing nappes of the

characteristic cone are computed.

-VI-

Section 1. Introduction

We treat steady, three dimensional_, isentropic_,

magnetohydrodynamic flow past a point source â€ž The flow

is governed by the Lundquist equations [ 1 ] which represent

a non-dissipative fluid with scalar pressure tensor _, infinite

conductivityj and in which displacement currents have been

ignored. We examine only small steady disturbances imposed

on a fluid initially with constant velocity, matter density _,

and magnetic field; that is, we employ the Lundquist equations

linearized about these quantities. It is assumed that the

flow is hyperliptic, i.e., the negative of the free stream

velocity lies within the fast wave front of the time-

dependent characteristic surface but outside the two cusped

slow wave fronts of this surface. The equations for steady

flow will then have both real and complex characteristics.

We shall find the fundamental solution for the steady flow,

i.e., the disturbance field generated by a point source.

The field quantities must be regarded not as ordinary functions

but as distributions. The fundamental solution will have

both elliptic properties (e.g. the disturbance will extend

throughout space) and hyperbolic properties (e.g. singulari-

ties will be carried on the real branches of the characteristic

cone) .

We introduce new dependent variables, following

H. Grad [ 2 ], which simplify the analysis by separating

-1-

the transverse and compressive components of the flow.

The transverse component is propagated one dimensionally

with the Alfven velocity and its contribution to the

steady flow characteristic cone is two degenerate Alfven

cones^ i.e. half lines which are the loci of Alfven wave

fronts .

The determination of the compressive component is

reduced to finding the fundamental solution of a fourth

order homogeneous partial differential equation with

constant coefficients. The solution is required to satisfy

three conditions. The first condition is a causality

condition which states that the solution should not be

singular on the backward-facing nappes of the character-

istic cone. This type of condition is well known in gas

dynamics. The second condition is that the solution is

required to be smooth in all of space except on the

forward-facing nappes of the characteristic cone. Finally ;,

we require the compressive component to vanish at infinity

as the inverse square of the distance, i.e. as â€”r^ â€” ?^ â€” py >

x^+y^+z^

except J of course J on the forward-facing nappes. We

expect the fundamental solution satisfying these three

conditions will be unique; however we do not give a unique-

ness proof.

Hyperliptic steady flow has the property that the

forward-facing nappes of the characteristic cone lie in a

half space, and we choose coordinates so that this is the

space z > 0. The causality condition is then that the

solution be smooth for z < 0. The characteristic polynomial

gives two real and two non-real complex conjugate wave speeds

in each direction. There are two directions where the real

wave speeds coincide.

We find the fundamental solution by extending the

plane wave representation for the fundamental solution of

hyperbolic equations to the hyperliptic case. If our

equation were hyperbolic, i.e. if all the wave speeds were

realj the fundamental solution would be H(z)u, , where H is

the Heaviside function. The function u-, consists of the

Laplacian (in x and y) iterated three times applied to a

superposition of plane wave functions (functions of

X cos + y sin 9 + A.(9)z where 'K.{9), i = 1,2,3^'^ is a

wave speed in the 0-direction) . The plane wave functions

used include log \x cos + y sin 9 + }\^{9)z\ , 1 = 1,2,3,4.

We begin modifying u-, by extending these logarithms into

the complex plane for the plane waves with complex wave

speeds. The resulting expression, uâ€ž, will be discontinuous

across the plane z = 0. We correct this by finding a

"smoothing" function, M, where M is a solution of the

homogeneous equation for z ^ and u^ = H(z)u2 + M is

2 2

smooth across the plane z = 0, x +y y' 0. The function M

is found by assuming it to be a superposition of plane waves

similar in form to uâ€ž. However, we only include plane waves

with complex wave speeds so as to satisfy the causality

â– 3-

condition^ i.e. so as not to introduce singularities on

the backward-facing nappes of the characteristic cone.

The weight functions included in the superposition

composing M are determined by the condition that u^ and

its first three normal derivatives he continuous across

p ?

z = Oj X +y 4 0' The function u^ and its first two

normal derivatives will then be continuous across z :=

while its third normal derivative 1:bs a jump of a constant

tim.es 5(x)5(y)^ the two-dimensional delta function^ across

this plane. We then easily exploit this property to show

that u, is a fundamental solution.

The proof that u^ satisfies the three conditions

above is straightforward. The causality condition is

satisfied by virtue of our method of selecting M and we

show u.^ is (real) analytic for z < 0. We use the Cauchy-

KowalaAiski theorem to show u^ is analytic in a neighborhood

of z = 0, x^+y^ ^ 0. We finally show by a displacement of

contour method that u is analytic for z > except on the

forward-facing nappes of the characteristic cone. The

fundamental solution is a function homogeneous of degree

one. The compressive component is composed of linear

combinations of derivatives of third and fourth order of

the fundamental solution and will thus have the desired

rate of decay at infinity.

The forward-facing nappes of the characteristic

cone consist of the two surfaces ruled by the tangent

-4-

half lines from the negative of the free stream velocity

to the slow wave fronts of the time -dependent characteristic

surface. It also contains the part of the plane ruled by

the tangent half lines to the line segment joining the slow

wave fronts. This planar segment arises because of the

presence of real double points in the characteristic

polynomial. The singularity across it is computed and shown

to vanish with the application of any derivative tangential

to the planar segment. The compressive component is composed

of linear combinations of derivatives of the fundamental

solution and the fundamental solution is differentiated

tangentially at least once in every term of these linear

combinations. The compressive component is thus smooth

across the planar segment.

For our method of solution^ we use coordinate systems

which have the forward-facing nappes of the characteristic

cone in the space z > 0. A Cartesian coordinate system

with positive z-axis in the direction of the free stream_5

such as is often used in steady flow problems, is then

suitable only if the forward-facing nappes lie completely

downstream. This will occur for equilibria for which the

negative of the free stream velocity lies outside the slow

normal speed loci. A Cartesian coordinate system with z-axis

perpendicular to the unperturbed magnetic field has the

forward-facing cone in z > for every hyperliptic flow.

Various properties of the characteristic polynomial are

-5-

derived in these coordinate systems.

The behavior of the fundamental solution near the

characteristic cone (excluding the planar segment) has

been treated in the hyperbolic case^ [3 ].

The behavior here is similar since the terms of the

fundamental solution which become singular in each case

have the same form. Our discussion is therefore brief

and serves chiefly to show the similarity to the

hyperbolic case.

-6-

Section 2. Properties of the Lundquist Equations

a; The Lundquist Equations

We treat a perfectly conducting, isentropic fluid

described by a velocity field u , density p, and scalar

pressure p which is a function of the density alone_, i.e.,

P â– = P(p)' The electric field E is given by

r= rx u^

where B is the magnetic field, since the fluid is a

perfect conductor. We ignore the displacement current in

Maxwell's equations, so

\i j^^ V X B ^ ,

where J is the current per unit area and M- is the specific

inductive capacity. The remaining equation of Maxwell's

equations is

(lA) ^B^t + V X(b'x"u) = ,

where x\fe have used E = B X u . Equation (lA) implies that

VÂ«B = for all time provided it is zero initially.

We neglect dissipative effects such as heat

conduction and viscosity. The equations of conservation

of mass and momentum are then

(IB) |^+ div (pif) =

and

-7-

(IC) P If + p(u^V)u^+ a^Vp + ^ b'x(Vxb') =

respectively, where a = ySp/dp is the speed of sound and

1

â– i- B^X (V x"^ is the Lorentz force per unit volume,

Equations (l) are the Lundquist equations for

an isentropic fluid and form a nonlinear, first order,

Galilean invariant system for the functions u, B and p.

An equilibrium solution to this system is given by

u = Up|, B = B-^ , and p = p^, where Uq, Bq and p^ are

any set of constants. We consider a subset of these

constant equilibria which we will describe later in this

section. We linearize about any equilibrium taken from

this subset and look for steady flows past a point source.

B. The Characteristic Cone of the Lundquist Equations

The subset of equilibria we consider may be

described geometrically with the aid of the characteristic

cone. This cone is the set of surfaces of singularity, or

wave fronts, which propagate from a point source in the

initial plane t = 0. We shall also use this cone to

construct the singularity surfaces for the steady flow and

to determine the proper causality condition needed to complete

the mathematical formulation of the steady flow problem. The

characteristic equation of equations (l) (the equation for

the characteristic surfaces) will also be useful. We give

a brief summary of relevant results on the characteristic

-8-

equation and cone using [^,5,6] and then describe

the equilibria we consider.

A characteristic surface {x,Y ,z ,t) = for a first

order system and for a given solution thereof is a surface

across which the solution is continuous while its normal

derivative (i.e. the normal derivative of each component

of the solution) is singular. We consider a point on the

characteristic surface where the solution has the value

u = U-, , B = B-, and p -= p-, . We use equations (l) to find

a linear algebraic system of equations for the normal

derivative of the solution at the point in terms of

u -, , B-, J and p. . This system may be found by introducing

a coordinate system near the given point with one set of

coordinate surfaces given by (t> (x_5y,z^ t ) = constant^, and

the other coordinates varying in each such surface.

The normal derivative of the solution is then u". ^ B, ^ and p^.

This normal derivative will be continuous across (x,y_, z ^ t) =

unless the determinant of the algebraic system vanishes.

The determinant is a homogeneous polynomial, ?(*,, 4> , , , ) depend on u -, , B-, and p.,, but not on

the coordinates of the point, i.e. not on the independent

variables (x,y,z,t), since the equations (l) do not depend

explicitly on the independent variables. Setting the

determinant to zero, we get the characteristic equation.

P(^^,^y>\) = [*' - (Ao-Vo)^][(0')^-(a^+A^)(

^. B' -> u' p'

|bq| Po

The motion of the fluid past a source,

(pOPs(^'')^Po^oC('^^)^^0 ^s^^^))^ ^^ Siven by the following

equations :

(4a) UqV-u^ + Uq. Vp = pg

(4b) Ug(uQ.V)u% a^Vp + AqAq x(Vxr) = u^M^

(4C) UqV X(AqXu^) + AqIuq-V)!"^ = AqB^

-14-

where Pg(x^)j, M^(x^) , Bg(x^) are C^ functions^ i.e. they

are infinitely diff erentiable with compact support^ and

B (x ) satisfies V.B^(x^) = 0. We also have the equation

V-B - 0, which is a boundary condition at infinity. If

V'B = at infinity^ then;, hy taking the divergence of

equation (^c), we find V'B^^ everywhere.

It would be tempting;, in order to find the fundamental

solution of equations (4), to consider the flow past a point

source J i.e. to replace the source terms p , M , and B^

f^s â€¢' s ^ s

in equations (4) by the functions p^5(x^), M^5(x^);, and

B 5(x ), where p^_, M . and B are constants. However_, we

would then have a contradiction, since, from equation (4g),

V.(uQVX(AjXu^)+AQ(u^.V)r) = OyAQV.r5(r) ,

unless B = 0. We will therefore proceed by showing that

every component, v., of the solution to equations (4)

satisfies the equation Lv. = q-, where L is the same

differential operator for all i and q. is a linear combina-

tion of various derivatives of the given source functions.

We then solve the equation

dÂ°^"^5(x.)

Lv =. 5^(x) 6P(y) 5^(z) , where b'^lx.) = ^-^ ,

^ dx? ^

X

and the solution to equations (4) will follow by forming

the proper superpositions. (An alternate approach for

-15-

resolving the above difficulty is given by H. Weitzner [ 8 ]â€¢)

B. The Compressive Component

We start by introducing the new dependent variables

>

0'

'a,^,^), following H. Grad [ 2 ]> with a = V-u , p - ^n''^

and Y = A^- B . These three variables and the density p

satisfy the following equations. Equation (4a) gives directly

(5A) UqQ + Uq â€¢ Vp .= p^ .

The divergence of equation (4b) gives

(5B) UQ(uQ.V)a + a^Ap + Aq Ay = u^V â€¢ M^

The inner product of A"^ with equations (4b) and (4c) gives

(5C) UqU^. Vp + a^A^. Vp = UqAJ.M^

and

,5D) u^A^a - UqAJ.VP + AquJ.VY . Aq^q * ^s

These equations are equivalent to a single fourth

order equation for each dependent variable of the form

Lv. = q., as indicated above. These equations may be found

by first writing equations (5) in matrix form as

-16-

â€” >

where A. are constant matrices^ v

} os \ rh\

a

and

r =

Rr

R-.

\ij

We may find the equation satisfied by each component

in a systematic way "by formally solving the algebraic system

1 X

'2 y

:) z-

where the differential operators D , D , D are treated as

constants [7 ]. The solution is

det (A^D^+ A^Dy+A^D^)?^^ (A^D^ + A2Dy + A-^D^ f r"

where the matrix of the transposed cofactors of the elements

of a matrix B is denoted by B . Each component of v then

satisfies the same equation but with a different forcing

term, namely'.

4

h

(6) det (A.D^ + A^D + A^Djv = y2_ 2

1 X

2 y p z^ 1

b . D, R.

imn Kmn j

operator

where b. are constants, and D.

j=l Â£+m+n=^

i = 1,2,3,4

is the differential

In other words, the right hand side of equation (6)

consists of a linear combination of the partial derivatives

â– 17-

of the R. J i = 1,2,5,4, where the degree of each term is

three or four. We will explicitly give some of the forcing

terms when we discuss the singularity surfaces of the solution

in Section 6-B.

It will then be sufficient to solve the equation

(7) det (A^D^+ A2Dy+ A-jD^)v = 5Â«(x) b^iy) 5^(z)

where a + p + Y=6 or 7.

We then obtain the solution to equations (5) by forming the

proper superpositions of these solutions, using various

values of a, p and y.

The characteristic equation for equations (7) (their

characteristic equations are identical since they do not

depend on the forcing term) is the same as that for

equations (5). (The characteristic equation for a single

Â»EW yORK UNIVERSJTY

â€¢50UÂ«AMT )NSTITUTE-liRP*RV

Courant Institute of

Mathematical Sciences

Magneto-Fluid Dynamics Division

Hyperliptic Magnetohydrodynamic

Steady Flow Past a Point Source

Stanley Friedlander

Air Force Office of Scientific Research Report

March 1970

New York University

New York University

Courant Institute of Mathematical Sciences

Magneto -Fluid Dynamics Division

MF-61 AFOSR-7O-O527

HYPERLIPTIC MAGNETOHYDRODYNAMIC STEADY FLOW

PAST A POINT SOURCE

Stanley Frledlander

March I97O

Research sponsored by the Air Force Office

of Scientific Research, Office of Aerospace

Research, United States Air Force under

Grant No. AF-AFOSR-815-67. ,

Table of Contents

Page

1. Introduction 1

2. Properties of the Lundquist Equations

A, The Lundquist Equations 7

B. The Characteristic Cone of the Lundquist Equations 8

3. The Steady Flow Equations

A. Derivation l4

B. The Compressive Component l6

C. The Transverse Component l8

D. Inverting the Solution 19

E. Construction of the Steady Flow Characteristic

Cone 20

F. A Note on Uniqueness of the Steady Flow 24

4. The Characteristic Polynomial 26

5. The Derivation of the Solution

A. Statement of the Problem 35

B. The Method of Fourier Transforms 37

C. The Method of Solution and an Example 43

D. Derivation of the Solution 47

E. Verification of th. Solution 62

P. The Singularity S':!.'^faces 68

6. The Solution Near tie c laracteristic Cone

A. The Solution Near ^he Planar Segment 76

B. Steady Flow Across the Planar Segment 80

C. Solution Near Remainder of Characteristic Cone 82

-111-

ABSTRACT

The disturbance field generated by a point source is

obtained for steady, three dimensional, isentropic, magneto-

hydrodynamic flow. The flow is governed by the Lundquist

equations linearized about constant velocity, matter density,

and magnetic field. The flow is assumed to be hyperliptic,

i.e., the negative of the free stream velocity lies within

the fast wave front of the time-dependent characteristic

surface but outside the two cusped slow wave fronts of this

surface. There are both real and complex characteristics.

New dependent variables are introduced which separate

the flow into transverse and compressive components. The

transverse component is propagated one-dimensionally with

the Alfven velocity. The determination of the compressive

component is reduced to finding the fundamental solution of a

fourth order homogeneous differential equation with constant

coefficients. The solution is required to satisfy smoothness

and causality conditions and a condition on the growth at

infinity.

The fundamental solution is found by extending the

plane wave representation for the fundamental solution of

hyperbolic equations to the hyperliptic case. The fundamental

solution for a hyperbolic equation is extended to include plane

-V-

waves with complex wave speeds. This function will not be

continuous across the plane z = 0. A solution for z :/ of

the homogeneous equation is then added so that the resulting

2 2

function is smooth across z = 0, x + y / 0. This function

is shown to he the fundamental solution for the compressive

component of the steady flow.

Since there are two directions for which the real wave

speeds coincide^ it is possible that the solution is singular

across the part of the plane ruled by the tangent half lines

to the line segment joining the slow wave fronts of the

time-dependent characteristic surface. The steady flow is

shown to be smooth across this surface. The singularities

of the steady flow across the forward-facing nappes of the

characteristic cone are computed.

-VI-

Section 1. Introduction

We treat steady, three dimensional_, isentropic_,

magnetohydrodynamic flow past a point source â€ž The flow

is governed by the Lundquist equations [ 1 ] which represent

a non-dissipative fluid with scalar pressure tensor _, infinite

conductivityj and in which displacement currents have been

ignored. We examine only small steady disturbances imposed

on a fluid initially with constant velocity, matter density _,

and magnetic field; that is, we employ the Lundquist equations

linearized about these quantities. It is assumed that the

flow is hyperliptic, i.e., the negative of the free stream

velocity lies within the fast wave front of the time-

dependent characteristic surface but outside the two cusped

slow wave fronts of this surface. The equations for steady

flow will then have both real and complex characteristics.

We shall find the fundamental solution for the steady flow,

i.e., the disturbance field generated by a point source.

The field quantities must be regarded not as ordinary functions

but as distributions. The fundamental solution will have

both elliptic properties (e.g. the disturbance will extend

throughout space) and hyperbolic properties (e.g. singulari-

ties will be carried on the real branches of the characteristic

cone) .

We introduce new dependent variables, following

H. Grad [ 2 ], which simplify the analysis by separating

-1-

the transverse and compressive components of the flow.

The transverse component is propagated one dimensionally

with the Alfven velocity and its contribution to the

steady flow characteristic cone is two degenerate Alfven

cones^ i.e. half lines which are the loci of Alfven wave

fronts .

The determination of the compressive component is

reduced to finding the fundamental solution of a fourth

order homogeneous partial differential equation with

constant coefficients. The solution is required to satisfy

three conditions. The first condition is a causality

condition which states that the solution should not be

singular on the backward-facing nappes of the character-

istic cone. This type of condition is well known in gas

dynamics. The second condition is that the solution is

required to be smooth in all of space except on the

forward-facing nappes of the characteristic cone. Finally ;,

we require the compressive component to vanish at infinity

as the inverse square of the distance, i.e. as â€”r^ â€” ?^ â€” py >

x^+y^+z^

except J of course J on the forward-facing nappes. We

expect the fundamental solution satisfying these three

conditions will be unique; however we do not give a unique-

ness proof.

Hyperliptic steady flow has the property that the

forward-facing nappes of the characteristic cone lie in a

half space, and we choose coordinates so that this is the

space z > 0. The causality condition is then that the

solution be smooth for z < 0. The characteristic polynomial

gives two real and two non-real complex conjugate wave speeds

in each direction. There are two directions where the real

wave speeds coincide.

We find the fundamental solution by extending the

plane wave representation for the fundamental solution of

hyperbolic equations to the hyperliptic case. If our

equation were hyperbolic, i.e. if all the wave speeds were

realj the fundamental solution would be H(z)u, , where H is

the Heaviside function. The function u-, consists of the

Laplacian (in x and y) iterated three times applied to a

superposition of plane wave functions (functions of

X cos + y sin 9 + A.(9)z where 'K.{9), i = 1,2,3^'^ is a

wave speed in the 0-direction) . The plane wave functions

used include log \x cos + y sin 9 + }\^{9)z\ , 1 = 1,2,3,4.

We begin modifying u-, by extending these logarithms into

the complex plane for the plane waves with complex wave

speeds. The resulting expression, uâ€ž, will be discontinuous

across the plane z = 0. We correct this by finding a

"smoothing" function, M, where M is a solution of the

homogeneous equation for z ^ and u^ = H(z)u2 + M is

2 2

smooth across the plane z = 0, x +y y' 0. The function M

is found by assuming it to be a superposition of plane waves

similar in form to uâ€ž. However, we only include plane waves

with complex wave speeds so as to satisfy the causality

â– 3-

condition^ i.e. so as not to introduce singularities on

the backward-facing nappes of the characteristic cone.

The weight functions included in the superposition

composing M are determined by the condition that u^ and

its first three normal derivatives he continuous across

p ?

z = Oj X +y 4 0' The function u^ and its first two

normal derivatives will then be continuous across z :=

while its third normal derivative 1:bs a jump of a constant

tim.es 5(x)5(y)^ the two-dimensional delta function^ across

this plane. We then easily exploit this property to show

that u, is a fundamental solution.

The proof that u^ satisfies the three conditions

above is straightforward. The causality condition is

satisfied by virtue of our method of selecting M and we

show u.^ is (real) analytic for z < 0. We use the Cauchy-

KowalaAiski theorem to show u^ is analytic in a neighborhood

of z = 0, x^+y^ ^ 0. We finally show by a displacement of

contour method that u is analytic for z > except on the

forward-facing nappes of the characteristic cone. The

fundamental solution is a function homogeneous of degree

one. The compressive component is composed of linear

combinations of derivatives of third and fourth order of

the fundamental solution and will thus have the desired

rate of decay at infinity.

The forward-facing nappes of the characteristic

cone consist of the two surfaces ruled by the tangent

-4-

half lines from the negative of the free stream velocity

to the slow wave fronts of the time -dependent characteristic

surface. It also contains the part of the plane ruled by

the tangent half lines to the line segment joining the slow

wave fronts. This planar segment arises because of the

presence of real double points in the characteristic

polynomial. The singularity across it is computed and shown

to vanish with the application of any derivative tangential

to the planar segment. The compressive component is composed

of linear combinations of derivatives of the fundamental

solution and the fundamental solution is differentiated

tangentially at least once in every term of these linear

combinations. The compressive component is thus smooth

across the planar segment.

For our method of solution^ we use coordinate systems

which have the forward-facing nappes of the characteristic

cone in the space z > 0. A Cartesian coordinate system

with positive z-axis in the direction of the free stream_5

such as is often used in steady flow problems, is then

suitable only if the forward-facing nappes lie completely

downstream. This will occur for equilibria for which the

negative of the free stream velocity lies outside the slow

normal speed loci. A Cartesian coordinate system with z-axis

perpendicular to the unperturbed magnetic field has the

forward-facing cone in z > for every hyperliptic flow.

Various properties of the characteristic polynomial are

-5-

derived in these coordinate systems.

The behavior of the fundamental solution near the

characteristic cone (excluding the planar segment) has

been treated in the hyperbolic case^ [3 ].

The behavior here is similar since the terms of the

fundamental solution which become singular in each case

have the same form. Our discussion is therefore brief

and serves chiefly to show the similarity to the

hyperbolic case.

-6-

Section 2. Properties of the Lundquist Equations

a; The Lundquist Equations

We treat a perfectly conducting, isentropic fluid

described by a velocity field u , density p, and scalar

pressure p which is a function of the density alone_, i.e.,

P â– = P(p)' The electric field E is given by

r= rx u^

where B is the magnetic field, since the fluid is a

perfect conductor. We ignore the displacement current in

Maxwell's equations, so

\i j^^ V X B ^ ,

where J is the current per unit area and M- is the specific

inductive capacity. The remaining equation of Maxwell's

equations is

(lA) ^B^t + V X(b'x"u) = ,

where x\fe have used E = B X u . Equation (lA) implies that

VÂ«B = for all time provided it is zero initially.

We neglect dissipative effects such as heat

conduction and viscosity. The equations of conservation

of mass and momentum are then

(IB) |^+ div (pif) =

and

-7-

(IC) P If + p(u^V)u^+ a^Vp + ^ b'x(Vxb') =

respectively, where a = ySp/dp is the speed of sound and

1

â– i- B^X (V x"^ is the Lorentz force per unit volume,

Equations (l) are the Lundquist equations for

an isentropic fluid and form a nonlinear, first order,

Galilean invariant system for the functions u, B and p.

An equilibrium solution to this system is given by

u = Up|, B = B-^ , and p = p^, where Uq, Bq and p^ are

any set of constants. We consider a subset of these

constant equilibria which we will describe later in this

section. We linearize about any equilibrium taken from

this subset and look for steady flows past a point source.

B. The Characteristic Cone of the Lundquist Equations

The subset of equilibria we consider may be

described geometrically with the aid of the characteristic

cone. This cone is the set of surfaces of singularity, or

wave fronts, which propagate from a point source in the

initial plane t = 0. We shall also use this cone to

construct the singularity surfaces for the steady flow and

to determine the proper causality condition needed to complete

the mathematical formulation of the steady flow problem. The

characteristic equation of equations (l) (the equation for

the characteristic surfaces) will also be useful. We give

a brief summary of relevant results on the characteristic

-8-

equation and cone using [^,5,6] and then describe

the equilibria we consider.

A characteristic surface {x,Y ,z ,t) = for a first

order system and for a given solution thereof is a surface

across which the solution is continuous while its normal

derivative (i.e. the normal derivative of each component

of the solution) is singular. We consider a point on the

characteristic surface where the solution has the value

u = U-, , B = B-, and p -= p-, . We use equations (l) to find

a linear algebraic system of equations for the normal

derivative of the solution at the point in terms of

u -, , B-, J and p. . This system may be found by introducing

a coordinate system near the given point with one set of

coordinate surfaces given by (t> (x_5y,z^ t ) = constant^, and

the other coordinates varying in each such surface.

The normal derivative of the solution is then u". ^ B, ^ and p^.

This normal derivative will be continuous across (x,y_, z ^ t) =

unless the determinant of the algebraic system vanishes.

The determinant is a homogeneous polynomial, ?(*,, 4> , , , ) depend on u -, , B-, and p.,, but not on

the coordinates of the point, i.e. not on the independent

variables (x,y,z,t), since the equations (l) do not depend

explicitly on the independent variables. Setting the

determinant to zero, we get the characteristic equation.

P(^^,^y>\) = [*' - (Ao-Vo)^][(0')^-(a^+A^)(

^. B' -> u' p'

|bq| Po

The motion of the fluid past a source,

(pOPs(^'')^Po^oC('^^)^^0 ^s^^^))^ ^^ Siven by the following

equations :

(4a) UqV-u^ + Uq. Vp = pg

(4b) Ug(uQ.V)u% a^Vp + AqAq x(Vxr) = u^M^

(4C) UqV X(AqXu^) + AqIuq-V)!"^ = AqB^

-14-

where Pg(x^)j, M^(x^) , Bg(x^) are C^ functions^ i.e. they

are infinitely diff erentiable with compact support^ and

B (x ) satisfies V.B^(x^) = 0. We also have the equation

V-B - 0, which is a boundary condition at infinity. If

V'B = at infinity^ then;, hy taking the divergence of

equation (^c), we find V'B^^ everywhere.

It would be tempting;, in order to find the fundamental

solution of equations (4), to consider the flow past a point

source J i.e. to replace the source terms p , M , and B^

f^s â€¢' s ^ s

in equations (4) by the functions p^5(x^), M^5(x^);, and

B 5(x ), where p^_, M . and B are constants. However_, we

would then have a contradiction, since, from equation (4g),

V.(uQVX(AjXu^)+AQ(u^.V)r) = OyAQV.r5(r) ,

unless B = 0. We will therefore proceed by showing that

every component, v., of the solution to equations (4)

satisfies the equation Lv. = q-, where L is the same

differential operator for all i and q. is a linear combina-

tion of various derivatives of the given source functions.

We then solve the equation

dÂ°^"^5(x.)

Lv =. 5^(x) 6P(y) 5^(z) , where b'^lx.) = ^-^ ,

^ dx? ^

X

and the solution to equations (4) will follow by forming

the proper superpositions. (An alternate approach for

-15-

resolving the above difficulty is given by H. Weitzner [ 8 ]â€¢)

B. The Compressive Component

We start by introducing the new dependent variables

>

0'

'a,^,^), following H. Grad [ 2 ]> with a = V-u , p - ^n''^

and Y = A^- B . These three variables and the density p

satisfy the following equations. Equation (4a) gives directly

(5A) UqQ + Uq â€¢ Vp .= p^ .

The divergence of equation (4b) gives

(5B) UQ(uQ.V)a + a^Ap + Aq Ay = u^V â€¢ M^

The inner product of A"^ with equations (4b) and (4c) gives

(5C) UqU^. Vp + a^A^. Vp = UqAJ.M^

and

,5D) u^A^a - UqAJ.VP + AquJ.VY . Aq^q * ^s

These equations are equivalent to a single fourth

order equation for each dependent variable of the form

Lv. = q., as indicated above. These equations may be found

by first writing equations (5) in matrix form as

-16-

â€” >

where A. are constant matrices^ v

} os \ rh\

a

and

r =

Rr

R-.

\ij

We may find the equation satisfied by each component

in a systematic way "by formally solving the algebraic system

1 X

'2 y

:) z-

where the differential operators D , D , D are treated as

constants [7 ]. The solution is

det (A^D^+ A^Dy+A^D^)?^^ (A^D^ + A2Dy + A-^D^ f r"

where the matrix of the transposed cofactors of the elements

of a matrix B is denoted by B . Each component of v then

satisfies the same equation but with a different forcing

term, namely'.

4

h

(6) det (A.D^ + A^D + A^Djv = y2_ 2

1 X

2 y p z^ 1

b . D, R.

imn Kmn j

operator

where b. are constants, and D.

j=l Â£+m+n=^

i = 1,2,3,4

is the differential

In other words, the right hand side of equation (6)

consists of a linear combination of the partial derivatives

â– 17-

of the R. J i = 1,2,5,4, where the degree of each term is

three or four. We will explicitly give some of the forcing

terms when we discuss the singularity surfaces of the solution

in Section 6-B.

It will then be sufficient to solve the equation

(7) det (A^D^+ A2Dy+ A-jD^)v = 5Â«(x) b^iy) 5^(z)

where a + p + Y=6 or 7.

We then obtain the solution to equations (5) by forming the

proper superpositions of these solutions, using various

values of a, p and y.

The characteristic equation for equations (7) (their

characteristic equations are identical since they do not

depend on the forcing term) is the same as that for

equations (5). (The characteristic equation for a single

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