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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

1 3 4 5

Online LibraryStanley FriedlanderHyperliptic magnetohydrodynamic steady flow past a point source → online text (page 1 of 5)