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^ I ^ TY ^ Institute of Mathematical Sciences

Division of Electromagnetic Researdi



Electrohydrodynamics I.

The Equilibrium of a Charged Gas in a Container




CONTRACT No. AF 19(604)-926


Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. MH - 1


Joseph B. Keller

d K^^^^^

osepli B. Keller

Morris Kline
Project Director

The research reported in this document has been made possible
through support and sponsorship extended by the Geophysics Re-
search Directorate of the Air Force Cambridge Research Center,
under Contract No. AF 19(604)-926. It is published for technical
information only and does not necessarily represent recommenda-
tions or conclusions of the sponsoring agency.

New York, 1955


- X -


The equilibrium of a uniformly charged gas in
a perfectly conducting container is studied. Equilibrium
occurs when the electric forces in the gas just balance
the pressure forces. It is found that for any container
and for each total mass of gas there is exactly one equili-
brium distribution of that mass of gas in the container.
In equilibrium the density and pressure attain their maxima
at the container surface, and they are both constant on
this surface. Furthermore, the density and pressure in-
crease at each point as the total mass of gas increases.
However, inside the container the density and pressure do
not increase indefinitely as the total mass of gas does.
Instead at each inner point the density and pressure both
approach finite upper limits as the total mass of gas be-
comes infinite* Most of the gas accumulates in a thin
layer near the surface when the total mass is large. The
fact that the density and pressure cannot be made arbitrar-
ily large at inner points by putting more gas into the con-
tainer is considered to be the main physically interesting
result *



Abstract i

1« Introduction •*•

2, Formulation of the problem 2

3. Existence, uniqueness and raonotonicity of U
the solution

U» Bovinds on the solution 9

5. Example: The ideal gas 12

References ■*■'

- 1 -

1« Introduction

By 'electrohydrodynamics' we mean the science of the motions of fluids
under the influence of electric fields. Such motions occur in vacuum tubes, where
the fluid is an electron gas, in plasmas where the fluid is a neutral mixture of
ions, in the ionosphere, etc. Most of the previous studies of these motions have
been based upon kinetic theory - the motions of individual particles have been
determined and from them the macroscopic motions have been computed. The pre-
sent investigation, however, is completely macroscopic, the fluid being treated
by the methods of continuvm mechanics. In this way it is possible to deal with
more general fluids and to obtain more complete results.

In this first article we determine the equilibrium of a uniformly
charged gas (e.g., an electron gas) within a container. This problem is some-
times treated in elementary physics, where it is concluded that all the charge
comes to rest on the walls of the container, due to the mutual repulsion of the
charges constituting the gas. This solution, however, neglects the effect of
the gas pressure, which is to prevent the gas from being entirely concentrated
in a layer at the container surface. In terms of kinetic theory, the elementary
solution neglects the thermal motion of the particles constituting the gas.

Since our analysis includes the pressure as well as the electric
forces, we do not obtain this elementary solution. Instead, we find that in
equilibrium the gas is distributed throughout the container, but its density
is greatest at the wall. In fact the density has the sane value at all points
of the wall (i,e., of each connected component of the wall), Fvirthermore , as
the total amount of gas within the container is increased, the density at every
point within the container is also increased. These results are physically

- 2 -

Our main result, however, is not physically obvioiis* It is that at
every interior point of the container the density does not increase indefinitely
with the total mass of gas* Instead, at each inner point the density has an
upper boTond which depends upon the shape of the container and the location of
the point. As the point approaches the boundary, the bound on the density in-
creases indefinitely. Thus as the total mass of gas in the container increases,
most of the gas accumulates in a thin layer near the surface. This behavior
is similar to that described in the elementary solution, which neglects the
pressure. The fact that the density cannot be made arbitrarily large at points
inside a container by putting more gas into the container may be of seme prac-
tical importance*

We also find that there exists a solution in which the mass of gas

within the container is infinite but the density at each interior point is

finite. This is a generalization of the previovis res\ilts of L. Bieberbach'- -^

and of H» Rademacher ^ -' , who proved the corresponding result for a charged
ideal gas» Various other problems pertaining to a charged ideal gas, includ-
ing a method of approximate solution for certain two-dimensional cases, were
treated by M. von Laue' - ^. In Section 5 we present some additional results
for an ideal gas*

2, Formulation of the problem

We consider the equilibriim of a charged fluid within a container.

Such a fluid achieves equilibrium when the pressxire forces and electrostatic

forces in it balance each other. In terms of the pressure p, the mass density

p, the charge density ap, and the electric field vector E, this eqiiilibrium con-
dition is

- 3 -

(1) Vp = apE ,

The constant a in (l) is the ratio of electric charge density to mass density
in the fluid. It is constant because we assume that the fluid is uniformly
charged. If each molecule of the fluid has mass m and charge e than a = e ra ,
which is the case when the fluid is an electron gas*

The fact that the charge is a source of the field is expressed by
the equation

(2) v? • E = Unap .

If the container is a perfect conductor then the tangental component of E must
vanish on the container surface S, which yields

(3) Exn»OonS,

n being normal to S»

The fluid is characterized by an equation of state which expresses
p as a function of p and the temperature. When the temperature is constant,
as we assume it to be, then the equation of state takes the form

ik) P = P(p) •

The f\inction p(p) is a non-negative increasing function defined for a ll non-
negative p, and becomes infinite as p does* The total mass of fluid within
the container D is given by

(5) M = / p dV .

Physcially it seems that the preceding conditions are the only ones
which mxist be satisfied by a cJiarged gas in a perfectly conducting container.
If this is so, then equations (1) - (5) shoiild have one and only one solution

'h '

p, p, and E for each value of the total mass M. In the next section we will
show that this is the case, and we will also determine various properties of
the solution.

3* Existence J \iniqueness and monotonicity of the solution
By eliminating E from (l) and (2) we obtain

(6) V • (p"-"- Vp) = lin a^ p ,

Since (U) expresses p as a function of p, (6) is an equation involving p alone.
Alternatively, since p(p) is monotonic, p may be expressed in terms of p by
means of (h) and then (6) becomes an equation for p. It is more convenient,
however, to introdixce instead a new function v defined by



/ *

Here p is an arbitrary constant greater then or equal to p(0). In terms of
V, (6) becomes

(8) V^ V = f(v) .

The function f(v) is defined in tenms of the inverse functions p(p) and p(v) by

(9) f(v) = Un a^ p[p(v)] .

The function f(v) is a non-negative increasing function of v.

From (3) and (l) we see that Vp is normal to the container surface
S, which is therefore a level surface of p. Then from (U) and (?) we conclude
that p, p and v are constant on S if S is connected, or on each connected com-
ponent of S if S is not connected. Assuming for simplicity that S is connected

- 5 -

we find that v is a solution of (8) which is constant on 3, Now let a denote
the value of v on S» From the existence, uniqueness, and monotonicity theorems
for solutions of (8) when f is increasing we have the following result. There
is one and only one solution v of (8) having the value a on the boundary S of
the container. This solution increases at every point as a increases. Further-
more, from the fact that f is non-negative it follows that v is a subharmonic
function. Therefore v < o throughout the container, by the maximum principle
for subharmonic functions. Corresponding results also apply to p and p since
they are increasing functions of v.

It remains to be shown that a can be vmiquely chosen so that (5) is
satisfied for any given non-negative number M, To this end we note that the
right side of (5) is an increasing function of a, so that (^ has at most one
solution a for each M. Thus we need only show that the right side of (5) is
zero for some choice of a, and that it becomes arbitrarily large as a increases.
Then it will follow that there is one and only one a for each M and thus one
and only one solution to equations (l) - (^. From the preceding remarks it
is already clear that a is an increasing function of M, and thus v, p and p,
which are increasing functions of a, will also be increasing functions of M.

To prove that the right side of (^ is zero for some choice of a, we
first let v be the value of v corresponding to p =0. This value of v is de-
termined by setting p = in (U) , which yields p(0) , and then inserting this
value of p into (?)• Then we have f(v ) = 0, as we see from (9). Th\is v ■= v
is a solution of (8) with the constant boundary value a = v , and for this
solution p = Oo But the right side of (5) is zero if p •» 0. Thus we have
shown that for a ■ v the right side of (5) is zero.

- 6 -

The preceding argument fails when v = -oo, i.e., when the integral
in (7) diverges for p » p(0) . In this case we may solve (l) - (^ directly
without introducing v, and the solution is p = E = 0, p = p(0). For this
solution the right side of (5) is zero, as was to be shown. Furthermore,
as a tends to -oo, the right side of (5) tends to zero since p tends to zero
uniformly (this is true because the inaxinium of p occtirs on S and tends to
zero) . Therefore we have shown that in this case, too, the right side of
(5) tends to zero as a tends to -oo, smd that it can be made equal to zero
by choosing an appropriate solution which, we may say, corresponds to a = -oo.

The above results, together with the fact that v is an increasing

function of a, show that for a > v we have ▼ > v • Prom this it follows
' o o

that p > and p > p(0) when a > v . Thus, as was to be expected, the density
is non-negative.

It now remains to be shown that the right side of (^ increases
indefinitely with increasing a. In order to show this we convert (8) into
an integral equation by using the Green's function G(r,r ) for Laplace's
equation. This function is defined by

(10) V^ G(r,r') = , r ^ r ,

(11) G(r,r') = , r* on S ,

(12) G(r,r') = -^^-j- + R(r,r') , |r| < B .

|r-r I

In (10) - (12) r and r denote two points in the container and R denotes a
bounded fiinction, whose bound is B. It is known that such a function G exists
and is unique, that G(r,r ) > if neither r nor r is on S, and that
G(r,r ) » G(r ,r) . By applying Green's theorem to v and G and using (8)
we obtain

- 7 -

▼(r) = a- / f [v(r')] G(r,r')dr' .


From the definition of f(v) this can be rewritten as

(lU) v(r) «= a - I411 a^ / p(r') G(r,r')dr' .

Now, Tising (Hi), we shall show that the right side of (5) becomes
arbitrarily large as a increases. Suppose, on the contrary, that there is a
constant C which bounds the right side of (5), i.e.,

(15) / p(r')dr' 00 as a -> 00 .
(/D |r-r I

To prove that (18) contradicts (l5) we introduce a sphere Q' of
radius c > and center r, and denote by D - (?" the part of D which is out-
side G" and by D /^ O the part of D inside G" . Then we have, recalling
that p > and using (l5) ,

- 8 -

». . P . \

(19) / £(i4 dr' - / £(i4 dr' . / £^4- dr'

|r-r I (/D - O-g |r-r | J^H^l |r-r |

- - + / p(r ) |r-r | d|r-r | d« ,



In the last integral da indicates the angular part of the integration. Now
from (19) and (18) we conclude that for almost all r in D, for all s > 0,

(20) / p(r ) |r-r | d|r-r jda— >oo asa-^oo»

Thus for almost all r in D, p(r) — ^ oo as a -^ oo. But then

t. I

(21) / p(r )dr->oo asa— ^oo.

Since (21) contradicts (l5) we conclude that (l^ is false. But the alterna-
tive is (21), which must therefore apply since we have shown that the right
side of (^ is zero for some value of a and that it increases monotonically
to 00 as a does*

In summary, we have shown that equations (1) - (^ have one and
only one solution for any M > 0» In this solution p > 0, p > p(0) if M >
and both p and p are constant on S and attain their maximxan values on S#
Furthermore, both p and p are increasing fxinctions of M at each point of D.
By slightly modifying the foregoing proofs, the corresponding results can be
obtained for two-dimensional containers. For a one-dimensional container (i*e.,
the region between two parallel planes) the boundary is not connected. Never-
theless the same results are obtained if it is assumed that p has the same
value on the two planes.

- 9 -

ii. Boxmds on the solution

We have just seen that p and p increase as M increaseso We will

now show that for a certain class of equations of state, both p and p are

bounded above at every inner point of D, independently of M« Th\is although

p and p increase with M, they both tend to finite limits as M becomes infinite,

at all inner points of D, This result is a consequence of the follovdng theorem,

which will not be proved here (see [I4]) :

Theorem I ; If v is a solution of (8) in D and if f(v) is positive
and increasing and



/ /


dx < 00

then there exists a function g(R) determined solely ty f(v) such

(23) v(P) < g[R(P,S)] .

In (23) P denotes a point in D and R(P,S) denotes the distance
from P to S. The function g(r) is decreasing and g(R) -^ 00 as R — f 0,
g(R) -^ -00 as R — ^ oo«

To apply this theorem to the present problem we note that if M >

then o > V and v > v , and hence f(v) > 0. F\irthermore, f is increasing.

Condition (22) can be simplified if we introduce p = p(p) as the inverse

function to (U) • In terms of p(p) , (22) becomes

/>°° dp
(2U) / — ^ < c» .

c/p(l) P^P^ ^P-Po^

Condition (2U) is obviously independent of the choice of p^ and involves
only the behavior of p(p) as p -f oo» ^or example, if p(p) increases faster

- 10 -


than a constant multiplied by p ' then (21;) is satisfied and the botrnd given

by (23) applies. In the case of a polytropic gas with adiabatic exponent y>


p(p) is proportional to p ' , Thus if y < 2, (2li) is satisfied and the bound
in (23) applies. This indicates that the bo\ind applies for most real gases.

When (2U) is satisfied, v, and therefore also p and p, are bounded above
at each inner point of D and the bounds are independent of M. Thus inside D,
as M increases p and p increase, approaching positive limits, while on S,
they tend to infinity with M, Since the bound g(R) increases as R decreases,
most of the mass is concentrated near S for large values of M. Thus the solu-
tion behaves somewhat like the elementary solution mentioned in the Introduc-
tion. The interesting result of these considerations is that no matter how
much fluid is put into the container, the density at any given inner point
cannot be made to exceed a finite limit.

The bound g(R) in (23) is determined by the following theorem (see


Theorem II ; The function g(E) in (23) is defined for any R > by
(2^ g(R) «= v(0) ,

where v is the solution of the following problem:

(26) V * I ^r ' ^(v) * < r < R ,

(27) v^ (0) = ,

(28) v 0.

- 11 -

Whenever (2I4) is satisfied, there exists a unique solution to the
problem (l) - (^ with M infinite, i.e., a solution in which the container
contains an infinite mass of gas» In this solution v, p, and p are infinite
on the boundary of the container. In fact v becomes infinite at the bound-
ary in the seune way as does the solution of t = f(v) . This latter function
can be computed explicitly by quadratures. These results follow from the
following theorem, which will not be proved here (see \ii]) :

Theorem III ; If f(v) is positive, increasing and satisfies
(22), then there exists one and only one solution v of (8)
in D which becomes infinite on the boundary of D. If s
denotes the distance of a point in D from the boundary,
then v becomes infinite at s = as does the function u(s)
defined by


(29) s

^00 r~


(y-a l_



Other bounds, both upper and lower, can be obtained from the integral
equation (13) which is satisfied by v. These bounds will involve a, the value
of V on the boundary. J'lrst we introduce the notation F [vj for the right side
of (13):

(30) F[v] = " " / ^[^r')l G(r,r') dr' .

Since f ( v) is an increasing f xmction, we see at once that F [v] is a decreas-
ing functional, i.e., if v. > Vg then F jvl < yr"»'2l» ^°" ^^^^ "^^ ^® written
as V = F [v] , and v < a since f > 0. We now define the sequence of functions

V by
n ''

(31) T^ - a, v^ - ^[Vl] * n-1,2,... .

- 12 -

From the decreasing property of F and the fact that v < v we find v, < v»

— o 1 —

By induction we conclude that the v- form a decreasing sequence of upper
botmds on v and the v„ ^- form an increasing sequence of lower bounds. These
bounds may be shown to converge to v, and in fact the existence of v can be
proved in this way.

5» Example: The ideal gas

To exemplify the preceding considerations, let us consider an
ideal gas, for which the equation of state (h) is

(32) p - ^ p .

In (32) T is the constant temperature, R is the gas constant, and m is the
average mass of the molecules in the gas« Then v becomes, if p = 1,

(33) '' ° ¥" "^^^ P •
From (9), (32) and (33), we see that f(v) is now

(3U) f(v) = ^ ^P[^^] •

Then (8) becomes


(3^ y% = !iS|il expj^ ^

If we define a new variable u by

(36) « ' H" * ^OS '"' [S]'
then (3^ becomes

(37) V^ u » e^ .


Equation (37) was studied by Mo von Laue*- -' in connection with the
equilibrium of an electron gas. He dedued (37) by using the statistical

-13 -

mechanical result that in equilibrium the density at any point is proportional
to an exponential function in which the exponent is the negative of the po-
tential en»'i'gy ftmction at the point divided by KT. The potential was then
assumed to be just the electrostatic potential, which limited the considera-
tions to an ideal gas.

Laue considered tlie problem of solving (37) for the potential u
in a half -space, where u is a given constant on the boundary plane and a
linear function of distance from the plane at infinity. He also treated
the corresponding problem for the exterior of a circle, appropriate
conditions on u at infinity* In both these problems (37) becomes an ordin-
ary differential equation which can be solved explicitly. In addition, he
observed that by means of conforraal mapping, the corresponding problem for
the exterior of any cylinder could be obtained from that for a circular cy-
linder if p = 00 (i.e., u = oo) on the cylinder. Trora our results we see
that this solution provides an upper bound for any solution with p finite
on the cylinder.

L. BieberbachL -■ proved that in the two-dimensional case within
a smooth closed curve, (37) has a xmique solution which becomes infinite
to the same order as s as the boundary curve is approached, where s
denotes the distance of a point from the boundary curve. H. Rademacher >- -J
extended Bieberbach's result to three dimensions. Both of these results
ai^ contained in Theorem III of the preceding section, which applies to
any gas for which (2U) is satisfied, and not merely to an ideal gas.
Furthermore, this theorem makes the stronger assertion that there is only

one solution which becomes infinite on the boundary at all, and not merely

that there is only one solution which becomes infinite as s •


If the container is a sphere, a cylinder, or a pair of parallel
planes, we may assume that the solution u of (37) is a function of one vari-
able only. This variable, which we will denote by r, is the distance from
the center of the sphere, from the axis of the cylinder, or from the median
plane in the three-, two-, and one-dimensional cases respectively. If u = u(r)
and n denotes the dimension, equation (37) becomes

(38) u^ + 2^ u = e^ .

rr r r

The regularity of u at the center of the sphere or axis of the cylinder re-
qtiires that

(39) u (0) = .


If we also assume (39) for n = 1, the solution in that case will be symmetric
in the median plane.

Although we have proved the existence of a solution of the equili-
brium problem for each of the three containers under consideration here, we
have not shown that the solutions for the sphere and cylinder are functions of
r only. However, this can be shown by directly proving the existence of such
solutions, making use of the ordinary differential equation (38) or the cor-
responding equation for any other gas. Then by the loniqueness theorem, the
symmetric solution is the only solution in each of these cases.

In the one-dimenaional case (38) can be sol-zed explicitly and we


(UO) p(r) - p^ sec

Here p is the value of p at r = 0. The total mass per unit area between the

-15 -

median plane and the plane r = constant is

(la) M(r) = / p(x)dx » / -^-r ^"

2na m

/ — 2 '

'2n a ra D


If the distance between the planes is 2L, and if M(L) is given, then p^ is
detenained by setting r = L in (Ul) and s
and (Ul) that p and M become infinite at

detenained by setting r = L in (Ul) and solving for p • We see from (UO)


RT ' n

• —

"5 2

2n a m p

But any finite M will be attained for some value of r less than this. Thus.

L must be less than this value, and we conclude that no matter how large M(L)

is, we have

U2; P^ < — n V •

° "" 8 a^ m L

This bound on p is a special instance of the type of bound on p found in
Section U»

An explicit solution can also be obtained in the two-dimensional
case, by following the method of [2], but no explicit solution has been found
for the spherical case. However, in this case, as well as in the others,
upper and lower bounds on the solution can be obtained from equation ( 31) •
To construct these bounds we first introduce the Green's fimction for equa-
tion (38) which satisfies (39) and which also vanishes on the container r « L.
[rhis is a simplification of the method outlined in Section h in which the
three-dimensional Green's function was employed and the integration vras over
the three-dimensional region. When, as in the present case, the solution is

- 16 -

a fimction of r only, the angular integration can be performed on the three-
dimensional Green'3 function yielding the Green's function considered above.
Alternatively, the method of deducing the bounds can be applied directly to
(38) .1 The Green's f unctiors for n = 1,2,3 are

n = l n = 2 n-3

(U3) Tl - r' r'log L/r' (r') f i^ - ^ ) , r < r'


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