Transport phenomena in ionized gases as initial value proble

Holt, E. H.

Huchital, D. A.

English

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NASA CON-I-R 
REPOR 
TRANSPORT PHENOMENA 
IN IONIZED GASES AS AN 
INITIAL VALUE PROBLEM 
by D. A. Hnch’tul und E. H. Halt 
? .” ‘, ’ /, ,j ‘, ‘: .., _I /, 
Prepared by .‘., 
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.P, ’ ,, ‘7 ; -. ,,, _.’ 
RENSSELAER POLYTECHNIC INSTITUTE i ,.r.. 
,.$;.. 
‘_ 1 .I ,- ‘.,, “. I ‘/ / : .’ Troy, N. Y. ,; I. ’ :;>y , 
f Or \.*I, ‘. ‘-Q ,’.._ ’ :
NATIONAL AERONAUTICS AND’ SPACE ADMINISTRATION . WASHINGTON, D. C. . SEPTEMBER 1966 
https://ntrs.nasa.gov/search.jsp?R=19660026033 2020-03-16T20:36:37+00:00Z
TECH LIBRARY iiFB, NM 
NASA CR-559 
TRANSPORT PHENOMENA IN IONIZED GASES 
AS AN INITIAL VALUE PROBLEM 
By D. A. Huchital and E. H. Holt 
Distribution of this report is provided in the interest of 
information exchange. Responsibility for the contents 
resides in the author or organization that prepared it. 
Prepared under Grant No. NsG-48 by 
RENSSELAER POLYTECHNIC INSTITUTE 
Troy, N.Y. 
for 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sale by the Clearinghouse for Federal Scientific and Technical Information 
Springfield, Virginia 22151 - Price $1.00 
- - 
TRANSPORT PBENOMENA IN IONIZEDGASESASAE IEITIALVAIUE PROBIEM 
by D= A. Huchital and E. H. Rolt 
Rensselaer Polytechnic Institute 
In many situations plasma phenomena vary rapidly in time. The theoret- 
ical formalism required in these cases must be capable of solving the 
revelant initial value problem. This paper discusses certain transport 
phenomena in plasmas in terms of initial value problems. Plasma diffusion 
is discussed from two points of view. If the moments of the Boltzmann 
equation are taken there results the macroscopic equations. This approach 
is shown to yield a diffusion equation which differs from the conventional 
equation which is based on Fick's law. Some gross aspects of the appropriate 
initial value problem are thereby revealed. A second point of view is 
developed by solving the Boltzmann equation using Fourier and Laplace trans- 
form techniques which permit the initial conditions to be inserted into the 
solution. This latter approach is shown to be the superior one. It is also 
used to study plasma transport in the presence of an applied electric field. 
In treating the collision operator a Lorentz gas model of the plasma is 
chosen. The solution is obtained in terms of odd and even components of the 
distribution function. It is shown that the initial condition propagates 
along a single particle orbit in phase space but the number of particles is 
depleted at the rate exp (-ut/2) where v is the electron collision fre- 
quency. Those particles which have suffered a collision are deposited in a 
residue in phase space. An asymptotic formula for the solution for relatively 
long times has been obtained. The collisional residue then contains the 
following elements: 1. the Maxwell-Boltzmann distribution, 2. a term 
describing the diffusion in configuration space independent of the electric 
field and 3. a term describing the heating of the electron gas by the 
electric field and the resulting diffusion in velocity space. 
IRTROIJJCTION 
The theory of transport phenomena in gases is largely based upon certain 
linear constitutive relations such as Fick's law, 
i; = -D VN 
which relate transported quantities to gradients in the state variables. The 
validity of such laws has been well established for steady flows due to con- 
stant gradients. 
However, many current plasma applications involve situations that may 
vary very rapidly in time, and there seems to have been little effort devoted 
2 
toward determining the range of applicability of the linear laws or to extend- 
ing the theory beyond this range. 
For example, a common description of the time varying characteristics 
,of the diffusion phenomenon is obtained by combining Fick's law with the 
equation of continuity to yield the well known diffusion equation, 
$f = D$N (2) 
Recently, this result has been criticized on the basis of the macroscopic 
plasma equations (Refs. l-3). In order to review this approach we note that 
the first moment of the Boltzmann equation is commonly written in the form 
ar 
at+ 
v.5 = -"r 
Simplifying the pressure tensor on the basis of a locally Maxwellian dis- 
tribution and a uniform temperature (for example, see Huchital and Holt in 
ref. 1) yields 
ar - + 2- 
at 
' VN=-vr m 
which, combined with the continuity equation, results in 
a$ aN 
at2 
+v,t= 2m g V2N 
(4) 
(5) 
Equation (5) is thena description of the diffusion process in terms of the 
telegrapher's equation. To more clearly illustrate the difference between 
equations (2) and (5), they have been solved as initial value problems for a 
specific initial condition as shown in Figure 1. The solutions are 
Equation (2) 
N(x,t) = 
Equation (5) 
N(x,t) = $ e- v t/2 N(x - vt,O) + N(x + vt,O) 1 
(6) 
1 - +-e 
2 ‘“‘2J,x_‘;t bt 710) & Jo( c) + .v N( Y ,O) Jo( C )] dY 
(7) 
Figure 1. Comparison of the Solutions of the 
Telegrapher's Equation. 
x 
Diffusion Equation and the 
where 
v = (2s/m)l'2 and {= i 
The major points to be gathered from the solution for equation (5) me 
1) the initial distribution splits and propagates with a finite speed 
along the positive and negative x axis, and 
2) these propagating groups leave behind a bell-shaped residue which 
flattens out slowly. 
3) the effect of collisions is to remove particles from the propagat- 
ing groups at a rate e- vt/2 and deposit them in the residue. It should 
be noted that the residue vanishes if v = 0. 
While the description afforded by these analyses represents an improve- 
ment over the conventional theory, it is still open to objections. We 
have interpreted the initial propagating groups as consisting of particles 
that stream directly through the background gas. However, the solution, 
wgy92(7L implies that this group propagates intact at a velocity 
even though we have postulated a distribution of velocities among 
the gasparticles. The reasons for this deficiency are apparent when one 
considers the nature of the flow equation. The chief problem in obtaining 
a solution was the assumption necessary to reduce the variable R . While 
the macroscopic approach can probably be improved to a certain extent by a 
more careful truncation of the series of equations , the point we wish to make 
here is that while these equations are attractive because they deal with 
physical quantities, the approach as a whole is difficult to apply to the 
problem at hand. 
An alternative, which has remained relatively unexplored, is to study 
the Boltzmann equation itself and to obtain solutions for the macroscopic 
variables by integration of the result. It appears to us that a microscopic 
approach, directly from the distribution function, is considerably more 
promising. The Boltzmann equation is, after all, a single equation in one 
unknown. To be sure, it is also a nonlinear, integro-differential equation, 
but in this case, it requires fewer simplifying assumptions to reduce it to 
a tractable form than are required by the moment equations. 
4 
SOLUTION OF TRE BOLTZMANN EQUATION FOR 
FREE DIFFUSION OF AN ELECTRON GAS 
The problem of transport variables is directly related to the symmetry 
or anti-symmetry of the distribution function in velocity space. No flow 
can result in, say, the x direction if the distribution function is symmetric 
about the v axis. 
X 
For gradients only in the x direction, the force free Boltzmann equa- 
tion becomes 
af af 
at + Vx-p 
= J(f) 
Let us split the distribution function into odd and even components 
with respect to vx9 lee-, 
f(vx) = fO(v,) + fe(vx) 
We can then write the Boltzmann equation in the form 
afO afe 
at' at+ 
af" afe If-++- = 
x ax x ax 
J(f" + fe) 
and we can show by expansion in spherical harmonics that 
J(f") w - v f 
0 
J(fe) = 0 
We now equate even and odd components of equation (10) to obtain 
af" 
at'" 
afe= -vf 0 
x ax 
afe 
at'" 
are =o 
x ax 
(10) 
Equations (11) and (12) enable us to solve the Boltzmann equation as an 
initial value problem. 
We can combine equations (11) and (12) to obtain partial differential 
equations in f0 and fe individually, of the form 
& af a2f 
at2 
+ v at = vx2 2 (13) 
- 
5 
However, it is important to note that f0 and fe are still related 
by the initial conditions. If these conditions specify the values of 
fO(x,vxA, feeb,vx~o) 
lim a/a t fO(x,vx,t), and tit o a/a t fe(x,vx,t) 
t--r0 
their relationship to each other is defined by equations (11) and (12). 
Let us consider the problem of an initially symmetric distribution func- 
tion. Then f"(x,vx,O) = 0 and fe(x,vx,O) = F(x,vx). 
The remaining initial conditions are determined from equations (11) and 
(12) to be 
lim t-0 a/ at f"(X,vx,t) = -vx a/ ax F(x,v~) 
lim a/ at fe(x,vx,t) = 0 
t-0 
Equation (13) is a form of the telegrapher's equation. Combining the 
results for f" and fe yields 
f(x,vx,t) = e- vt/2 F(x - vxwx) 
+ 
y e- 42 
2 “x 
where 
F(y,vx) &I, (A) + IO(A) do 1 
In contrast with the previous result, equation (7), it may appear that 
equation (14) predicts propagation of the free streaming group only along 
the positive x axis. In this connection, it is necessary to recall that V takes both positive and negative value so that propagation in both direc- 
t:ons is implied. 'Second and most important, equation (14) predicts that 
each velocity class diffuses at its own intrinsic speed. Therefore, a 
diffusing gas smears out due to a distribution of initial velocities. In 
addition, we note that the rate of development of the diffusion phenomenon 
is determined by the initial distribution of velocities, F(vx)> so that it 
becomes questionable to try to express the expansion of a diffusing gas in 
terms of macroscopic parameters. 
A further implication of equation (14) that might, under some circum- 
stances, be important, is the point that "hot" particles diffuse more quickly 
than do "cool" ones. Therefore, temperature gradients are immediately set 
up in a diffusing gas so that it is unrealistic, though not necessarily 
inaccurate, to discuss density gradients in the nonsteady state without con- 
sidering the associated temperature gradients. 
6 
S6LUTION OF THE BOIXZMANN EQUATION FOR 
AN ELECTRON GAS IN AN ELECTRIC FIEXO 
If we include the effects of an external field on the previous model, 
the Boltztuann equation takes the form 
*+ af+,k= at "x x avx J(f) (15) 
where R = eE/m. By a method entirely analogous to that used in the previous 
section, we find we can write 
m&f + Rg = at mv .h? -,,f” X x av, 
So that if we take the initial condition to be 
fe(x,vx,O) = F(x,vx); fO(x,vx,O) = 0 
we find 
f(x,vx,t) = e- vt/2 Rt2 F(x - vxt - 2, v - Rt) X 
y e- 42 
vx+Rt 
J 
v2 
+ 
2R vx-Rt 
F(x-& - > + QI,(A )]dy 
(16) 
(17) 
where 
A=&-dw; 
v 
Q= 
C 
(VxX‘ 
-Y+Rt 
Y)2 + R2t2 1 
112 
(19) 
Equation (18) demonstrates the role of collisions in flow in an accel- 
erating field. We note that the initial condition migrates along a single 
particle orbit in phase space but that the number of particles in the class 
is depleted at a rate e- vt/2 . These particles appear in a phase space 
residue described by the two integral terms. 
7 
Asymptotic Formula for Long Time 
In order to discuss equation (18) more fully, we will perform an 
asymptotic expansion in the neighborhood of the origin in phase space. I;et US simplify the process by writing F(x,vx) = n 6 (x)G(vx). Making use of 
the fact that G(vx) = G(-vx) and the well-known asymptotic formulas for 
modified Bessel functions of large argument, we find 
f(x,vx,t) N n G(vx - ") (z) v1/2 x X 
2 
ed- &) + exp(- yp) X R2t I 
(20) 
It is interesting to note that the distribution function is now even in velocity 
space because 
o(vx - Rx/(-vx)] 
The situation is somewhat more difficult to discuss in configuration 
space as the most interesting characteristics of the solution are contained 
in the term G(v X - Rx/vx). In order to illustrate these points, let us 
choose the specific example of a Gaussian initial velocity distribution, 
that is 
G(vx) = A exp(-Kvx2) 
Then 
fhvx,t> - nA tJrRt ec)U2 exp 2K~x - Kvx2 - KR~X~/V 2 X X 1 
1 
The first term in square brackets is the normal diffusion term referring to 
the thinning out of the distribution in space due to its initial thermal 
energy. The term in exp(2KRx) is most interestingas it is odd in x whereas 
all other terms are even. Therefore the asymptotic expansion implies more 
particles for positive x than negative x. This is certainly a reasonable 
conclusion, as the external force has been postulated in the positive x 
dirPon* 
Of course, for large x, the solution is dominated by terms in 
-x e so the distribution becomes vanishingly small at large distances from 
the origin. We have attempted to illustrate these characteristics in Figure 2 
_._._.. -.. . -... .._ -- 
l 
8 
where we have considered the initial condition of a rectangular pulse in con- 
figuration space with a Gaussian distribution of velocities. We have 
illustrated the result in terms of the particle density, 
N(x,t) = Jfdvx 
and the.total velocity distribution 
z(y) = 
f 
fdx 
Energy Considerations 
The asymptotic formula for the residue can be put in terms of familiar 
quantities if we consider the second term in square brackets of equation (20). 
This is essentially a description of the modification of the distribution in 
velocity space by the electric field. Now we can estimate the average power 
delivered by the field to the electrons by the formula 
P= u E2 
gas 
where Q 
gas 
is the conductivity of the electron gas. As an order of magni- 
tude estimate, we take 
so that the energy delivered per particle by the electric field is 
2 
<E> = +t 
We can therefore write the second term in square brackets of equation (20) as 
exp [- $1 - exp[-*] 
We see, then, that the distribution falls off according to the ratio of 
kinetic energy to energy delivered by the field. Now since all the particles 
in the residue have suffered collisions, the velocities, in this central 
portion of phase space are completely random. We can therefore define a new 
thermal energy, e(t), as being equal to the energy delivered by the field 
into random motion of the particles. 
We can continue this procedure to put the entire residue in terms of 
particle energies. Let us define the reference level of potential energy at 
x = 0, so that 
potential energy = P. E. = - mBx 
N(x,t) = jf dv 
I 
Figure 2. Development of the Initial Condition F(x,vx) = n a(x)exp(-Kvz) 
in an Electric Field. 
Then if we write the'initial distribution of velocities in the form 
-Kvx2 
e = exp(- * vx 2> 
0 
where 0 is the initial thermal energy, 
for the gesidue as 
we can write the complete expression 
f m n A & exp [- K.E* it*"*] X 
exp ( - - KR2X2) vX2 + exp (- 
2 K.E. 
e(t)) 1 (20) 
We note that the first element of this result is the conventional Maxwell- 
Boltzmann distribution, that is exp(-total energy/thermal energy) where the 
thermal energy is the value determined by the initial conditions. And as noted 
above, the last term is a modified Msxwellian distribution referred to a 
monotonically increasing thermal energy representing the randomization of 
energy delivered by the field. 
REFERENCES 
1. Huchital, D. A. and E. H. Halt, "Initial Value Aspects of Plasma 
Diffusion", Bull. Amer. Phys. Sot. 10, 212 (1965), also Tech. Rep. No. 15, 
Plasma Research Laboratory, R.P.I. n964). 
2. Shimony, Z. and J. H. Cahn, "Time-Dependent Ambipolar Diffusion Waves", 
Phys. Fluids, & 1704 (1965). 
3. Sandler, S. I. and J. S. Dahler, "Nonstationary Diffusion", Phys. Fluids, 
IJ 1743 (1964). 
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Transport phenomena in ionized gases as an initial value problem

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