Diffusion Slip for a Binary Mixture of Hard—Sphere
Molecular Gases: Numerical Analysis Based on the
Linearized Boltzmann Equation
Shigeru Takata
Department of Aeronautics and Astronautics, Graduate School of Engineering,
Kyoto University, Kyoto 606-8501, Japan
Abstract. The diffusion-slip problem for a binary mixture of gases is investigated on the basis of the
linearized Boltzmann equation for hard-sphere molecules with the diffuse reflection boundary condition. The
problem is analyzed numerically by the finite-difference method, where the collision integrals are computed
by the numerical kernel method first introduced by Sone, Ohwada and Aoki for single-component gases
[Sone et. a/., Phys. Fluids A, 1, 363 (1989)]. This is the first report in which the method is extended and
applied to the case of mixtures. The analysis is carried out for several combinations of the component gases
and the behavior of the mixture is clarified at the level of the velocity distribution functions. As a result,
the coefficient of the diffusion-slip and the associated Knudsen-layer functions are obtained.
INTRODUCTION
As is well known, if there is a concentration gradient of a component gas in a binary mixture, the diffusion
takes place. It is a relative flow of a component gas to the other, and is not in itself related to whether or not the
total mixture flows. On the other hand, if the gradient is established along the surface of a body in a mixture
of slightly rarefied gases, a flow of the total mixture is induced along the surface. This phenomenon is called
the diffusion slip (creep) and the induced flow is called the diffusion-slip flow. Since its physical mechanism
was first given in Ref. [1], the phenomenon has attracted much interest of researchers in the field of kinetic
theory [2-6].
Theoretically, the study of the phenomenon is reduced to that of a half-space boundary-value problem of
the linearized Boltzmann equation, the "diffusion-slip problem". Various approaches have been so far taken
for the analysis of the problem. They are, however, limited to those based on model equations, which are not
as successful as the BGK model for single-component gases, or those based on rough approximations such as
variational and moment methods.
In the meantime, we have recently shown [7] that the diffusion slip is a source of the ghost effect [8,9] in gas
mixtures. Thus, it can cause the failure of the classical fluid dynamics for the description of gas mixtures even
in the continuum limit. This fact gives a new importance to the diffusion-slip problem and stimulates us to
study it in detail. In the present study, in order to understand the behavior of the mixture comprehensively,
we carry out an accurate numerical analysis of the linearized Boltzmann equation for a binary mixture of
hard-sphere molecular gases. The numerical method is the combination of the finite-difference and numerical
kernel methods, the latter of which was introduced in Ref. [10] for single-component gases. The extension of
the method to the case of mixtures is the other aspect of the present work.
PROBLEM AND NOTATION
Consider a semi-infinite expanse (X\ > 0) of a binary mixture of gases, gas A and gas B, over a plane
wall (Xi = 0), where Xi is the rectangular coordinate system. The wall is at rest and is kept at a uniform
temperature TQ. Far from the wall, the mixture is at the same temperature TO and has a uniform molecular
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
22
number density no, but there is a uniform gradient of the concentration XA [or XB (= 1 — XA)] of the
component gas A (or gas B) in the X% direction. We will investigate the steady behavior of the mixture under
the following three assumptions, (i) The behavior of the mixture is described by the Boltzmann equation for
hard-sphere molecules, (ii) The gas molecules are diffusively reflected on the wall, (iii) The magnitude of the
concentration gradient of each component gas is so small that the equations and the boundary conditions can
be linearized around a reference state. The reference state is the absolute equilibrium state at rest characterized
by temperature TO, molecular number density no of the mixture, and concentrations XA and X$(= 1 — XA)
of the individual component gases.
We here introduce the notation of basic quantities: mA (or mB) and dA (or dB) are the mass and diameter
of a molecule of gas A (or gas B); IQ = [\^27r(dA)2nQ\~l is the mean free path of the molecules of gas A at the
equilibrium state at rest with the molecular number density no; Xi = JQ£^~ 1 (<y/7r/2)~ 1 is the nondimensional
space rectangular coordinate system; (2fcTo/ra A ) 1//2 (^ [or (2fcTo/m A ) 1//2 £] is the molecular velocity, where k is
the Boltzmann constant. With a = A,B, ma = ma/mA, da = da/dA, C = |CI> EC* = (m a /7r) 3 / 2 exp(-m a C 2 ),
and riQ(2kTQ/mA)~3/2Ea(XQ + 4>a) is the velocity distribution function of the molecules of gas a.
The macroscopic variables of interest are expressed in terms of (f>a as follows: for each component gas a
(a = A, B), the molecular number density nopQj*-hATa), flow velocity (2fcT 0 /m A ) 1 / 2 wf , temperature T 0 (l+T a ),
pressure PQ(XQ -fP a ), stress tensor PQ(XQ Sij -f P/p, and heat-flow vector pQ(2kTQ/mA)l/2Qf are written as
a a
E d<:, ra = H-L
a
2
f
= ^ J Cf^'X,
°
r
P?j = 2m" J CiC^-E'X,
°
F
5
Q? = J C*(m%2 - -WEad£,
and, for the total mixture, the molecular number density no(l + ^/V), density
=,J
A 1//2
velocity (2/cTo/m ) ^, temperature TQ(! + T), pressure po(l + -P)? stress tensor po(<% + Pij}->
vector po(2fcTo/m" 4 ) 1 / 2 Qz are written as
i heat-flow
= f
N
=
an(
a=A,B
^ a=A,B
a=A,B
E
a=A,B
a
a a
Q
a=A,B
a=A,B
CiO E ™ <t> E d<;, Q,= fa E ™ (c|-|^— l
J
a^,B
2
a^,B
E0=A,B
where d£ = d^id&d^s and the integrations are performed for the whole space of £.
FORMULATION
Let us first introduce the following function (j)^ (a = A, B) which is the solution of the linearized Boltzmann
equation expressing the state of the component gas a far from the wall:
+ 2mQ6 &xi\ (OXA/dx2)Xl=<x,
(a = A, B),
(3)
where SAA = $BB = 1 and SAB = SB A = 0, 6 is an undetermined constant, and the set of the functions D^a
is the solution of the following simultaneous integral equations [11]:
JOL i j/3 \ ^
^a v
a
3
3
a
3
a
/•oo
subsidiary condition:
^ ™PX% I
P=A,B
^°
'
(4)
C^D^E^d^ = 0.
The definition of the operator L@a is given in Eq. (8) below. The <^ is called the fluid-dynamic part of the
solution of the problem.
23
Then, if we put the function (j)a (a = A, B) in the form <fia = cj)% + (dXA/dx2)Xi=oo$(K,
problem to the following one-dimensional boundary- value problem for <1>^:
C,
0,
we can reduce the
Ci > 0, a* = 0,
xi ^ oo,
(6)
(7)
where
-0
(8)
Here e is a unit vector, £* is the variable of integration corresponding to £, ^C* = ^C*i^C*2^C*3> and
is the solid angle element in the direction of e. The integration in Eq. (8) is carried out for the whole space
of £* and for all directions of e. The (dXA/dx2)Xi=oc®(K is called the Knudsen-layer part of the solution of
the problem. When the boundary-value problem (5)-(7) is solved, the undetermined constant b in Eq. (6) is
simultaneously determined with the solution ^>^.
BASIC PROPERTIES OF MACROSCOPIC VARIABLES
Corresponding to the decomposition of 0 a , we can split each macroscopic variable into the fluid-dynamic part,
denoted by a subscripted F, and the Knudsen-layer part, denoted by a subscripted K. The former is defined
by Eqs. (1) and (2) with 0a replaced by <^, and the latter by those with 0a replaced by (dXA/dx2)xi=oo^<KThen, from Eq. (3) we have the following expression of the fluid-dynamic parts: for individual component gases
(x
(OaA
U
2F
\ ~o—— I
^w
A ~~ £
OaB)%2i
a
a
a
a
a
T^p —
— nU, ni
U^p —
— ii
U%p —
—D
(^ip —
— r)
^sp —
— ntl,
a
a
P
_~~ *F°ij->
PA
*ijF
\u
and for the total mixture
NF = PF=TF=Q,
UIF = u3F = QIF = QSF = 0,
r}Y A \
——
\
/^YA\
,
Q2F = -
——
i dX A\
UF = I -^— I
PijF = 0,
9X
(mA-mB)x2,
2 JX1=00
"i
[0TA-^TB) +
X
0(&«A ~ A Q B )],
where A a ^, F^^ a , and ^Ta (<^5 f3 = A,B)
are moments of the function D^a (see the Appendix A in Ref.
for their definitions). They are related to the transport coefficients of gas mixtures.
On the other hand, we can seek the solution <1>J> of the boundary- value problem (5)-(7) in the form
[11]
*£(*i> co = c2(c22 + c32r1/2^i, c, Ci/o-
It follows from this similarity that the Knudsen-layer part can appear only in u%, u%, Pfy (or ^21)? Pi2
^21), Q% , and Q 2 - That is,
Pi2K(= P?IK] =
-5—
Sa(Xl),
Pi2K(= PZIK) =
24
-5—
S'(xi),
(12)
where C/a, [/, Sa, 5, Ha, and H are called the Knudsen-layer functions and are defined as follows:
a
^aKEad<:,
(Xl) =
0
U(X1) =
J
P=A,B
C,
A
P=A,B
B
S(Xl) = S (Xl} + S (Xl},
(13)
H°(Xl)=
In the meantime, from the conservation law of the momentum, it is easily found that the stress tensor of the
total mixture (Pi2 or P^i) should be constant in the whole space. Therefore the Knudsen-layer function 5
should be equivalent to zero because of the condition at infinity [see Eq. (7)]. This property, the uniformity of
the stress tensor (or 5 = 0), is used as a measure of accuracy of the numerical computation.
To summarize, the Knudsen-layer part appears only in the flow velocities, stress tensors, and heat-flow
vectors except the stress tensor of the total mixture. The other macroscopic variables can be expressed by the
fluid-dynamic part of the solution. Among them, the molecular number density, pressure, temperature, and
stress tensor of the total mixture as well as the temperatures of the individual gases are in particular uniform
in the whole space.
Finally the expression of u^p in Eq. (10), if considered at x\ = 0, is no other than the slip condition for the
flow velocity of the mixture on the wall. We call the constant 6 the coefficient of diffusion slip in the present
paper.
NUMERICAL COMPUTATION
The adopted numerical method is the combination of the finite-difference and numerical kernel methods. For
the latter, we first transform the expression (8) of L@a into that in terms of integral kernels, and then, taking
into account the similarity (11), make the database of the kernels numerically before the computation of the
problem itself. The database is also available for analyses of other problems with the same similarity.
In the computation, the original region (0 < xi < oo, 0 < (" < oo, — 1 < Ci/C < 1) i§ truncated to a finite
region 0 < x\ < d(< oo), 0 < £ < Z(< oo), and —1 < Ci/C ^ 1? where the parameters d and Z are chosen as
d = 20.70 (or 26.40) and Z = 4.5/VVna for gas a (a = A, B). It is made sure that the error by this truncation
is negligibly small. The number of lattice points used in the present computation is 221 (or 241) for d = 20.70
(or 26.40), of which 201 points are in the region 0 < x\ < 15, in xi-space, 49 in ("-space, and 97 in £i/£-space,
respectively. The lattice system is nonuniform in 0 < x\ < 15 and in ("-space with a monotonically increasing
interval. It is uniform in 15 < x\ < d (with the same interval as the largest in 0 < x\ < 15) and in d /("-space.
Since mA = dA = 1 and XA + X$ = 1, the boundary- value problem (5)-(7) is characterized by the three
parameters: rhB (or mB /mA), dB (or dB /dA), and XA. We carried out numerical computations for dB = 1
and various values of mB and XA.
RESULTS AND DISCUSSIONS
The coefficient b of the diffusion slip is tabulated in Table 1. Although b is non-zero for XA = 0 and XA = 1,
the diffusion slip itself does not take place for these cases because no concentration gradient is possible. Since
TABLE 1.
A
X
0
0.1
0.3
0.5
* The
Coefficient b of the diffusion slip for dB /dA = 1.
mB/mA
1*
2
4
5
0.1002 0.1141 0.1124
0
0
0.1098 0.1309 0.1299
0
0.1330 0.1769 0.1793
0
0.1637 0.2513 0.2633
data are analytically obtained.
10
0.0987
0.1154
0.1659
0.2618
0.7
0.9
1
25
0
0
0
2
0.2059
0.2666
0.3078
mB/mA
4
5
0.3864 0.4276
0.6804 0.8365
0.9881 1.3424
10
0.4870
1.3108
3.1722
^.0.05
- -0-05
--0.1
-0.15
FIGURE 1. Profiles of the Knudsen-layer functions of flow velocities, (a) U for various values of the reference
concentration XA of gas A in the case mB /mA = 2 and dB /dA = 1. (b) C7, UA , and UB for various values of mass
ratio mB/mA in the case XA = 0.5 and dB /dA = 1.
b is positive, the slip flow is induced in the direction of the higher concentration of the lighter component gas
(gas A).
The profiles of the Knudsen-layer functions for flow velocities are shown in Figure 1. Figure la shows the
profiles of U for various values of XA in the case mB /mA = 2, and Figure Ib those of [/, UA and UB for
various values of mB /mA in the case XA = 0.5. Although UA and UB are monotonic with respect to #1, their
combination U is not; that is, U increases from a negative value at x\ = 0 to a small positive and then decays to
zero at infinity. The tendency is increasing for larger difference of molecular mass (see Figure Ib). Comparing
with the values of b in Table 1, the negative value of U near the wall (x\ = 0) implies the deceleration of the
induced slip- flow there because b is positive. The profiles of other Knudsen-layer functions SA(= —5 s ), HA,
and HB are shown in Figure 2 in the case XA = 0.5. From Figures Ib and 2, the contributions of J7, UA and
UB to H are found to be larger than those of HA and HB [see the last equation in Eq. (13)].
Figure 3 shows 3>^EA in the case mB /mA = 2 and XA = 0.5 as an example of the behavior of the Knudsenlayer parts of the velocity distribution functions. It is discontinuous at £1 = 0 on the wa^ (xi = 0> Figure 3a).
The discontinuity disappears inside the gas. There is, however, a steep gradient around (j = 0 in the vicinity
of the wall (Figure 3b), and the difference between the features in £1 < 0 and in £1 > 0 seen on the wall
26
0.05
-0.05
FIGURE 2. Profiles of Knudsen-layer functions SA (or — SB), HA, and HB for various values of mass ratio mB/mA
in the case of XA = 0.5 and dB/dA = 1.
remains almost unchanged. Going away from the wall, the ^^EA becomes milder and decays with keeping the
difference of the features between the two regions. The Knudsen-layer part of the other component — ^^EB
shows the similar behavior except the higher peak and the narrower width of its form. These differences are
due to the larger mass of a molecule B (mB /mA > 1).
In Figure 4 the present result of the coefficient b is compared with those in Refs. [5,6], where the data
are given by the (dimensional) coefficient divided by the diffusion coefficient. In the figure, according to the
description in Refs. [5,6], comparison is made using the first approximation of the diffusion coefficient. The
relative error of Loyalka's formula [5], which is obtained by the variational method, is larger for smaller XA
and becomes 10% for the worst case (mB /mA = 10 and XA = 0). With more reliable data of the diffusion
coefficient, however, Loyalka's formula gives better agreement with the present result. The relative error of the
formula using our numerical data of the diffusion coefficient [12] is less than 4.1% for mB/mA = 2 and 2.2%
for the other cases.
Finally, we give some pieces of information about the accuracy of the present computation. The values
of b with different lattice systems in ("- and (j/£-spaces are tabulated in Table 2. The values are computed
using a coarser system (101 lattice points) in xi-space with d = 15 in the case mB/mA = 2 and XA = 0.5.
Table 3 shows the values of b in the same case using different lattice systems in xi-space with different d and
the common systems in £- and (j./("-spaces (49 points for the former and 97 points for the latter). Another
measure of accuracy is, as stated before, the computed value of the Knudsen-layer function S. The computed
|5|, which should theoretically be zero, is less than 1.7 x 10~4 for XA = 0.9 and mB/mA = 10, less than
2.6 x 10~4 for XA = I and mB/mA = 10, and less than 1.1 x 10"4 for the other cases.
The present computation was carried out on Fujitsu VPP800/12 computer at the Institute of Space and
Aeronautical Science and on VT-Alpha 533, 600 and VT-Alpha6 500 workstations in our laboratory.
ACKNOWLEDGMENTS
The author is grateful to Professor Kazuo Aoki for valuable discussions and helpful comments on the
manuscript. This work was partially supported by the grant-in-aid (No. 10750045) for Scientific Research
from the Japan Society for the Promotion of Science.
27
(b) xi = 0.0375
(e)
xi = 1.6731
FIGURE 3. Knudsen-layer part 4^.EA of the velocity distribution function of gas A and its contour plot in the case
mB/mA = 2, dB/dA = 1, and XA = 0.5 at (a) xi = 0, (b) xi = 0.0375, (c) xi = 0.1904, (d) xi = 0.5649, and (e)
xi = 1.6731. Here £r = (f 2 — Ci) • In ^ne contour plots the curves are drawn with the interval 0.01 in (a)—(d) and
0.005 in (e); the outermost curves in (a)-(d) indicate 0.01.
28
• present results
10C
O XA = 0.999 1 4-moment
o x£ = 0.5 J method [6]
— variational method [5]
8 910
FIGURE 4. Coefficient 6 of the diffusion slip for dB/dA = 1: Comparison with previous works. Dashed lines indicate
the spline curves connecting the present results for the same X^. The others are explained in the figure.
TABLE 2. Accuracy test I. Dependence of the value of
6 on the lattice systems in f- and Ci/C-spaces. The values
are computed using a system with 101 points in x\-space
with d = 15 in the case raj 3 /mA = 2, dB /dA = 1, and
TABLE 3. Accuracy test II. Dependence of the value
of 6 on d and the lattice system in xi-space. The values
are computed using a system with 49 points in f-space
and 97 points in fi/f-space in the case mB/mA = 2,
dB/dA = 1, and XA = 0.5.
XA = 0.5.
N^ *
"ci/c-
25
0.163044
0.163032
37
0.163591
0.163579
0.163573
49
Lattice system*
d
S-I
S-II
S-III
15
0.163670 0.163689
0.163664 0.163683 0.163685
20.70
0.163664
26.40
* S-I: The system with 101 points for d = 15, 111 points
for d = 20.70, and 121 points for d = 26.40.
S-II: The system with 201 points for d = 15 and 221
points for d = 20.70.
S-III: The system with 331 points for d = 20.70.
61
73
97
0.163670 0.163695
121
* The Nc and N^/^ are the numbers of lattice points in
f - and fi /("-spaces, respectively.
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29