Various Transport Coefficients Occurring in Binary Gas
Mixtures and Their Database
Shigeru Takata,∗1 Shugo Yasuda,† Kazuo Aoki,† and Takumi Shibata†
∗
Département de Mathématiques et Applications, École Normale Supérieure,
45, rue d’Ulm, 75230 Paris Cedex 05, France
†
Department of Aeronautics and Astronautics, Graduate School of Engineering,
Kyoto University, Kyoto 606-8501, Japan
Abstract. A steady state of a binary mixture of hard-sphere gases in the near continuum regime, where
the Knudsen number is small, is considered in the case where the density and temperature variations are
large, but the Mach number of the flow is as small as the Knudsen number. Thus, the flow vanishes in the
continuum limit where the Knudsen number goes to zero. The set of fluid-dynamic-type equations in this
case was derived by Takata and Aoki [Transp. Theor. Stat. Phys. 30, 205 (2001)] by means of a systematic
asymptotic analysis of the Boltzmann equation. This set gives the correct behavior of the mixture in the
continuum limit, i.e., it describes the ghost effect discovered by Sone et al. for a single-component gas [Phys.
Fluids 8, 628 (1996)]. The set contains various transport coefficients that depend on the local properties
of the gas. In particular, their dependence on the local concentration of one of the components cannot be
obtained explicitly. In this paper, this unknown dependence is established numerically by a direct numerical
analysis of the basic integral equations, and a database that provides the numerical values of the transport
coefficients immediately for an arbitrarily specified local state of the mixture is constructed. The database
makes the fluid-dynamic-type equations applicable to practical problems.
INTRODUCTION
According to the classical fluid dynamics, the steady temperature field in a gas at rest with a uniform
pressure (e.g., the gas in a closed container with an arbitrary wall-temperature distribution in the absence
of an external force) is described by the steady heat-conduction equation. The classical fluid dynamics is
generally considered to be valid in the continuum limit where the Knudsen number, defined by the ratio of
the mean free path of the gas molecules to the characteristic length of the system, approaches zero. Contrary
to this common belief, the heat-conduction equation fails to give the correct temperature field in the gas at
rest even in the continuum limit. This is due to the fact that the flows with Mach number being as small
as the Knudsen number, i.e., the flows that disappear in the continuum limit, still give a finite effect on the
temperature field in this limit. On the basis of a systematic asymptotic analysis of the Boltzmann system, Sone
et al. [1] discovered this striking fact and derived the fluid-dynamic-type system (fluid-dynamic-type equations
and their boundary conditions) that gives the correct temperature field in the continuum limit. Such an effect
of vanishing flows in the continuum limit was termed ghost effect [2–4]. Subsequently, the fluid-dynamic-type
system was extended to the case of a multi-component gas mixture [5–7], and the cause of the ghost effect in
the mixture was clarified.
The above-mentioned fluid-dynamic-type system contains the velocity field of the first order in the Knudsen
number. The ghost effect was found by considering the continuum limit on the basis of this system. If the
limit is not considered, the original fluid-dynamic-type system describes the steady behavior of the gas (or a
mixture of gases) at small Knudsen numbers in the situation that the variations in the temperature and density
are large, but the Mach number of the flow is as small as the Knudsen number. Since such a situation is quite
usual in many applications, the system is also of practical importance.
1)
Permanent address: Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University,
Kyoto 606-8501, Japan
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
106
The fluid-dynamic-type equations for the mixture derived in [7] contain various terms originating from the
so-called non Navier–Stokes terms in the stress tensor (the terms corresponding to the thermal stress and
concentration stress). In consequence, there appear various coefficients in addition to the usual transport
coefficients, i.e., the coefficients of viscosity, thermal conductivity, mutual diffusion, and thermal diffusion. We
will call all these coefficients the transport coefficients for short. The transport coefficients generally depend on
the local concentration of each component as well as the local density and temperature of the total mixture. In
particular, the functional form of their dependence on the local concentrations cannot be obtained analytically
even for simple molecular models such as hard-sphere molecules. Even without the explicit functional form,
one can obtain important information on the fundamental properties of the mixture [7]. However, in order to
apply the fluid-dynamic-type equations to practical problems, one needs the explicit functional form.
In the present study, we are concerned with a binary mixture of hard-sphere gases, for which the dependence
of the transport coefficients on the density and temperature of the total mixture is explicit. We are going to
establish their dependence on the local concentration of one of the components numerically and construct a
database that provides the numerical values of the transport coefficients immediately for an arbitrarily specified
local state of the mixture.
BASIC EQUATIONS
We first describe the physical situation that we are going to consider and summarize the the governing
fluid-dynamic-type system. Then, we give the complete definitions of the transport coefficients occurring in
the system.
Fluid-dynamic-type equations
Let us consider a binary mixture of hard-sphere gases (say, A–component and B–component) in contact
with the boundary at rest (when an infinite domain is considered, the mixture is supposed to be at rest with
a uniform pressure at infinity). The boundary may be the ordinary solid body or the condensed phase of one
of the component gases. In the latter case, evaporation or condensation of the component (vapor) may take
place on the boundary.
We first summarize the main notation. The reference quantities are as follows: L0 is the reference length of
the system, m0 the reference molecular mass, d0 the reference molecular diameter, T0 the reference temperature,
n0 the reference molecular number density, ρ0 the reference mass density (ρ0 = m0 n0 ), p0 the reference
pressure
√
(p0 = kn0 T0 , where k is the Boltzmann constant), l0 is the reference mean free path (l0 = 1/ 2πd20 n0 ), and
Kn = l0 /L0 the Knudsen number. Then, Xi is the rectangular coordinate system in space; nα , pα , viα , mα ,
and dα are the molecular number density, pressure, flow velocity, molecular mass, and molecular diameter of
the α–component (α = A, B), respectively; n, ρ, p, vi and T are the molecular number density, mass density,
pressure, flow velocity, and temperature of the total mixture, respectively. Then, we introduce the following
dimensionless quantities:
xi = Xi /L0 ,
n̂α = nα /n0 ,
n̂ = n/n0 ,
dˆα = dα /d0 ,
= viα /(2kT0 /m0 )1/2 ,
v̂i = vi /(2kT0/m0 )1/2 ,
m̂α = mα /m0 ,
p̂α = pα /p0 ,
ρ̂ = ρ/ρ0 ,
p̂ = p/p0 ,
v̂iα
(1)
T̂ = T /T0 .
We consider a steady state in the following situation:
• The Knudsen number (Kn) is small.
• The variations in the densities and the temperature are large [i.e., the variations of n̂α , n̂, ρ̂, and T̂ are
of O(1) ], whereas the Mach number of the flow is small [ i.e., v̂iα and v̂i are of O(Kn) ].
This situation is commonly encountered because it is realized whenever the variation of the boundary temperature is large for both types of boundary (see [7] for the details). It should be noted that in this situation the
flow vanishes in the continuum limit (Kn → 0).
√
According to [7], the physical quantities are expanded in power series of a small parameter = ( π/2)Kn:
h = h(0) + h(1) + · · · ,
(h = n̂α , p̂α , v̂iα , n̂, ρ̂, p̂, v̂i , or T̂ ),
α
with v̂i(0)
= v̂i(0) ≡ 0.
(2)
Note that the expansion for the velocities v̂iα and v̂i starts from the first order. Then, the first terms of the
expansion (2) (the first three terms for p̂) are governed by the following fluid-dynamic-type equations.
107
p̂(0) = const,
p̂(1) = const,
∂
∂
A
(n̂A
(n̂B v̂ B ) = 0,
(0) v̂j(1) ) = 0,
∂xj
∂xj (0) j(1)
(3)
(4)
!
∂ p̂(2)
∂v̂i(1)
∂
∂
1/2 ∂v̂i(1)
=−
+
µ̂T̂(0)
−
2ρ̂(0) v̂j(1)
∂xj
∂xi
∂xj
∂xj
∂xj

!
∂ T̂(0)
∂  1
∂
− ∂
Υ̂2
−
∂xj n̂(0) ∂xj
∂xi
∂xj
!
"
T̂(0)
Υ̂4 ∂X A ∂ T̂(0)
∂
∂
−
−
∂xj n̂(0) ∂xj ∂xi
∂xj n̂(0)
∂
∂xj
1/2 ∂ T̂(0)
λ̂T̂(0)
∂xj
A
B
v̂i(1)
− v̂i(1)
!
Υ̂1 ∂ T̂(0) ∂ T̂(0)
p̂(0) ∂xi ∂xj
!
T̂(0) ∂X A ∂X A
Υ̂3
n̂(0) ∂xi ∂xj
#
∂
∂X A
,
Υ̂5
∂xj
∂xi
!
∂ T̂(0)
∂
5
A
B
A
B B
[kT p̂(0) (v̂j(1)
− v̂j(1)
)] − (n̂A
= 0,
(0) v̂j(1) + n̂(0) v̂j(1) )
∂xj
2
∂xj
!
1/2
T̂(0) D̂AB
∂ ln T̂(0)
∂X A
=−
+ kT
,
n̂(0) X A X B
∂xi
∂xi
−
(5)
(6)
(7)
where
B
n̂(0) = n̂A
(0) + n̂(0) ,
p̂(0) = n̂(0) T̂(0) ,
B B
ρ̂(0) = m̂A n̂A
(0) + m̂ n̂(0) ,
X α = n̂α
(0) /n̂(0) ,
A
B B B
v̂i(1) = (m̂A n̂A
(0) v̂i(1) + m̂ n̂(0) v̂i(1) )/ρ̂(0) ,
Aij = Aij + Aji − (2/3)Akk δij .
(8)
Here, X α is the local concentration of the α–component in the zeroth order (X A + X B = 1). The coefficients
µ̂, λ̂, D̂AB , kT , Υ̂1 , Υ̂2 , Υ̂3 , Υ̂4 , and Υ̂5 (transport coefficients for short), which depend on m̂α and dˆα , are the
functions of the local concentration X A . The µ̂, λ̂, and D̂AB correspond to the viscosity
√ µ, thermal conductivity λ, and mutual-diffusion coefficient DAB ; more specifically, they are related as µ = ( π/2)(2kT0 /m0 )−1/2
√
√
1/2
1/2
1/2
×p0 l0 T̂(0) µ̂, λ = ( πk/m0 )(2kT0 /m0 )−1/2 p0 l0 T̂(0) λ̂, and DAB = ( π/2)(2kT0 /m0 )1/2 l0 (T̂(0) /n̂(0) )D̂AB . The
kT is the thermal-diffusion ratio, i.e., the ratio of the thermal-diffusion coefficient to the mutual-diffusion coefficient. The definitions of these transport coefficients are given in the next subsection. The boundary condition
for Eqs. (4)–(8), which is given in [7], is omitted here because of the limited space.
The p̂(0) and p̂(1) are constants to be determined from the condition proper to each problem. Equations
B
A
B
(4)–(7), with the help of Eq. (8), form a closed set of equations for n̂A
(0) , n̂(0) , n̂(0) , ρ̂(0) , T̂(0) , v̂i(1) v̂i(1) , v̂i(1) ,
A
and p̂(2) , which can finally be reduced to a set of partial differential equations for n̂A
(0) (or X ), T̂(0) , v̂i(1) , and
α
α
p̂(2) . The partial pressure p̂α
(0) of each component is then given by p̂(0) = n̂(0) T̂(0) .
We conclude this subsection with a brief comment on the ghost effect. When the Knudsen number (or )
going to zero, the flow velocities vanish (v̂iα → 0) and the number densities and temperature reduce to their
α
respective leading-order terms (n̂α → n̂α
(0) , T̂ → T̂(0) ) because of the form (2). However, n̂(0) and T̂(0) are
always determined together with v̂i(1) by Eqs. (4)–(7), irrespective of . This means that the densities and
temperature in the continuum limit ( → 0) are affected by the velocity field in spite of the fact that the flow
itself vanishes in the same limit. This statement is always true when v̂i(1) does not vanish identically. In fact, it
was shown that the flow corresponding to v̂i(1) , that is, the flow of O(), is caused generally by various effects,
such as the thermal creep and the diffusion creep [7]. Such an effect of the vanishing flow in the continuum
limit is called the ghost effect. It was first pointed out in [1] and discussed in [2–4] for a single-component gas.
By the way, equations similar to Eqs. (4)–(7) had been derived in a more intuitive way by Galkin et al. [8],
but the ghost effect was not noticed then.
Transport coefficients
Let us denote by Aα (ζ), B α (ζ), and D(β)α (ζ) the solutions of the following Eqs. (9), (10), and (11), respectively, where the independent variable ζ is the magnitude of a vector (ζ1 , ζ2 , ζ3 ), i.e., ζ = (ζi2 )1/2 .
108
X
β=A,B
X
β=A,B
X
β=A,B
e βα (ζi Aβ , ζi Aα ) = −ζi (m̂α ζ 2 − 5/2),
K̂ βα X β L
e βα (cij B β , cij B α ) = −2m̂α cij ,
K̂ βα X β L
subsidiary condition:
X
m̂β X β I4β (Aβ ) = 0,
(9)
β=A,B
with cij = ζi ζj − (1/3)ζ 2 δij ,
e βα (ζi D(γ)β , ζi D(γ)α ) = −ζi (δαγ − m̂α n̂α /ρ̂),
K̂ βα X β X α L
X
subsidiary condition:
m̂β X β I4β (D(α)β ) = 0,
(10)
(11)
β=A,B
where
e βα (f, g) = [Jˆβα (f E β , E α ) + Jˆβα (E β , gE α )](E α )−1 ,
L
Z
√
βα
)g(ζiβα ) − f (ζ∗i )g(ζi )] |ej Vj | dΩ(ei ) d3 ζ∗ ,
Jˆβα (f, g) = (1/4 2π) [f (ζ∗i
Z ∞
α
In (F ) = (8π/15)
ζ n F (ζ)E α (ζ)dζ,
E α (ζ) = (m̂α /π)3/2 exp(−m̂α ζ 2 ),
0
α
βα
ζiβα = ζi + (µ̂βα /m̂ )(ej Vj )ei ,
ζ∗i
= ζ∗i − (µ̂βα /m̂β )(ej Vj )ei ,
Vi = ζ∗i − ζi ,
βα
β
α
2
βα
β
α
β
α
3
K̂ = [(dˆ + dˆ )/2] ,
µ̂ = 2m̂ m̂ /(m̂ + m̂ ),
d ζ∗ = dζ∗1 dζ∗2 dζ∗3 .
(12)
(13)
(14)
(15)
(16)
In Eq. (13), which is the collision term of the Boltzmann equation, ζ∗i is the integration variable for ζi , ei is
the unit vector, dΩ(ei ) is the solid-angle element in the direction of ei , and the domain of integration is all the
e βα is the linearized collision operator. Here, we should recall
directions of ei and the whole space of ζ∗i . The L
A
B
A
that X and X are not independent (X + X B = 1) and note that Aα , B α , and D(β)α depend on X A (or
X B ), m̂α , and dˆα .
(γ,κ)α
(γ,κ)α
(γ)α
(γ)α
α
Then we define the functions Aα
(ζ), D2
(ζ), DA1 (ζ), and DA2 (ζ), which also
1 (ζ), A2 (ζ), D1
depend on X A (or X B ), m̂α , and dˆα , by the following integrals.
X
α
ζi ζj Aα
K̂ βα X β Jeβα (ζi Aβ , ζj Aα ),
(17)
1 + A2 δij =
β=A,B
(γ,κ)α
ζi ζj D1
+
(γ,κ)α
D2
δij
=
X
β=A,B
(γ)α
(γ)α
ζi ζj DA1 + DA2 δij =
X
β=A,B
where
K̂ βα X β Jeβα (ζi D(γ)β , ζj D(κ)α ),
K̂ βα X β [Jeβα (ζi Aβ , ζj D(γ)α ) + Jeβα (ζi D(γ)β , ζj Aα )],
Jeβα (f, g) = Jˆβα (f E β , gE α )(E α )−1 .
(18)
(19)
(20)
The transport coefficients are defined with the aid of the functions introduced above. The µ̂, D̂AB , kT , and
λ̂ are defined as follows:
X
ˆ AA + ∆
ˆ BB − ∆
ˆ AB − ∆
ˆ BA ),
µ̂ =
m̂β X β I6β (B β ),
D̂AB = X A X B (∆
β=A,B
(21)
0
A B
kT = D̂T /D̂AB ,
λ̂ = λ̂ − kT D̂T /X X ,
where
ˆ αβ = 5 I4α (D(β)α ),
∆
2
5
D̂T = X A X B [I4A (AA ) − I4B (AB )],
2
5 X
5 β
β β
β 2
λ̂ =
X I4 (m̂ ζ − )A ,
2
2
0
(22)
β=A,B
√
1/2
and D̂T corresponds to the thermal-diffusion coefficient DT , i.e., DT = ( π/2)(2kT0/m0 )1/2 l0 (T̂(0) /n̂(0) )D̂T .
The Υ̂1 , Υ̂2 , Υ̂3 , Υ̂4 , and Υ̂5 are defined by
109
ζ ∂Aβ
1 X
β
β 2
β
β
β β
,
Υ̂1 =
X I6 B A1 + (m̂ ζ − 3)A −
2
2 ∂ζ
Υ̂3 =
β=A,B
(A,A)
Υ̂3
+
where
(B,B)
Υ̂3
−
(A,B)
Υ̂3
−
(B,A)
Υ̂3
,
Υ̂4 =
(A)
Υ̂4
−
Υ̂2 =
(B)
Υ̂4 ,
1 X
X β I6β (Aβ B β ),
2
β=A,B
(A)
Υ̂5 = Υ̂5
−
(23)
(B)
Υ̂5 ,
β
1 X
β β
β (α,γ)β
(α)β ∂B
X I6 B D1
−D
=
,
2
∂X γ
β=A,B
ζ ∂D(α)β
∂B β
1 X
(α)β
X β I6β B β DA1 + (m̂β ζ 2 − 2)D (α)β −
− Aβ
,
=
2
2 ∂ζ
∂X α
β=A,B
1 X
=
X β I6β (B β D(α)β ).
2
(α,γ)
Υ̂3
(α)
Υ̂4
(α)
Υ̂5
(24)
β=A,B
It should be noted that the partial derivatives of B α with respect to the concentration X A or X B appear in
Eq. (24). Their meaning is as follows. Since Eq. (10) is also valid for X A and X B that are independent each
other, the solution B α can be regarded as functions of two independent variables X A and X B . We first take
partial derivatives of such B α and then introduce the restriction X A + X B = 1.
NUMERICAL ANALYSIS AND RESULTS
We first note that the equations corresponding to Eqs. (9)–(11), (21), and (22) also appear in the Navier–
Stokes level in the framework of the Chapman–Enskog expansion [9]. In the literature [9–11], the solutions of
Eqs. (9)–(11) are expanded in the Sonine polynomials, and the first four terms at most (only a single term in
[9]) are used to approximate the solutions. Then, the approximate coefficients µ̂, λ̂, D̂AB , and kT , are obtained
(γ,κ)α
from Eqs. (21) and (22). We may use the same approximations in Eqs. (17)–(19) to obtain Aα
, etc.,
1 , D1
which give the corresponding approximations of Υ̂1 , ..., Υ̂5 from Eqs. (23) and (24). In this method, however,
the process of analysis, as well as the resulting formulas for the transport coefficients, becomes increasingly
complicated with the increase of the number of terms used for the approximation, even though only a few
terms are used.
In the present study, in contrast to the classical method mentioned above, we take a more direct approach,
that is, we solve Eqs. (9)–(11) numerically to obtain accurate values of Aα , B α , and D(β)α . Our goal is to
establish the functional forms of the transport coefficients with respect to the concentration X A numerically.
A method that is suitable for this purpose is to expand these functions as power series of σ = X A − 1/2
(−1/2 ≤ σ ≤ 1/2), i.e.,
∞
X
1
Xα
(β)α
B
A
Gn (ζ)σn , σ = X A −
D
with
X
=
1
−
X
, (25)
G = Aα , B α , or
G(ζ; X A ) =
2
1 − Xβ
n=0
which are expected to converge for all σ in −1/2 ≤ σ ≤ 1/2. If we substitute the expansion (25) into Eqs. (9)–
(11), we obtain a sequence of integral equations for Gn , which is solved numerically from the lowest order.
Then, using Eq. (25) with the obtained Gn in Eqs. (21) and (22), we obtain µ̂, λ̂0 , D̂AB , and D̂T in the form
of power series in σ. The substitution of the same result in Eqs. (17)–(19) yields the corresponding expansions
(γ,κ)α
for Aα
, etc. Thus, from Eq. (23), we obtain the power-series expansion of Υ̂1 , ..., Υ̂5 in σ. [Because
1 , D1
of the reason that was mentioned at the end of the previous section, we also need the power-series expansions
of ∂B α /∂X A and ∂B α /∂X B . They are obtained directly from their integral equations that are derived by
differentiating Eq. (10) with respect to X A or X B .] In the present analysis, we have taken 71 terms (for
1 < mB /mA ≤ 5) or 101 terms (for 5 < mB /mA ≤ 10) in the expansion (25) and obtained the coefficients of
the corresponding expansions for all the transport coefficients. Once the coefficients of these expansions are
stored, one can readily obtain accurate values of the transport coefficients for an arbitrarily given X A , just by
summing up the series, without using interpolation. We have built such a database, which makes it possible
to apply the fluid-dynamic-type equations (4)-(8) to practical problems. The expansion (25) was inspired by
the modified Knudsen number expansion in [12].
The advantages of the present approach can be summarized as follows: (i) the method is conceptually
simple and straightforward, so that it is suitable for numerical computation; (ii) since the convergence within a
110
1.5
10
18
12
6
8
6
4
798
1.5
HG I
J
A
B
CD
E
-0.05
-0.1
t
vw=xvy
z{|
}
~

-5
0
12
[Z \
8
4
0
0
25
2

„†ƒ … 10 …
5
0
0.5
Ž
15
€
oqp
‡†ˆ‰Šˆ‹
Œ
20
{
-2.5
16
e e e ]f ^=_]`Pacbd
e e g
ei
ee h
\kj lnm
b
@#F
2.5
0
1
0
2 3 34
5
/6
1
0.5
5
sr t
u
1
K L=MKNPO I Q
U R RTS
X R RWV
Y
I
2
0
% $ &)1.25
'
(
#"
0
:;=<:>
?@
!
2
0.05
*+,*-
./
1
-5
0
‘
1
–• —

˜™š˜›
œž
Ÿ
¡
0
-1
’
0.5
q‚
-2
1
0
0.5
“q”
¢
—

1
FIGURE 1. Transport coefficients µ̂, λ̂, D̂AB , kT , Υ̂1 , Υ̂2 , Υ̂3 , Υ̂4 , and Υ̂5 versus X A for dB /dA = 0.5.
specified accuracy of the power series can easily be checked numerically, high accuracy has been accomplished in
the result; (iii) thanks to the power-series expansion, the derivatives (of any order) of the transport coefficients
with respect to the concentration X A are readily obtained from the same database. The (iii) is important in the
following aspect. In the fluid-dynamic-type equations (4)–(7), the transport coefficients undergo differentiation
with respect to the space coordinates xi , which lead to their derivatives with respect to X A . When the
equations are solved numerically, we do not need to use finite-difference approximations for these derivatives
because of (iii), so that the equations become more flexible for application of various numerical schemes.
Although the present method is conceptually simple, it contains some technical difficulties in the numerical
computation. For example, (a) in order to obtain Gn in Eq. (25), we need to solve many (71 to 101) linear
e βα ,
integral equations, corresponding to each of Eqs. (9)-(11), consisting of the linearized collision integrals L
(γ,κ)α
α
which originally contain five-fold integrals; (b) in the process of obtaining A1 , D1
, etc. in Eqs. (17)–(20),
we need to compute the complicated nonlinear collision integrals Jeβα of the Gn ’s, which originally contain
five-fold integrals. These difficulties have been overcome by various analytical and numerical devices. The
e βα and Jeβα have been performed by means of the method similar to those proposed in [13]
computation of L
for the linearized collision integral for a single-component gas. According to the method, each collision term
e βα
is essentially expressed in the form of the product of a matrix composed of the values of the integrals L
βα
e
or J
of given sets of functions (the sets of basis functions in terms of which Gn are expanded) at the grid
points in ζ and a vector composed of the values of Gn at the grid points. We call the former matrix numerical
kernel for short. Because of the inherent complexity of the collision integrals, the computation required for the
111
3
4
3
2
2.5
2
1.5
1
1
0.8
%'$ &)(
"!
#
0.5
0
798
NOPNQ
R LS
T
U
V
W
X
L
KMJ L0.15
B
C D E
FHG G @I
-0.1
0.1
1
0
2 3 34
5
/6
\]=^\_
`a
[
0.95
ZY [
b
0.9
c
0.85
-0.15
dfe
ahg
0.8
4
3
3
ml n
o
1
0.5
0.2
A
-0.05
0.7
0.6
0.05
:;=<:>
?@
*+,*-
./
pqrpsutwvx
0.5
ikj
3
‹
“’ ”
‚
Ž
1
z
v | } | {n
| |
1
0
0
€ ‚
ƒ
y
2
„…†„‡uˆw‰ŒŠ‹‹
2

0
1
0
0.5
~k
2
•–—•
˜
™š›œ œ
žœ œ
1
‰
1
0
0
”
Ÿ ¡
0.5
k‘
š
1
FIGURE 2. Transport coefficients µ̂, λ̂, D̂AB , kT , Υ̂1 , Υ̂2 , Υ̂3 , Υ̂4 , and Υ̂5 versus X A for dB /dA = 1.
numerical kernels is relatively heavy. However, the kernels can be prepared prior to the solution process. Once
they have been constructed, the process of solving Eqs. (9)–(11), as well as that of obtaining the functions Aα
1,
(γ,κ)α
D1
, etc. from Eqs. (17)–(20), reduces to simple and straightforward computation. As the result, in spite
of the fact that we have to solve many (71 to 101) integral equations for each of Aα , B α , and D(β)α for various
values of m̂α (or molecular mass ratio mB /mA ) and dˆα (or molecular diameter ratio dB /dA ), we are able to
construct the database with reasonable computer time. In this way, the difficulties in the numerical analysis
are compressed in the numerical kernels. On the other hand, there appear only three types of numerical kernel:
the first one is associated with Eqs. (9) and (11), the second one with Eq. (10), and the last one with Eqs. (17)–
(19). Furthermore, concerning the molecular parameters, these kernels depend only on m̂α , independent of
dˆα . These advantages, which reduce the load of computation significantly, have been taken into account in
the construction of the numerical kernels. It should also be mentioned that the fact that the kernels can be
prepared separately enables us to concentrate on the careful and accurate construction of the kernels.
In the present study, the database has been constructed for the diameter ratio dB /dA = 0.5, 1, and 2 and
for many values of mass ratio mB /mA . The transport coefficients µ̂, λ̂, D̂AB , kT , Υ̂1 , Υ̂2 , Υ̂3 , Υ̂4 , and Υ̂5
are shown as functions of X A for mB /mA ≥ 1 in the case of dB /dA = 0.5 in Fig. 1, dB /dA = 1 in Fig. 2,
and dB /dA = 2 in Fig. 3. The figures show that the dependence of the transport coefficients on X A are quite
different depending on the diameter ratio. For mB /mA = 1 in Fig. 2 (dB /dA = 1), which is the case where the
molecules of the two components are mechanically identical, the coefficients reduce to
112
2.5
1.5
1
0.5
2
1.5
1
0.5
0
BCDBE
FG
0
?A@
H
-0.1
~1€~
‚„ƒ
}…
† †ˆ
‡‰
Šˆ ‹
ƒŒ
0
-1
" " "$
!#
%
&
'
VW1XVY
Z R\[ [
]\^ _`^ a`^ Rbc
c[ c
5
798
77;: 6 7
:=
> 4<
0.25
0.75
d
gf h
e
0.5
ij1kil
mnph o o
q oo o o
rps s s
ts s s
s
w w nu v
0.25
0
0
0.5
2.5
’“1”’•
–—˜
š ‘™
› › › ›9œ
 ›› ››
—
  ‘ -0.5
9
./10.2
34
0.3
1
0
-2
) ( *-,
+
0.2
0.05
-0.2
0
I
JK
LNM M GO
-0.15
0.35
0.1
Q P RUT
S
-0.05
|{ }
0.4
2
¡ ¢
£¤1¥£¦¨§©«®ª­¬ ¬
¬ ¬
1.5
-1
1
-1.5
0.5
¯°
-3
0
0.5
xzy
1
-2
0
0.5
zŽ
1
0
0
0.5
žzŸ
±
¢
©
1
FIGURE 3. Transport coefficients µ̂, λ̂, D̂AB , kT , Υ̂1 , Υ̂2 , Υ̂3 , Υ̂4 , and Υ̂5 versus X A for dB /dA = 2.
µ̂ = 1.270042,
Υ̂2 = 0.973953,
λ̂ = 2.402855,
D̂AB = 0.764215,
kT = Υ̂3 = Υ̂4 = Υ̂5 = 0.
Υ̂1 = 0.094600,
(26)
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Sone, Y., Aoki, K., Takata, S., Sugimoto, H., and Bobylev, A. V., Phys. Fluids 8, 628 (1996).
Sone, Y., in Rarefied Gas Dynamics, edited by Shen, C., Beijing: Peking University Press, 1997, p. 3.
Sone, Y., in Annual Review of Fluid Mechanics, Palo Alto: Annual Reviews, 2000, Vol. 32, p. 779.
Sone, Y., Kinetic Theory and Fluid Dynamics, Boston: Birkhäuser, 2002.
Takata, S., and Aoki, K., Phys. Fluids 11, 2743 (1999).
Takata, S., Aoki, K., and Muraki, T., in Rarefied Gas Dynamics, edited by Brun, R., Campargue, R., Gatignol,
R., and Lengrand, J.-C., Toulouse: Cépaduès–Éditions, 1999, Vol. 1, p. 479.
Takata, S., and Aoki, K., Transp. Theor. Stat. Phys. 30, 205 (2001).
Galkin, V. S., Kogan, M. N., and Fridlender, O. G., Fluid Dyn. 7, 287 (1972).
Chapman, S., and Cowling, T. G., The Mathematical Theory of Non-uniform Gases, 3rd ed., Cambridge: Cambridge
Univ. Press, 1995.
Devoto, R. S., Physica, 45, 500 (1970).
Bonnefoi, C. L., and Fauchais, P., Phys. Fluids 19, 1827 (1976).
Sone, Y., and Itakura, E., J. Vac. Soc. Jpn. 33, 92 (1990) (in Japanese).
Sone, Y., Ohwada, T., and Aoki, K., Phys. Fluids A 1, 363 (1989).
113