The Behavior of a Vapor-Gas Mixture in the
Continuum Limit: Asymptotic Analysis
Based on the Boltzmann Equation
Kazuo Aoki
Department of Aeronautics and Astronautics, Graduate School of Engineering,
Kyoto University, Kyoto 606-8501, Japan
Abstract. A binary mixture of a vapor and a noncondensable gas in contact with the condensed phase of
the vapor, on the surface of which evaporation or condensation of the vapor can take place, is considered.
The steady behavior of the mixture in the continuum limit with respect to the vapor is investigated on
the basis of kinetic theory by means of an asymptotic analysis for small Knudsen numbers. First, the case
where the mixture is confined in the gap between two parallel plane condensed phases, one of which may
be moving in its surface, is considered (two-surface problems), and it is shown that there are two different
types of continuum limit depending on the amount of the noncondensable gas, i.e., (a) a local equilibrium
state of the mixture in which no evaporation or condensation takes place but the temperature, number
densities, and flow velocity along the condensed phases are still affected by the vanishing evaporation and
condensation (ghost effect); (b) a uniform flow of the vapor with the noncondensable gas being confined in
the infinitely thin Knudsen layer on the condensing surface. Then, the mixture around arbitrarily shaped
condensed phases at rest is considered, and the fluid-dynamic type system describing the continuum limit
corresponding to type (a) mentioned above is derived. The cause of the ghost effect in this general case is
clarified with the help of the explicit form of the fluid-dynamic type system.
INTRODUCTION
Vapor flows with evaporation or condensation on the boundary have been one of the important subjects in
rarefied gas dynamics. For single-component systems consisting of a pure vapor and its condensed phase, many
successful results have been obtained. For example, a new type of gas dynamics (i.e., fluid-dynamic equations
and their boundary conditions) describing the vapor flows around arbitrarily shaped condensed phases in the
continuum limit (the limit where the Knudsen number tends to zero) has been established, together with
its higher-order corrections in the Knudsen number, by means of a systematic asymptotic analysis of the
Boltzmann equation for small Knudsen numbers [1-5].
In practical situations, however, evaporation and condensation often take place in the presence of other gases
that do not participate in evaporation or condensation (noncondensable gases). Such two- or multi-component
systems (vapor-gas mixture) have also been investigated (e.g., [6-9]). But, because of the complexity, the level
of understanding is still unsatisfactory. For instance, the behavior of the mixture in the continuum limit has
not been clarified yet.
In the mean time, a study based on kinetic theory revealed a serious defect contained in the classical fluid
dynamics [10]. That is, contrary to common belief, the Navier-Stokes system fails to describe the behavior of
a gas (a simple gas around solid boundaries without evaporation or condensation) even in the continuum limit
in some important situations. This is due to the fact that the gas flows of the order of the Knudsen number,
which therefore vanish in the continuum limit, still give a finite effect on the behavior in this limit (ghost effect)
(see [11-13]). The effect is expected to manifest itself in a wider class of problems for the vapor-gas mixture.
For the reasons mentioned above, it is an important issue to clarify the behavior of a vapor-gas mixture in the
continuum limit on the basis of kinetic theory. In this paper, we summarize recent results [14-17] relevant to
this subject obtained by a systematic asymptotic analysis and a numerical analysis of the Boltzmann equation.
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
565
We consider a binary mixture of a vapor (^4-component) and a noncondensable gas (J5-component), both of
which are composed of hard-sphere molecules, in contact with the boundary consisting of the condensed phase
of the vapor, and we investigate the steady behavior of the mixture in the continuum limit with respect to the
vapor on the basis of the Boltzmann equation (the results based on model Boltzmann equations will also be
used in the following discussions). The conventional complete condensation condition is assumed for the vapor
and the diffuse reflection for the noncondensable gas on the boundary. In the former, the vapor molecules
are assumed to be emitted according to the corresponding part of the Maxwellian distribution characterized
by the temperature and velocity of the boundary and by the saturation density of the vapor at the boundary
temperature.
The main notation used in this paper is summarized here: raa and d^ are the mass and diameter of a
molecule of a-component (a = A, B', hereafter, the letter a is used to represent the labels A and B of the
components), L is the reference length, TO is the reference temperature, no is the reference molecular number
density (a representative number density of the vapor), pQ = fcn0T0 (k: the Boltzmann constant) is the reference
pressure, and nBv is the average number density of B-component in the domain; IQ is the mean free path of
the molecules of A-component in the equilibrium state at rest with number density no and temperature TO
[/o = l/V27r(d£) 2 no], and Kn= 10/L is the Knudsen number; Xi = Lxi is the space coordinate system,
& = (2fcT0/mA)1/2Ci is the molecular velocity, and Fa = n0(2kTQ/mA)-3/2Fa is the velocity distribution
function of a-component; na — n0na, pa = mAnopa, vf = (2kTQ/mA)l/2vf, Ta = T0Ta, and pa = popa
are the molecular number density, mass density, flow velocity, temperature, and pressure of a-component,
respectively; n = n0n, p = m A n 0 p, vt = (2kTo/mA)l/2Vi, T = T0T, and p = Pop are the molecular number
density, mass density, flow velocity, temperature, and pressure of the mixture, respectively. The dimensionless
variables (xi, C*> and the quantities with a hat) are mainly used in the following discussions.
TWO-SURFACE PROBLEMS
We first investigate the two-surface problem of evaporation and condensation. That is, we consider the
mixture in the domain 0 < X\ < L (or 0 < x\ < 1) between two parallel plates of the condensed phase. Let
us suppose that the surface at X\ = 0 (or x\ — 0) is kept at temperature T/ and is set at rest, whereas that
at Xi = L (or x\ = 1) is kept at temperature T// and may be moving in its surface in the x2 direction with a
constant speed C///. We denote by n/ and n// the saturation number density of the vapor (A-component) at
temperature T/ and that at temperature T//, respectively. Here, we take T/ and n/ as our reference quantities,
i.e., TO = T/, n0 = n/, and p0 = fcn/T/. This problem is characterized by the following seven parameters:
mB/mA, d%/dA, T/j/T/, n///nj, UII/(2kTI/mA)1/2, Kn, and nBv/nx. The parameter nBv/m (= finBdxi)
controls the amount of the noncondensable gas contained in the domain. Our interest is to clarify the behavior
of the mixture in the continuum limit with respect to the vapor (Kn— 0+).
For this purpose, we carry out a systematic asymptotic analysis of the Boltzmann system for small Kn,
following [18,19,3,5,10] as a guideline. First, we seek a moderately varying solution F% (Hilbert solution) in
the form of a power series of Kn: Fg=FgQ-\-FglKn+- - •. Let h be any macroscopic quantity (h=ha, n, pa,
p, vf , Vi, Ta, T, pa, or p) and HH be the same quantity corresponding to Fg. Then hH is also expanded as
hH=hHo+hHiKn+' • -. By substituting the expansion of F% into the Boltzmann equation, one finds that the
leading-order terms Fg0 are the local equilibrium distributions, i.e.,
( -m°[(Ci - vino) 2 + (C2 - v2m)2 + ^}/fHO
),
(1)
where mA = 1 and mB = mB/mA. It should be noted that i)Am = v?HQ = viHQ and f$0 = fjQ = fm
for these distributions. On the other hand, the particle conservation and the noncondensable condition for
the 5-component lead to nBvB = 0, therefore, nf QvBHO = 0. If nf v /n/ = O(l), then nB 0 does not vanish
identically, and therefore we have v£HO = VBHO = VIHO = 0, that is, the flow velocities normal to the condensed
phases vanish at the leading order; if nf v /n/ < O(Kn), then we have ng0 = 0. We will discuss these two cases
separately. In the latter, we restrict ourselves to the case nBv/nj = O(Kn).
566
Continuum limit when n^v/71*
By continuing the usual Hilbert procedure with the result VAHQ = v^HO — VIHO = 0, we obtain the equations
of fluid-dynamic type for n# 0 , n^ 0 , THO, and v2HQ contained in Fg0.
Furthermore, thanks to VIHO = 0> the
FffQ can be made to satisfy the kinetic boundary conditions (complete condensation for the vapor and diffuse
reflection for the noncondensable gas) at the order of Kn° by the suitable choice of the boundary values of
n#0, n§0, THO» and V2HO- This choice gives the boundary condition for the fluid-dynamic type equations. To
summarize, hA, nB , T1, and vi (#3 — 0) in the continuum limit Kn= 0+ (i.e., n# 0 , n#0, THO, and VIHO) are
governed by the following equations and boundary conditions [16]:
«t(=tf = «f) = 0,
= 0,
2
DAB
dx
<ftV)=0,
(2)
dXl
i
(5)
pA = f i A f ,
p = pA+pB,
pB = nBf,
(6)
and
hA = l,
A
f = 1,
n = n j //n/,
v2 = 0,
f - T///T/,
at xi = 0,
(7)
A 1 2
02 - UII/(2kTI/m ) / ,
at x x - 1-
(8)
Here, </>A is an auxiliary unknown function, the physical meaning of which will be clarified below, and /}, A,
DAB, and D^B are given functions of the local concentration of A-component X A (or that of jB-component
XB) (Xa = na/h
= pa/p
with h = nA + nB; X A + X B = 1), depending on mB /mA and d^/dj*, and
are related to the conventional transport coefficients, i.e., (^yiT/2)(2kT/mA)l^2l(mAn(l/21
knX, DAB, D^B)
are the coefficients of viscosity, thermal conductivity, mutual diffusion, and thermal diffusion, respectively,
where T = T/T, n = n/n, and / = l/V^7r(G^) 2 n (see [16,17] for the details). However, since the explicit
functional forms of these functions are not known, we have to use approximate formulae [20] or numerical
results in order to make the system (2)-(8) solvable. We should mention here that a database which provides
accurate numerical values of these functions immediately for any XA was constructed recently [21]. Although
the auxiliary function (f>A can be eliminated from Eqs. (3)-(6), we retain it for convenience in the following
discussions. It is to be noted that the boundary conditions (7) and (8) are valid for more general kinetic
boundary conditions [16].
The first equation of Eq. (2) indicates that evaporation or condensation does not take place in the continuum
limit. Therefore, in this limit, the problem appears to be identical with the ordinary plane Couette flow with
heat transfer between two (non evaporating or non condensing) plates (i.e., the case where both of A— and
^-component are noncondensable). However, it is not true. In fact, the solution of the latter problem in the
continuum limit is given by Eqs. (2)-(6) with (j)A = 0 and boundary conditions (7) and (8) with the condition
for nA being discarded. (In this case, we have to take an appropriate reference number density no, such as the
average number density nAv of A-component.) Hence, the presence of (f)A in Eqs. (2)-(6) makes the behavior
in the present problem different from that of the ordinary plane Couette flow with heat transfer. The physical
meaning of (f)A is not clear if one considers only the continuum limit. In the framework of the asymptotic
analysis for small Kn, it is identified as (f)A = (2/^)vAHl or (pA = lim (2/^)(vA/Kn), which is of 0(1)
Kn —>0
(recall that the vapor flow velocity normal to the boundary VAH is expanded as VAH = vAHlKn + • • •; in the
second expression of <f>A, the correction to vAHl in the Knudsen layers, which degenerate on the surfaces of the
condensed phase in the limit Kn= 0+, is not taken into account). Therefore, in the continuum limit, although
VA (the flow of the vapor, normal to the boundary, with evaporation and condensation) itself vanishes, its effect
remains finite in the form of <j>A. This is one of the examples of the ghost effect found and discussed in [10-13].
It is obvious from Eq. (5) that the cause of the ghost effect in the present problem is the mutual diffusion and
thermal diffusion [the terms containing DAB and t>TAB, respectively, in Eq. (5)].
567
\
-10
\
0.2
\
0.5
\
\
\
l
n*Jn, = 0.1
-20 -
0.5
Xi/L
(a)
FIGURE 1. The behavior in the continuum limit for n%v/m = O(l) in the case Un = 0. (a) The result obtained by
Eqs. (2)-(8) for mB /mA = 2, d^/d^ = 1, T///T/ = 1.1, and n///n/ = 4. (b) Comparison with the DSMC result for
small Kn for mB /mA = 5, d^/d^ = 2, T///T/ = 1.3, n///n/ = 5, and nf v /nj = 0.5. In (a) and (b), the solid line
indicates the result by Eqs. (2)-(8), and dotted line the result for the ordinary heat transfer (HT).
Now, we show some examples of the behavior in the continuum limit. We restrict ourselves to the case where
both plates are at rest (£/// = 0) but their temperatures are different (Tj ^ T//) and focus our attention to
the ghost effect on the temperature field [15].
Figure l(a) shows the profiles of the number densities nA (vapor) and nB (noncondensable gas), the temperature T (total mixture), and the auxiliary function <f)A in the continuum limit, obtained from Eqs. (2)-(8), for
various values of n^v/nj in the case mB /mA = 2, d^/d^ = 1, T///T/ = 1.1, and n///n/ = 4. The dotted line
indicates the corresponding result for the ordinary heat-transfer problem (between ordinary plates), in which
the parameter n///n/ is discarded, and n/ is a reference number density determined in such a way that nAv/nj
is the same as that in the original problem. In spite of the fact that there is no evaporation or condensation in
the present problem, the temperature profile is quite different from that of the ordinary heat- transfer problem.
As n^v decreases, J5-component tends to concentrate near the cold wall, and the gradients of nA and nB
become steeper there. Correspondingly, the magnitude of (f>A, which is a measure of the ghost effect and is
essentially controlled by the gradient of nA [see Eq. (5)], increases, and thus the deviation of the profile of T
568
from that in the ordinary heat-transfer problem becomes more significant. When n^v is small (n^v/nj = 0.1
and 0.2), there appears the region with a negligibly small amount of B-component, where A-component is
almost in a uniform saturated equilibrium state at rest with temperature T// and number density n//. In the
ordinary heat-transfer problem, on the other hand, the temperature is almost independent of nf v . It should
be noted that the temperature profile in Fig. l(a) is not monotonic [see also Fig. l(b)]. This is caused by
the effect of thermal diffusion [15]. On the other hand, Fig. l(b) shows the numerical solution of the original
boundary- value problem of the Boltzmann equation for small Kn obtained by means of Bird's DSMC method
[22] for mB/mA = 5, d*/d£ = 2, T///T/ = 1.3, n///n/ = 5, and nf w /n/ = 0.5. Instead of (f>A, the vapor-flow
velocity vf is shown in Fig. l(b). The figure shows that, as Kn —> 0, the vapor flow tends to vanish, and the
profiles of the number densities and of the temperature approach those obtained from Eqs. (2)-(8).
As is evident from the asymptotic analysis, T [= 0(1)] is affected by v\ of O(Kn). Therefore, as Kn decreases,
it becomes increasingly difficult to obtain the correct temperature T by the DSMC method, since v\ becomes
small in proportion to Kn. In fact, the result for Kn= 0.02 in Fig. l(b) was obtained by using 2000 (uniform)
cells in #1 and 105 simulation particles for the quantity n/L for each component.
Continuum limit when n^/n/ = O(Kn)
Let us put n^v/nj = AKn, where A is a given constant, and investigate the behavior in the continuum limit
with respect to the vapor (Kn= 0+). It should be noted that n^v/nj vanishes in this limit. Since njjQ = 0
or FjJQ = 0 in this case, we only need to deal with F^0. If we proceed following the Hilbert procedure, we
can easily show that n#0, #mo> ^2HQ, and THQ in F^0 are all constant, and furthermore FArn=FBrn— 0 for
m > 1. At this point, one might conclude that the present case is trivial because no noncondensable gas is
contained in the vapor. However, one should notice that, unlike the previous case, F^0 with VIHQ / 0 cannot
be made to satisfy the boundary condition on the condensed phases. In order to obtain the solution satisfying
the boundary condition, we have to seek the solution in the form Fa=F§Q-\-F^Q (FHQ = 0), where F£0 are the
correction terms appreciable only in the thin layers (Knudsen layers) with thickness of the order of the mean
free path [or 0(Kn) in xi] adjacent to the condensed phases. In other words, the noncondensable gas can be
present inside the Knudsen layers. However, it cannot stay in the Knudsen layer at the evaporating surface
[23], i.e., FKQ = 0 there, and can be present only in the Knudsen layer adjacent to the condensing surface. The
situation that n^0 (i.e., nB corresponding to FB0) is of O(l) in the Knudsen layer with thickness of O(Kn)
(in #1) is consistent with the condition nf v /n/ = 0(Kn).
The analysis of these Knudsen layers are essentially the same as that of the evaporating or condensing flow
in a half space (the so-called half-space problems). That is, F^0 for the evaporating surface is given by the
solution for the steady flow of a pure vapor evaporating from a plane condensed phase [24-26], while F^0 and
FKO f°r the condensing surface are given by the solution for the steady flow of a vapor condensing onto a
plane condensed phase in the presence of the noncondensable gas [27,28]. The conditions on the condensed
phases that determine the constants ?V#0, VIHQ, ^2HO> and THQ are obtained together with the solution for the
Knudsen-layer corrections. We summarize the conditions below.
Let us suppose that evaporation is taking place on the surface at X\ = L, and condensation on the surface
at Xi — 0, namely, T/ < T//, n/ < n//, and VIHQ < 0- Then, the boundary conditions for n"4, T, and vi
(1)3 — 0) in the continuum limit (i.e., n^0, THQ, and Vino) are given as follows.
On the evaporating surface (at xi = 1):
t>2 - (2kTI/mArl/2Un,
nA = (n///n/)[MMi)/MMi)],
f = (T///T/)/i2(M1),
f- 1 / 2 ,
Mi<l,
(9)
(10)
where MI is the Mach number based on the normal component of the flow velocity, and hi and h^ are given
functions of MI. Accurate numerical values of these functions were obtained by using the BGK model [29-31]
by Sone and Sugimoto [24].
On the condensing surface (at x\ = 0):
n^f =F s (Mi,M 2 ,f,r),
A
u T> F 6 (Mi,M2,f,r),
(forMi<l),
nAf > F s (l,M 2 ,f,r),
(for Ml > 1),
(for MI - 1),
(lla)
(lib)
569
(12)
M2 =
where MI is the same as Eq. (10), M% is the Mach number based on the tangential component of the flow
velocity, Fs and F^ are given functions of four variables, and Fn/ is a quantity of the order of the average
number density of ^-component in the Knudsen layer; in Eq. (12), e — 0 for hard-sphere molecules and e = 1
for the model Boltzmann equation proposed by Garzo et al. [32]. The functions Fs and Ft, were constructed
numerically in [27] on the basis of the model of [32] for the case where the molecule of the vapor and that of the
noncondensable gas are mechanically identical. (In this case, the problem is essentially decomposed into two
problems: one is for the total mixture, which is equivalent to the half-space problem for a pure vapor [33-35],
and the other is for the noncondensable gas. In [27,28], this property being fully exploited, the F-dependence
of Fs and Fb is obtained explicitly.) In [27,28], however, only the case of M% = 0 was considered. The analysis
of [27,28] for Fs was extended to the case of general M% by T. Taguchi recently (private communication), and
it was shown that the dependence of Fs on M<2 is rather weak, as in the pure vapor case (F = 0) [35] . It should
also be mentioned that some result of F s (Mi,0,T, F) for hard-sphere molecules was obtained recently by the
DSMC method in the case where the molecules of the two components are not identical [36].
Since hA, T, v\ (or MI), and t)2 (or M2) are all constant, they are determined by Eqs. (9)-(lla) and (12)
(note that MI < 1). Some examples of the numerical result, based on the functions hi, h^ for the BGK model
and Fs for the model of [32], for the case where both of the condensed phases are at rest (i.e., t/// = 0) are given
in Table 1 [14]. On the other hand, Fig. 2 shows the result by the DSMC method for hard-sphere molecules
for small Kn (i.e., the numerical solution of the original Boltzmann system) in the case where T///T/ = 1,
nii/ni — 2, UH = 0, and the molecules of both components are the same (mA = mB and d^ = d% for
hard-sphere molecules): the result for n^v/nj = Kn (A = 1) and that for n^v/nj — IKn (A = 2) are shown
in Figs. 2(a) and 2(b), respectively, and that for a pure vapor (n^v = 0 or A = 0) is shown in Fig. 2(c). The
corresponding result of Table 1 for the continuum limit (VA = vi in this limit) is also shown by the dotted line
in the figure. As shown by Figs. 2(a) and 2(b), the noncondensable gas, which is distributed over the whole
region at Kn = 0.1, is confined near the condensing surface at Kn = 0.01, and except for this region and for the
vicinity of the evaporating surface, the flow field is uniform. At Kn = 0.005, the nonuniform regions shrink,
but the uniform flow of the vapor does not change. Such behavior agrees with the result of the asymptotic
analysis described above.
In general, the way of comparison of the results based on different molecular models is not unique. If we suppose that the mutual diffusion coefficient is a basic and common quantity, we have the relation (/o) model =7 Go) hs
TABLE 1. The constants nA, T, vi, and Mi for given values of the parameters T/j/Tj,
nn /m, and A (Un = 0) when the molecule of the noncondensable gas is mechanically the
same as that of the vapor. The values in the parentheses are those obtained by the use of the
conversion of A explained in the main text.
Turn
i
ii
ii
ii
i
1.1
1.1.11
1.1.11
1.1
1.1
1.1
nn/m
1.2
1.2
1.2
1.2
2
2
2
2
5
5
5
5
10
10
10
10
A
0
0.5
1
2
0
0.5
1
2
0
0.5
1
2
0
0.5
1
2
nA
1.118
1.128 (1.131)
1.137 (1.142)
1.149 (1.154)
1.543
1.606 (1.622)
1.655 (1.679)
1.720 (1.747)
2.781
3.158 (3.245)
3.411 (3.533)
3.743 (3.881)
4.614
5.686 (5.913)
6.336 (6.632)
7.141 (7.470)
f
0.981
0.984 (0.984)
0.985 (0.986)
0.987 (0.988)
0.930
0.941 (0.944)
0.949 (0.953)
0.960 (0.964)
0.919
0.960 (0.969)
0.985 (0.996)
1.014 (1.024)
0.854
0.926 (0.939)
0.961 (0.976)
0.999 (1.013)
570
-Vi
0.0423
0.0367
0.0321
0.0259
0.1564
0.1318
0.1136
0.0902
0.3789
0.2942
0.2436
0.1833
0.5069
0.3641
0.2921
0.2138
(0.0351)
(0.0298)
(0.0233)
(0.1257)
(0.1048)
(0.0810)
(0.2761)
(0.2206)
(0.1601)
(0.3378)
(0.2620)
(0.1847)
Mi
0.0468
0.0405 (0.0387)
0.0354 (0.0329)
0.0285 (0.0257)
0.1777
0.1489 (0.1417)
0.1278 (0.1176)
0.1009 (0.0904)
0.4331
0.3289 (0.3073)
0.2689 (0.2422)
0.1994 (0.1733)
0.6009
0.4145 (0.3819)
0.3265 (0.2906)
0.2344 (0.2010)
z
Kn = 0.005
_ Kn = 0.005
/
^MC^O
ft^
1.6
1.6
1.2
1.2
0.4
aA^^
iia
"7
0.4
\ /0.005
i7*^W.?
B
n /ru
0
0
1
-.v^o.oi
10
1
AA
A
:° o.oos
S A /Kn
= 0.1
•° A&*4
*^^
0.96
0.96
0.15
0.15
^0.005
-
"'"'oof -"-
Kn = 0.1
0.005
0.05
1.6
0 05
5
0.5
(a)
Xl/L
1
0.005
i
i
0.5
i
Xl/L
......
1
(b)
Kn = 0.005
1.4
/Kn = 0.1
0.92
0.2
Kn = 0.
0.16
0.01 0.005
0.12
0.5
(c)
FIGURE 2. The behavior for small Kn in the case nf w /n/ = O(Kn) [T///T/ = 1, n///n/ = 2, t/// = 0, and the
molecules of both components are mechanically identical (mB/mA = 1 and d^/d^ — 1 for hard-sphere molecules)], (a)
riav/ni = Kn (A = 1), (b) nf v /nj = 2Kn (A = 2), (c) pure vapor case, i.e., nfv = 0 (A = 0). The symbols •, o, and A
indicate the DSMC result for hard-sphere molecules for small Kn, and the dotted line indicates the corresponding result
for the continuum limit given by Eqs. (9)-(lla) and (12) (the values in the parentheses in Table 1; see the main text).
571
(^ = 0.764215) when the molecules of A— and B— component are identical. Here, the subscripts model and
hs indicate, respectively, the quantity for the model of [32] and that for hard-sphere molecules. But, since
n
av/ni is a giyen quantity irrespective of the molecular model, we should have (AKn) mo d e /= : (AKn)^ s , or
(A) mo rf e /=(A)/ ls /7. This gives a conversion formula between the model of [32] and hard-sphere molecules. If
we regard the A in Table 1 as (A)^ s and use the resulting (A)mode/ (= A/7) in Eqs. (9)-(lla) and (12) (e — 1),
we obtain the values in the parentheses in Table 1. These values are shown by the dotted line in Figs. 2(a)
and 2(b).
To summarize, in the continuum limit, all the noncondensable gas is confined in the Knudsen layer with a
vanishingly small thickness (compared with L) at the condensing surface, and the vapor flow becomes uniform.
The uniform state depends on A. This means the following striking fact. Since we are considering the continuum
limit Kn= 0+ under the condition nf v /n/ = AKn, the average number density nfv of the noncondensable gas
is vanishingly small compared with the reference number density n/. Nevertheless, the vapor flow is affected
by the presence of the noncondensable gas through A and is different from that of the pure vapor case.
GENERAL GEOMETRY
As we have seen in the preceding section, even in the simplest two-surface problems, the continuum limit
with respect to the vapor is not trivial. In one of the limits where n^v/ni = 0(1), an infinitesimal flow of
the vapor gives a finite effect on the state at rest of the mixture (ghost effect), whereas in the other limit
where n^v/nj = O(Kn), an infinitesimal amount of the noncondensable gas gives a finite effect on the uniform
vapor flow. The corresponding limits are also present for the case of a general geometry, though the situation
becomes more complicated (the two types of the limit may coexist). In this section, we discuss the limit of the
first type for the general geometry [17].
Let us consider the case where the shape of the boundary (the condensed phase of the vapor) is arbitrary
but smooth. We investigate the behavior of the mixture in the continuum limit with respect to the vapor. For
simplicity, we consider the following situation: (i) the boundary is at rest; (ii) the noncondensable gas is present
everywhere in the domain; and (iii) the mixture is in a state at rest with a uniform pressure at infinity when
an infinite domain is considered. In this case, an asymptotic analysis similar to that in the previous section
can be carried out. Here, we give only the result for the fluid-dynamic system that describes the behavior in
the continuum limit.
The fluid-dynamic type equations are:
«,(=«* = t»f ) = 0 ,
^-(na<pf)=0,
|^ = 0,
(13)
p
_
dxj
2 dxj
n
4
dxt dxj
where
pa = naf,
a
a
X = h /h,
pa = (m°'/mA)na,
A
A
B
p<t,i=p <t> + p <t,f,
n = nA + nB,
P=pA+pB,
p = pA+pB,
(Aij) = Aij+Aji-(2/3)Aee6i:i.
(17)
(18)
A
Here, TI, T2, TS, T^, and T^ are given functions of the local concentration X of A-component, and 0^,
<f)f , fa, and II are auxiliary unknown functions (note that </>^, </>f , and fa are vectors).
The boundary conditions are:
nA = nw/n0,
T = TW/TQ,
572
(19)
on the boundary. In Eqs. (19) and (20), Tw is the temperature of the boundary, nw is the saturation number
density of the vapor at temperature Tw (Tw and nw may vary along the boundary), TO and no are, respectively,
the reference temperature and number density (see the last paragraph in Introduction), 67 and b£ are given
functions of XA on the boundary, and ef' and e^ are, respectively, the tangential and normal unit vectors
to the boundary.
To give the physical meaning of the auxiliary functions cj)f, $f, (f>i, and II, we mention that, in the asymptotic
analysis, any physical quantity g (g =Fa, na, n, vf, Vi, etc.) is expressed as a sum g=gn+9K with gH
being the moderately varying (Hilbert) part and QK the Knudsen-layer correction. Each part is expanded as
gH=gHQ+gHiKn-i- - • and #K=g*:iKn-|- - •. In particular, vim=vfm=vfHQ=Q in this case. [As in the preceding
section, na, T, £«, etc. in Eqs. (13)-(20) correspond, respectively, to n# 0 , THQ, ^HO, etc.] Then, the auxiliary
functions are identified as <J>f=(2/^/K)v?Hl, 4>i=(2/-\/n)viHi, and II=(4/7r)pH2» and they are of O(l) (PHI,
which does not occur in the above system, is also a constant). Incidentally, to derive the boundary condition
(20), we need the analysis of the Knudsen layer F^l.
As in the two-surface problem with nf v /n/ = O(l) (in the case £/// = 0), the flow of each component
vanishes in the continuum limit Kn = 0+. Nevertheless, as seen from the system (13)-(20), the number
densities, temperature, etc. in this limit are affected by the vanishing (or nonexisting) flow through c/)f, fa,
and II. That is, the ghost effect manifests itself. The cause of the ghost effect is the cause of the flow of O(Kn),
which will be discussed in the following. Let us first examine the boundary condition (20). If the temperature
of the boundary varies along it, a flow of 0(Kn) is induced along the boundary because of the second condition
of Eq. (19) and the first term on RHS of Eq. (20) [recall that fa = (2/A/5r)viHi]- This flow is known as the
thermal creep (see [37-39] for a single-component gas and [40] for a binary mixture). On the other hand, if the
concentration XA varies along the boundary, a flow of O(Kn) is induced because of the second term on RHS
of Eq. (20). This flow is called the diffusion slip (see [41-43,40,44]). Next, let us look into Eqs. (13)-(16). The
terms containing TI and T2 in Eq. (14), which originate from the thermal stress, can be the cause of a gas flow
of O(Kn) [45,46,10], and the terms containing TS and T^, which come from the concentration stress, can also
cause a flow of O(Kn) [47,46]. On the other hand, Eq. (16) indicates that the gradient of the concentration and
that of the temperature cause a relative flow of each component, which is also a possible cause of the flow of
O(Kn). These are known as the mutual diffusion and the thermal diffusion. To summarize, the ghost effect is
caused by (i) thermal creep, (ii) diffusion slip, (iii) thermal stress, (iv) concentration stress, and (v) diffusion.
The causes (i) and (iii) have been clarified in the case of a single-component gas [10]. The cause (v) has already
appeared in the two-surface problem. The rest is peculiar to the present case of general geometry.
The system (13)-(20) is still formal because accurate numerical values of TI, T2, T3, TA, and T^ are not
available at present. A database for these functions similar to that for /}, A, DAB, and DTAB [21] is under
construction. Concerning the coefficient of thermal creep 67 and that of diffusion slip b^, though approximate
results are available [40], we stick to accurate numerical results by a finite-difference analysis of the Knudsen
layer problem. Such a result has been obtained for b$ in [44].
ACKNOWLEDGMENT
The author's attendance at the 22nd International Symposium on Rarefied Gas Dynamics was supported by
the Ministry of Education, Science, Sports and Culture in Japan.
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