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On the diffusion-slip flow of a binary gas mixture over a plane wall with imperfect accommodation
Fluid Dynamics North-Holland
Research
2 (1987) 35-46
35
On the diffusion-slip flow of a binary gas mixture over a plane wall with imperfect accommodation Yoshimoto
ONISHI
Department of Mechanical Engineering. Received
University of Osaka Prefecture, Sakai 591, Japan
6 May 1986
Abstract. The diffusion-slip flow of a binary gas mixture over a plane wall is investigated analytically on the basis of the linearized Boltzmann equation of B-G-K type under the boundary condition of Maxwell’s type or of diffuse-specular reflection type. The velocity field together with the macroscopic slip coefficient is obtained explicitly by an accurate analysis of the Knudsen layer formed near the wall, and the effects of imperfect accommodation of the molecules of the component gases in the mixture are clarified in some detail. The values of the diffusion-slip coefficient for various pairs of actual gases are also given together with those calculated from the corresponding coefficients obtained by other authors.
1. Introduction Suppose a semi-infinite expanse of a slightly rarefied gas mixture bounded by a plane wall with constant temperature. When a non-uniformity in the concentration of the component gases exists on the wall, there occurs under the uniform total pressure a flow over it called diffusion sIip or diffusion creep (Kramers and Kistemaker, 1943; Hidy and Brock, 1970) which is induced by this non-uniformity. This phenomenon, which cannot be described within the ordinary continuum theory, may find various technological applications, one of which will be the collection of small particles (micron- and submicron-sized) floating in a gas mixture by the simple application of the concentration gradient of its component gases in the mixture. This diffusion-slip problem was first studied by Kramers and Kistemaker (1943) following Maxwell’s approach, and an approximate expression for the diffusion-slip coefficient was given. Later, various expressions concerning this coefficient have been presented, e.g., by Zhadnov (1967) using Grad’s 13-moment method (Grad, 1949), by Yalamov et al. (1968) and Lang and Loyalka (1970) based on a simple kinetic model, by Annis and Mason (1970) based on a dusty-gas model, and by Loyalka (1971a) and Lang and Muller (1975) based on the linearized Boltzmann equation. Loyalka used in his analysis a variational technique (e.g. Cercignani 1975) and Lang and Muller, the modified Maxwell method (Loyalka 1971b). Also the empirical expression for the coefficient is given by Schmitt and Waldmann (1960) in their experimental work on the diffusiophoresis problem for a spherical particle (see also Waldmann, 1961; Waldmann and Schmitt, 1966). In the theoretical treatment mentioned above, the analysis within the Knudsen layer, which is crucial in the proper and accurate determination of the slip velocity on the wall and the whole velocity distribution, has not been carried out explicitly. Recently, the diffusion-slip flow (together with the Couette flow and the thermal-slip flow) of a slightly rarefied binary gas mixture bounded by two parallel plane walls has been worked out by Onishi (1986a), who made an accurate analysis of the Knudsen layers associated with 0169-5983/87/$4.00
0 1987, The Japan
Society of Fluid Mechanics
36
Y. Onishi / Diffusion-slip
flow of a binary gas mixture
the flow field, and, as a consequence, the effects of the molecular mass ratio and the concentration ratio of the component gases on the velocity have been clarified in some detail. However, the interaction between the gas molecules and the walls employed in this analysis was of a diffuse-reflection type with perfect accommodation. In reality, the perfect accommodation is not so highly probable, and one may well expect that some of the incident molecules upon the wall are reflected specularly. This type of reflection, if it exists, may affect considerably the slip velocity on the wall and hence the velocity distribution in the whole flow region. The purpose of the present work is to investigate how and to what extent the imperfect accommodation of gas molecules to a wall may affect the velocity of the diffusion-slip flow of a binary gas mixture induced by the partial pressure (or density) gradient of the component gases on the wall when the total pressure is uniform. The analysis will be based on the Boltzmann equation of B-G-K type proposed by Hamel (1965) subject to the boundary condition of Maxwell’s type or diffuse-specular reflection (e.g. Cercignani, 1975) in which a certain fraction of molecules incident upon the wall are reflected specularly, while the remainder are re-emitted diffusely with a Maxwellian distribution characterized by the state of the wall, i.e., the velocity and temperature of the wall. The fraction of these molecules re-emitted diffusely may be called the accommodation coefficient. Incidentally, we note that accommodation coefficients of a more general type (see e.g. Kuscer (1974) for the definitions) simply reduce to this accommodation coefficient for the reflection probability function of Maxwell’s type (see also Kline and Kuscer, 1972). We further restrict ourselves to a case in which deviation of the system from a stationary equilibrium reference state is small and linearization of the problem is possible. A comment may be made briefly on the B-G-K type of equation adopted here. This model equation retains most of the essential features of the collision term of the original Boltzmann equation (the conservation of mass, momentum, and energy of the mixture; the correct collisional transfer of momentum and energy between the component gases), and is expected to produce at least the qualitatively correct behavior of the mixture and its component gases. However, from the quantitative viewpoint, the fact that the model cannot give the correct Prandtl number (it gives the Prandtl number unity) is one of the shortcomings associated with inaccurate parts of the description of the true relaxation processes. 2. Analysis Consider a binary mixture of non-condensable gases, gas A and gas B, over a stationary plane wall (X-Z plane) located at Y = 0. The temperature of the whole system is maintained uniformly at T,. The partial pressure gradient of gas A is imposed along X, while the total pressure of the mixture is kept everywhere uniform at PO. Let the partial pressures of gas A and gas B at X = 0 be denoted by Pt and PoB respectively, so PO= PO*+ P,f. The linearized Boltzmann equation of B-G-K type (Hamel 1965) for this problem may be written in terms of the non-dimensional perturbed quantities as follows:
IIE?M ayl
%I~AW~ S SE8 30 S~~tl3~~OUI aIjl30
@SUap
l2qLUtlU
ay$ %I+J
‘$N
‘S”N/(SoN
-
$T)
= “sZ.4
‘S&I 30 luamn&? ayi IIF UMO~S Lg~~~Idxa SF Lipo[a~ .w~n~a~ou~ 30 luauodmos luma~a~ ayl Quo puv ‘so% wauodwoD-s ~03 luapg3aoD uogepomu~om~ ayl sy (1 3 sx) > 0) so alaqM ‘0 e “3 .1o3
(6)
‘
‘+?-a ‘(9986‘1 ‘a9861 ‘~861 ‘ysyug fg961 ‘IaureH) saw% luauodtuo3 SIP 30 pm a.rnlxp ayl 30 swagg3aoD I.rodsum ayl 01 paleIaJ am sapuanbaq uorsgo~ ayL ~IB~SUOD aq 01 pawnsse alay a.m (vex =)av~ pm aan ‘“n ‘8 se8 30 a~ 30 wy~ 01 v SE% 30 saImaIour ayl 30 vl yled aal maw ayl 30 oyw ayl 1C~a~gw_uaqe .IO ‘sa~n~a~ow v ST&!103 days 01 8 se2 JO sapwaIow ayl 103 kmanbaq UO~SIJIO:,ayl JO O~LI ayl sl 10 ‘alo3alaqJ ‘a se8 30 aInDaIour B 103 huanbaq uoy~o:, a&lam ay] sluasaldal (IZt) + zzO)wN~~ = %l$,4J + BB~ao~ ‘,$I~~~~s fs.rauwd ~0~~~~03 sl! 30 aAysadsaJ.I! v w% 30 apmaIoru 1! .103 huanbaq UO~JIO:, a%lam ayy sluasaldal snql ( zl~ + ~)~?f& = BvWaoN+ “W&T ‘8 St?%30 saItmaIow aql qylrn wy~ ‘“U,“ly PIIt! ‘safmalotu lay10 SIT YlrM v SE% 30 ap-t3aIow I! 30 aug Irun lad suoysg~o:, 30 laqwnu ayl st “n$~ w?% ?uauodwoD-g 30 Llysuap _taqu.rnu amaJa3al aql SB ((OJT)/~~ =) sol pw ‘Llpo~aa asua.Ia3a.I ayl se O~~vzi\ ‘amle~admal amala3al ayl se OJ ‘y@uaI amala3al ayl SB ‘v SE%30 sapma-[ot.u ayl30 ywd aal maw ay$ Buraq vl ‘vet+ uayw amy aM alaH y~~lsuo:, uueurzl~o~ aql y put! v so%?30 SSBUIlpn sad UIWSUOD se% ayi SF vg pm ‘kja+padsal 8 se8 pm v s& 30 sassmu I~pI3a~our ayl ale au1 pm vzu ‘( k?,“iy = :d) amssald pgled ayl (,6+ ~),od = sd pue ‘dlysuap laqumu ayl (sz4 + 1):~ = so ‘(a ‘v = s) ,y witi 30 (luauodruo:, olaz-uou Quo ayl) &goIaA MOWwazu ayi 30 luauoduro+X aqi sn OJvvz/ = sfj ‘saieurp~oo~ .re@hmpa~ ayi (2 ‘A ‘x),1~Jf = (2 ‘A ‘x ) ‘topaz Ri;rDoIaA JepwaIour ‘w”IaA
ayl
(“3 ,x9 ‘“3) O~Vyzi\
.wnDaIo”
‘Qag3adsa~
8 sv2l pm
aqi aJe (am + @&/c_(o~vtr&oN
v
so% 30 suogmn3
puE ($
uoynqysIp
+ s)aZ/E_(o~vtrz)$v
aJaqM
Y. Onishi / Difjiiion-slip flow
38
surface after the interaction S through the wall, i.e.,
with it. n$
ofa binary gas mixture
is determined
by the no net mass flow condition
for gas
(13) We now assume 1973)
the solution
of the present
(PA= Q$(L
5,, 5,> v) -+-W$(L
cpB=cp~(s~~
5,, 5,, Y) + wF(L
I4A = U”(Y),
7AZZ. t
UB = zt”( Y),
pSO
>
problem
in the form (Sone,
1966; Onishi,
sy, Ez),
(14)
‘ty‘y,t,)3
05)
p=p“LyA& nB=pB=
1972,
(16) - ~~~/~~)YA~,
07)
The last expression of where y A is a given constant which is equal to $v’&ZA(l/PoA)(WA/aX). eq. (17) is a consequence of the total pressure of the mixture being kept uniform everywhere. Following the method of analysis in the previous papers (e.g., Onishi, 1973, 1986a), we obtain, after some laborious manipulations, the following set of integral equations for the macroscopic velocities for uA and uB: as \i;ruA = iWuABJ-r(
I Y -y,, I) dy, +
(1 -
aA)fom~ABJ-,(y
+Y,)
dya
(18)
where { = cuy, and J,(y)
is the Abramowitz
function
(Abramowitz
and Stegun,
1964) defined
by -r2-y’t
dt
In the above equations, we extract (Sone, 1966; Onishi, 1986a)
(n: integer). the Knudsen-layer
behavior
of the solutions
by putting
(2%21)
u
where kp* and k: are constants, and YPA and YPB are functions of y which are required to vanish exponentially rapidly as y -+ 00. Y,” and YPB may be called the Knudsen-layer correction functions in view of the nature of the corrections to k: and kp” for small y. It is noted that when the velocity gradient is imposed at infinity (y -+ co), the viscous slip part which has already been studied by Onishi (1986~) may also be derived. Substitution of eqs. (20) and (21) into eqs. (18) and (19) yields a set of coupled integral equations for (k,*, k:, YPA,Y,“), which is subject to the condition Y$
YpB+O sufficiently
rapidly
as y -+ 00.
(22)
Y. Onishi / Diffusion-slip flow of a binary gas mixture
Explicit evaluation of this set of equations in the limit which should be satisfied between kp” and k,*:
39
y ---*cc gives the following
k;=k;-1.
This relation equations:
reduces
relation
(23) the set of integral
equations
above mentioned
to the following
(qA- y,“) J-,(lS-Sol)
dlo
1
(qA-
-cxBk;JO({)
new set of
qB)]J-l(6+s,)
(%I
+ ct”J,({).
(25)
For a special case (YA = aB = 1 (perfect accommodation), these equations reduce exactly to those that have been studied by Onishi (1986a). The fairly accurate solutions to the above integral equations can also be obtained by the refined moment method (Sone and Onishi, 1973) for arbitrary values of aA and aB, except for the case aA = aB = 0 which is briefly discussed in Section 3. That is, YPAand YPBare obtained in series expansions of J,, of the following forms: b,,]J,(z)
withz=
n=O
i’=
a
y
9
zz:’
where a,‘s and b,‘s are constants to be found, and kp* is determined simultaneously constants. These solutions depend on m,/m,, NoB/NoA, K~B /K~ and K~~/K~ as cxA and (Ye. Some of the profiles of YPAand YPBare shown in Fig. l(a), and the values various sets of parameters are listed in Table 1. It may often be preferable to express the velocity field in terms of the mean velocity U = 1/-u, which is calculated from eqs. (20) and (21) as u=
I(mANoA PO
or in its dimensional
u A +mBNoBuB)
= - zyA[k:+
yr\l(z)]j
with these well as on of k: for mass flow
(27)
form,
(28) where k,M=kp*--
mBNOB
PO
ypM =
’
$
[ mANtYpA + msNoBYpB] T
(29)
and p. = m,Nt + mBNoB, p. being the reference density of the mixture (at X= 0). Here it is noted that P;‘(WA/aX) = No-‘(aNA/aX) holds. Some numerical values of kE are also listed in Table 1, and some profiles of YPM and the non-dimensional mass flow velocity, kg + YpM, are shown in Figs. l(b) and 2 respectively. It may be noted that, for aA = (yB = 1, the values of
Y. Omshi / Diffusion-slip flow of a binary gas mixture
40 .6 -
a
Fig. 1. Knudsen-layer correction functions for the diffusion-slip velocity..
k: and kg listed in Table 1 are identical to those given by Onishi (1986a) (the same symbols are used there). It is seen from Fig. 2 that for a special molecular interaction case (K~ = ~~~ = Key), (N~NoB/No2)U is in the direction of the diffusion velocity of the heavier component gas -in other words, in the opposite direction to the partial pressure (or concentration) gradient of
Table 1 The values of kp* and kg mE/mA
K?/%’
for aAA = kAB = kBa
aA=lO
k,*
,
2=1.0
kM I)
aA
=
1.0, aB = 0.8
k,*
kM D
aA= k,*
>cy’=l.O kM D
aA = aB= 0.8 k,*
kM D
0.1
0.1 1.0 10.0
0.06294 0.39792 0.86643
0.05304 0.30701 0.36643
0.05108 0.34477 0.83675
0.04118 0.25386 0.33675
0.06996 0.42666 0.88024
0.06006 0.33575 0.38024
0.05686 0.37203 0.85308
0.04696 0.?8112 0.35308
0.5
0.1 1.0 10.0
0.08005 0.46478 0.89653
0.03243 0.13144 0.06320
0.06594 0.41267 0.87487
0.01832 0.07934 0.04154
0.09388 0.50901 0.91208
0.04626 0.17567 0.07875
0.07754 0.45615 0.89326
0.02992 0.12282 0.05992
1.0
0.1 1.0 10.0
0.09091 0.50000 0.90909
0.0 0.0 0.0
0.07586 0.45086 0.89144
- 0.01504 - 0.04914 - 0.01766
0.10856 0.54914 0.92414
0.01766 0.04914 0.01504
0.09091 0.50000 0.90909
0.0 0.0 0.0
2.0
0.1 1.0 10.0
0.10347 0.53522 0.91995
- 0.06320 -0.13144 - 0.03243
0.08792 0.49099 0.90612
- 0.07875 - 0.17567 - 0.04626
0.12513 0.58733 0.93406
- 0.04154 - 0.07934 - 0.01832
0.10674 0.54384 0.92246
- 0.05992 - 0.12282 - 0.02992
10.0
0.1 1.0 10.0
0.13355 0.60207 0.93706
- 0.36645 - 0.30702 - 0.05304
0.11975 0.57333 0.93004
- 0.38025 - 0.33576
0.16323 0.65522 0.94892
- 0.33677 - 0.25387 - 0.04118
0.14691 0.62796 0.94314
-0.35309 - 0.28113 - 0.04696
Y. Onishi / Diffusion-slip flow of a binary gas mixture
0’
_P
C
\
\ \
C
13
*z
N
0:
Y. Onishi / Diffusion-slip
42
jlow of a binary gas mixture
the heavier component gas, although a slight directional reversion is observed near the wall for aA # aB. The effect of the concentration ratio of the component gases manifests itself mainly in the magnitude of the velocity. The coefficient - kzN:/( N$NeB), which we denote by a,&, i.e.,
* =-/g-
uAB
NOZ
(30)
N,AN,B ’
is the one commonly called the diffusion-slip coefficient for the mean mass flow velocity. Loyalka (1971a) has also obtained an expression for the corresponding coefficient (a,‘;) in his notation) on the basis of the linearized Boltzmann equation with the use of a variational technique, and for Maxwellian molecules (obeying the inverse fifth power force law), to which Hamel’s model equation corresponds (as far as the transport coefficients are concerned), his result is given as (in our notation) (8) -
U12
Noz
1
-
5mApo(N&+
NOB) [ (2 - CX^)N,j% + (2 - CX”)N,B] ( MaA - M1’2aB) N,AaA + NeaaBm
)>
(31)
with 2N, + 3A( M-‘N,A - N,B) + N,B( mA + ma)NoDAB/nB ‘=
2N,+3A(MN,B-N,A)+N;(mA+mB)NODAB/nA
(32)
’
where A = 0.517 (for Maxwellian molecules). It is noted that A and E which appear in his paper are identical to those in Chapman and Cowling (1970), and here E has been eliminated by the relation E = :No(mA + mB)DAB (see Eq. (9.81, 1) of Chapman and Cowling (1970) for the relation between E and DAB). Lang and Muller (1975), with the application of the modified Maxwell-method to the same equation, also obtained the coefficient. Their result, which should correspond to uzB, is given as (in our notation) N,2 s;, P: PO N,ANoB d,, ’
(33)
---
U LM
=
with -=_
s;,
1
61
2 (nA+qB)
na
A
f&
$B+
qB
P:
(M~/$-JL~B)
[(2- “A)(qA/vB)+ (2 -a”)] JMaA
+ (Pi/P;)
aB
I ’
(34) where l,, and Fiji are given in their paper, and here use has been made of the relation between (see Ferziger and Kaper, 1972). All the dir and DAB, i.e., DAB = (po/p~)2(NoANoB/No2)~11 accommodation coefficients for each component gas appearing there are taken to be equal, as they should be, to the present Maxwell’s type accommodation coefficient for the same component gas. The comparison among uAB, ~$28)and uLM are made in Table 2 for the following typical five cases, gas A-gas B = N,-CO,, N,-O,, N,-H,, Ne-Ar, and Ne-He. These diffusion-slip
1.0 0.8 LO 0.8 0.5 1.0
1.0 0.8 LO 0.8 0.5 1.0
1.0 0.8 1.0 0.8 0.5 1.0
1.0 0.8 1.0 0.8 0.5 1.0
1.0 I.0 0.8 0.8 0.5 1.0
1.0 1.0 0.8 0.8 0.5 1.0
1.0 1.0 0.8 0.8 0.5 1.0
1.0 1.0 0.8 0.8 0.5 1.0
0.5
1.0
2.0
IO.0
1.0 0.8 1.0 0.8 0.5 1.0
LO 1.0 0.8 0.8 0.5 1.0
0.1
aB +-TB
up OLM
N242 -* ~,=a
-1.793 0.218 0.068 - 2.592 -0.315 -1.168 0.386 0.284 - 1.886 -0.238 -2.027 -0.008 -0.190 -2.787 -0.390 -1.375 0.193 0.069 -2.059 -0.299 -0.749 0.153 0.070 -1.260 -0.274 -0.229 0.072
QLM
Nz-02
3.011 3.163 2.309 2.443 I.591
0.184 0.193 0.369 0.396 0.025 0.021 0.188 0.181 0.177 0.182 0.170
-0.070 0.01.5 0.138 0.248 -0.223 -0.169 -0.047 0.024 -0.010 0.038 0.061
0,778 0.008 0.028 0.954 0.216 0.257 0.49S -0.174 -0.17s 0.662 0.018 0.036 0.488 0.034 0.046 0.064 2.345 2.529 f.727 1.888 1.203
0.389 0.591 0.133 0.324 0.227 -5.562 -5.174 -5.704 -5.34% -4.930 -3.764
- 1.911 -1.639 -2.040 -1.770 -1.527 -1.005
-1.399 -0,989 -2.273 -1.874 -2,587
-3.384 -2.847 -3.95u -3.460 -3.574
0.299 0.266 0.470 0.450 0.143 0.106 0.285 0.257 0.263 0.244 0.221
0.459 0.338 0.637 0.538 0.262 0.144 0.426 0.325 0.312 0.306 0.273
-0.976 -0.760 -1.213 -0.981 -0,990 -0.906 -0.700 -1.148 -0.925 -0.955
3.358 3.491 2.612 2.730 1.790
0.999 1.158 0.702 0.854 0.636
0.659 0.847 0.404 0.591 0.489
0‘594 0.404 0.771 0.616 0.370 O.ISf 0.546 0.387 0.468 0.362 0.320
0.419 0.069 0.039 0.026 -1.124 -0.527 0.621 0.272 0.265 0.253 -0.923 -0.396 0.178 -0.132 -0.179 -0.191 -1.240 -0.687 0.379 0.069 0.044 0.034 -1.030 -0.537 0.319 0.068 0.053 0.046 -0.876 -0.552 -0.577 0.067 -0.825 -0.645 -0.970 -0.775 -0.701
-1.257 -0.672 -1.522 -0.904 -0.374
‘LM
0.376 0.615 0.129 0.377 0.378
-0.863 -0.679 -0.963 -0.765 -0.619
0.298 0.233 0.48110.445 0.10s 0.033 0.280 0.226 0.251 0.215 0.197
0.391 0.267 0.577 0.484 0.115 0.045 0.361 0.258 0.312 0.243 0.219
?&
op
-2.096 1.002 0.659 -1.669 1.154 0.901 -2.160 0.135 0.337 -1.723 0.924 0.629 - 1.162 0.789 0.584 0.508
flLM
Ne-Ar
0.147 0.488 0.917 0.712 0.501 0.231 0.684 0.467 0.582 0.435 0.381
-0.216 -0.157 -0.292 -0.219 -0.223
01’;’
-0.348 -0.257 -0.462 -0.353 -0.361
0.489 0.307 0.086 0.133 0.050 -0.356 -0.677 0.338 0.324 0.272 -0.077 -0.536 0.667 OS?0 0.256 0.059 -0.143 -0.083 -0.183 -0.557 -0.776 0.447 0.295 0.118 0.121 0.054 -0.271 -0.622 0.379 0.277 0.166 0.103 0.059 -0.143 -0.53s -0.375 0.069 0.246 -I
0.636 0.379 0.795 0.609 0.386 0.086 OS80 0.362 0.486 0.337 0.295
%?B
0p
N,-CO,
-2.112 -2.383 -2.882 -2.588 -2.364 -1.838
-1.369 -1.112 -1.534 -1.283 -1.138 -0,842
-0.914 -0.706 -1.070 -0.854 -0.757 -0.571
-0.612 -0.450 -0.754 -0.576 -0.519 -0.414 ^_-
-0.333 -0.227 -0.451 -0.324 -0.308 -0.281
%FB
Ne-He
-0.648 -0,423 -0.919 -0.687 -0.74s
-0.724 -0st9 -0.911 -0.693 -0.647
-0.869 -0.653 -1.010 -0.778 -0.641
-2.350 -1.851 -2.450 -1.936 -1.315
4LM
-1.713 0.775 -1.285 LOP4 -2.110 -0.044 -1.738 0.252 -1.775 -0.532
-0.770 -0.531 -1.029 -0.785 -0.806
-0.518 -0.344 -0.720 -0.529 -0.544
-0.374 -0.241 -0.536 -0.382 -0.394
-0.252 -0.158 -0.375 -0.258 -0.266
Up
ofthediffusion-slip coefficient calcuh&~I fromvarious expressions presented here. Theunderscored numerical vahwssrethose of a,'$) defined ineq.(36)
NOB/N: aA
The values
Table2
Y. Onishi / Diffiuion-slip Jlow of a binary gas mixture
44
coefficients have been evaluated all the pairs. In their evaluation, (ma/m,,
nA/nB,
at To = 20 o C and PO = 1 atm (thus N, = 2.504. 1019 cm-3) the values
D,,)
= (1.571,1.191,0.16
cm2/s)
for N, - CO,,
= (1.143,
cm2/s)
for N, - O,,
= (0.0714,
0.862,0.22
1.989, 0.76 cm2/s)
= (1.982,1.408,0.32
for
for N, - H,,
cm’/s)
for Ne - Ar,
= (0.198,
1.600, 1.07 cm2/s)
for Ne - He,
with 9* = 0.0175.
10e2 dyne.
s/cm2
mA = 4.649.
1O-23 g
for N,,
nA = 0.0314.
lop2 dyne.
s/cm21
mA = 3.351.
1O-23 g:
for Ne,
were used. The data for the viscosities were taken from the American Institute of Physics Handbook (1972) and those for the diffusion coefficients from the Handbook of Chemistry, Fundamentals II (edited by Chem. Sot. Japan, 1984). The values of ~~~~~~ and K~*/K~, which are needed for the evaluation of uzB, have been calculated from the relation (e.g., Onishi, 1986a) KAB -=
Cm*+
mB)9A
mAmBNoDAB
KAA
’
A
Key -=-
“I
KAA
qB
(35)
It is seen from Table 2 that, in general, the present result uzB and that of Loyalka ~$28)are in qualitative agreement; particularly when the amount of the heavier component gas is large, the two results may be said to agree quite well. However, Lang and Muller’s result gives a somewhat different aspect of the slip coefficient, especially when the amount of either of the component gases is large. In any case, the effects of the imperfect accommodation as well as of the molecular mass and the concentration ratios are not so simple as to be described in general terms, as the Tables listed here show. However, we note, in particular, that 1uzB 1 (and also 1ai!’ I) becomes large as the accommodation coefficient of the heavier component gas becomes poor, and also as its amount in the mixture decreases. Table 2 also contains, for comparison, the numerical values for the corresponding coefficient by Kramers and Kistemaker (1943) (see also Waldmann, 1961; Waldmann and Schmitt, 1966) (v) =
012
_
\/=crnA
-
mB)
(36) h/&)[hWo)
+
which was obtained on the basis accommodation (Y* = aB = 1.
,im,m,]
’
of the approximate
Maxwell’s
approach
under
the perfect
3. Discussion Here we examine some properties (i) For the case K~ = ~~~ = Key: kp*(M,
N,B/%“,
of the solutions to eqs. (24) and (25). Let these solutions be denoted by
aA, aB),
YpA(z; M, NOB/N;,
aA, aB),
YpB(z; M, NOB/No*, aA, aB),
(37)
Y. Onishi / Diffusion-slip flow of a binary gas mixture
where the arguments replacements
above
indicate
that
the
solutions
NoB/N,A + N,A/N,B )
M-M-‘,
in eqs. (24) and (25), we notice that the solutions given by the solutions (37) before the replacement, N,A/N$,
k;(M-I,
depend
on them.
(CIA, a”) + (2, to the resulting i.e.,
CY*,(Y“) = 1 - kp*(M,
45
the
a”) equations
N,B/N,A, a*,
N,A/N,B, a*, a”) = - yP*(z; M, N,B/N,A, a*, a*),
yP*(z;
N,A/N,B, a*, CY”) = - q*(z;
M, N,B/N,A,
(24) and (25) are
a*),
yp”(z; M-‘, M-i,
By making
(38)
aA, a*),
and N,A/N,B, a*, CY”) = -k;(M,
k,M(M-‘,
N,B/N,A,
aA, a*),
where the argument of space z is also given by eq. (26) with (Y= m. The last relation can be confirmed from Table 1. It may also be noted that for a particular case K~ = ICKY = K**, and aA = CY*, for which gas A and gas B are considered virtually identical in physical = mBy mA nature, we have k;
%JSB = -,
andhencekg=O,andYPM=O,
PO
(39)
as may be expected (see also Table 1). (ii) For the limiting case CX*, LX*-+ 0: When (Y* = CX*= 0 (complete specular reflection) in eqs. (24) and (25) then YF = YPB= 0, and k; becomes indeterminate. However, in the limiting case CX*,CX*-+ 0 with the ratio c = CX*/CX*being kept constant (c may be taken to be unity), we can obtain a definite value of kp* as k;
=
&i?~,Bc/(
N; + mN,Bc).
(40)
This is easily shown in the same way as that used by Sone (1970) for the thermal-creep flow case of a single component gas. That is, expand k;, Y* and YPBin the following power series of ff*, k;
=
K.
+
aAK1
+
. . . ,
Yp”= Q,” + CX”Q~ + . . . ,
YP”= Q,” + CY”Q,”+ . . . .
Substitution of these expansions into eqs. (24) and (25) yields a sequence of integral equations for Qf and QB. The first set of equations gives Q,^ = Q,” = 0, of course. The second set, which governs Qf and Q,“, involves K~. Integration of the second set of equations with respect to y from 0 to infinity (remember 5 = CYY)leads us to the definite value of ~~ [ = kp* (CX* -+ 0)] given in eq. (40).
References Abramowitz, M. and LA. Stegun (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series 55) p. 1001. American Institute of Physics Handbook (1972), D.E. Gray, ed. (McGraw-Hill, New York) pp. 2-232-2-248. Annis, B.K. and E.A. Mason (1970) Phys. of Fluids 13, 1452. Chapman, S. and T.G. Cowling (1970) The Mathematical Theory of Non-Uniform Gases (Cambridge Univ. Press, London) pp. 163, 171-172. Cercignani, C. (1975) Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh and London) pp. 118-122,212-215. Ferziger, J.H. and H.G. Kaper (1972) Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam) Section 6.5.
46
Y. Onishi / Diffusion-slip
flow of a binary gas mixture
Grad, H. (1949) Commun. Pure Appl. Math. 2, 331. Hamel, B.B. (1965) Phys. of Fluids 8, 418. Handbook of Chemistry, Fundamentals II (Kagaku Binran, Kiso-Hen II) in Japanese (1984), edited by Chem. Sot. Japan (Maruzen, Tokyo) pp. 11-68-H-69. Hidy, G.M. and J.R. Brock (1970) The Dynamics of Aerocolloidal Systems (Pergamon Press, Oxford) pp. 58-59. Khnc, T. and I. Kuscer (1972) Phys. of Fluids 15, 1018. Kramers, H.A. and J. Kistemaker (1943) Physica 10, 699. Kuscer, I. (1974) in: M. Becker and M. Fiebig, eds., Rarefied Gas Dynamics (DFVLR-Press, Porz-Wahn, Germany) Vol. II, pp.E.l-1-El-21. Lang, H. and S.K. Loyalka (1970) Phys. of Fluids 13, 1871. Lang, H. and W.J.C. Muller (1975) Z. Naturforsch. 3Oa, 855. Loyalka, S.K. (1971a) Phys. of Fluids 14, 2599. Loyalka, SK. (1971b) Phys. of Flui& 14, 2291. On&i, Y. (1972) Trans. Japan Sot. Aero. Space Sci. 15, 117. O&hi, Y. (1973) BUN. Uniu. Osaka Prefecture Ser. A 22, 91. Onishi, Y. (1984) in: H. Oguchi, ed., Rarefied Gas Dynamics (Univ. of Tokyo Press, Tokyo) Vol. II, pp. 875-884. O&hi, Y. (1986a) ZAMP 37, 573. O&hi, Y. (1986b) J. Fluid. Mech. 163, 171. Onishi, Y. (1986~) in: T. Matsui, ed., Proc. Third Asian Congress of Fluid Mechanics, pp. 523-526. Schmitt, K.H. and L. Waldmann (1960) Z. Naturforsch. IZa, 843. Sone, Y. (1966) J. Phys. Sot. Japan 21, 1836. Sone, Y. (1970) J. Phys. Sot. Japan 29, 1655. Sone, Y. and Y. Onishi (1973) J. Phys. Sot. Japan 35, 1773. Waldmann, L. (1961) in: L. Talbot, ed., Rarefied Gas Dynamics (Academic Press, New York) pp. 323-344. Waldmann, L. and K.H. Schmitt (1966) in: C.N. Davies, ed., Aerosol Science (Academic, New York) pp. 137-162. Yalamov, Yu.I., I.N. Ivchenko and B.V. Deryagin (1968) Sou. Pkys. Dokl. 13, 446. Zhdanov, V.M. (1967) Soo. Phys. Tech. Phys. 12, 134.
Research
2 (1987) 35-46
35
On the diffusion-slip flow of a binary gas mixture over a plane wall with imperfect accommodation Yoshimoto
ONISHI
Department of Mechanical Engineering. Received
University of Osaka Prefecture, Sakai 591, Japan
6 May 1986
Abstract. The diffusion-slip flow of a binary gas mixture over a plane wall is investigated analytically on the basis of the linearized Boltzmann equation of B-G-K type under the boundary condition of Maxwell’s type or of diffuse-specular reflection type. The velocity field together with the macroscopic slip coefficient is obtained explicitly by an accurate analysis of the Knudsen layer formed near the wall, and the effects of imperfect accommodation of the molecules of the component gases in the mixture are clarified in some detail. The values of the diffusion-slip coefficient for various pairs of actual gases are also given together with those calculated from the corresponding coefficients obtained by other authors.
1. Introduction Suppose a semi-infinite expanse of a slightly rarefied gas mixture bounded by a plane wall with constant temperature. When a non-uniformity in the concentration of the component gases exists on the wall, there occurs under the uniform total pressure a flow over it called diffusion sIip or diffusion creep (Kramers and Kistemaker, 1943; Hidy and Brock, 1970) which is induced by this non-uniformity. This phenomenon, which cannot be described within the ordinary continuum theory, may find various technological applications, one of which will be the collection of small particles (micron- and submicron-sized) floating in a gas mixture by the simple application of the concentration gradient of its component gases in the mixture. This diffusion-slip problem was first studied by Kramers and Kistemaker (1943) following Maxwell’s approach, and an approximate expression for the diffusion-slip coefficient was given. Later, various expressions concerning this coefficient have been presented, e.g., by Zhadnov (1967) using Grad’s 13-moment method (Grad, 1949), by Yalamov et al. (1968) and Lang and Loyalka (1970) based on a simple kinetic model, by Annis and Mason (1970) based on a dusty-gas model, and by Loyalka (1971a) and Lang and Muller (1975) based on the linearized Boltzmann equation. Loyalka used in his analysis a variational technique (e.g. Cercignani 1975) and Lang and Muller, the modified Maxwell method (Loyalka 1971b). Also the empirical expression for the coefficient is given by Schmitt and Waldmann (1960) in their experimental work on the diffusiophoresis problem for a spherical particle (see also Waldmann, 1961; Waldmann and Schmitt, 1966). In the theoretical treatment mentioned above, the analysis within the Knudsen layer, which is crucial in the proper and accurate determination of the slip velocity on the wall and the whole velocity distribution, has not been carried out explicitly. Recently, the diffusion-slip flow (together with the Couette flow and the thermal-slip flow) of a slightly rarefied binary gas mixture bounded by two parallel plane walls has been worked out by Onishi (1986a), who made an accurate analysis of the Knudsen layers associated with 0169-5983/87/$4.00
0 1987, The Japan
Society of Fluid Mechanics
36
Y. Onishi / Diffusion-slip
flow of a binary gas mixture
the flow field, and, as a consequence, the effects of the molecular mass ratio and the concentration ratio of the component gases on the velocity have been clarified in some detail. However, the interaction between the gas molecules and the walls employed in this analysis was of a diffuse-reflection type with perfect accommodation. In reality, the perfect accommodation is not so highly probable, and one may well expect that some of the incident molecules upon the wall are reflected specularly. This type of reflection, if it exists, may affect considerably the slip velocity on the wall and hence the velocity distribution in the whole flow region. The purpose of the present work is to investigate how and to what extent the imperfect accommodation of gas molecules to a wall may affect the velocity of the diffusion-slip flow of a binary gas mixture induced by the partial pressure (or density) gradient of the component gases on the wall when the total pressure is uniform. The analysis will be based on the Boltzmann equation of B-G-K type proposed by Hamel (1965) subject to the boundary condition of Maxwell’s type or diffuse-specular reflection (e.g. Cercignani, 1975) in which a certain fraction of molecules incident upon the wall are reflected specularly, while the remainder are re-emitted diffusely with a Maxwellian distribution characterized by the state of the wall, i.e., the velocity and temperature of the wall. The fraction of these molecules re-emitted diffusely may be called the accommodation coefficient. Incidentally, we note that accommodation coefficients of a more general type (see e.g. Kuscer (1974) for the definitions) simply reduce to this accommodation coefficient for the reflection probability function of Maxwell’s type (see also Kline and Kuscer, 1972). We further restrict ourselves to a case in which deviation of the system from a stationary equilibrium reference state is small and linearization of the problem is possible. A comment may be made briefly on the B-G-K type of equation adopted here. This model equation retains most of the essential features of the collision term of the original Boltzmann equation (the conservation of mass, momentum, and energy of the mixture; the correct collisional transfer of momentum and energy between the component gases), and is expected to produce at least the qualitatively correct behavior of the mixture and its component gases. However, from the quantitative viewpoint, the fact that the model cannot give the correct Prandtl number (it gives the Prandtl number unity) is one of the shortcomings associated with inaccurate parts of the description of the true relaxation processes. 2. Analysis Consider a binary mixture of non-condensable gases, gas A and gas B, over a stationary plane wall (X-Z plane) located at Y = 0. The temperature of the whole system is maintained uniformly at T,. The partial pressure gradient of gas A is imposed along X, while the total pressure of the mixture is kept everywhere uniform at PO. Let the partial pressures of gas A and gas B at X = 0 be denoted by Pt and PoB respectively, so PO= PO*+ P,f. The linearized Boltzmann equation of B-G-K type (Hamel 1965) for this problem may be written in terms of the non-dimensional perturbed quantities as follows:
IIE?M ayl
%I~AW~ S SE8 30 S~~tl3~~OUI aIjl30
@SUap
l2qLUtlU
ay$ %I+J
‘$N
‘S”N/(SoN
-
$T)
= “sZ.4
‘S&I 30 luamn&? ayi IIF UMO~S Lg~~~Idxa SF Lipo[a~ .w~n~a~ou~ 30 luauodmos luma~a~ ayl Quo puv ‘so% wauodwoD-s ~03 luapg3aoD uogepomu~om~ ayl sy (1 3 sx) > 0) so alaqM ‘0 e “3 .1o3
(6)
‘
‘+?-a ‘(9986‘1 ‘a9861 ‘~861 ‘ysyug fg961 ‘IaureH) saw% luauodtuo3 SIP 30 pm a.rnlxp ayl 30 swagg3aoD I.rodsum ayl 01 paleIaJ am sapuanbaq uorsgo~ ayL ~IB~SUOD aq 01 pawnsse alay a.m (vex =)av~ pm aan ‘“n ‘8 se8 30 a~ 30 wy~ 01 v SE% 30 saImaIour ayl 30 vl yled aal maw ayl 30 oyw ayl 1C~a~gw_uaqe .IO ‘sa~n~a~ow v ST&!103 days 01 8 se2 JO sapwaIow ayl 103 kmanbaq UO~SIJIO:,ayl JO O~LI ayl sl 10 ‘alo3alaqJ ‘a se8 30 aInDaIour B 103 huanbaq uoy~o:, a&lam ay] sluasaldal (IZt) + zzO)wN~~ = %l$,4J + BB~ao~ ‘,$I~~~~s fs.rauwd ~0~~~~03 sl! 30 aAysadsaJ.I! v w% 30 apmaIoru 1! .103 huanbaq UO~JIO:, a%lam ayy sluasaldal snql ( zl~ + ~)~?f& = BvWaoN+ “W&T ‘8 St?%30 saItmaIow aql qylrn wy~ ‘“U,“ly PIIt! ‘safmalotu lay10 SIT YlrM v SE% 30 ap-t3aIow I! 30 aug Irun lad suoysg~o:, 30 laqwnu ayl st “n$~ w?% ?uauodwoD-g 30 Llysuap _taqu.rnu amaJa3al aql SB ((OJT)/~~ =) sol pw ‘Llpo~aa asua.Ia3a.I ayl se O~~vzi\ ‘amle~admal amala3al ayl se OJ ‘y@uaI amala3al ayl SB ‘v SE%30 sapma-[ot.u ayl30 ywd aal maw ay$ Buraq vl ‘vet+ uayw amy aM alaH y~~lsuo:, uueurzl~o~ aql y put! v so%?30 SSBUIlpn sad UIWSUOD se% ayi SF vg pm ‘kja+padsal 8 se8 pm v s& 30 sassmu I~pI3a~our ayl ale au1 pm vzu ‘( k?,“iy = :d) amssald pgled ayl (,6+ ~),od = sd pue ‘dlysuap laqumu ayl (sz4 + 1):~ = so ‘(a ‘v = s) ,y witi 30 (luauodruo:, olaz-uou Quo ayl) &goIaA MOWwazu ayi 30 luauoduro+X aqi sn OJvvz/ = sfj ‘saieurp~oo~ .re@hmpa~ ayi (2 ‘A ‘x),1~Jf = (2 ‘A ‘x ) ‘topaz Ri;rDoIaA JepwaIour ‘w”IaA
ayl
(“3 ,x9 ‘“3) O~Vyzi\
.wnDaIo”
‘Qag3adsa~
8 sv2l pm
aqi aJe (am + @&/c_(o~vtr&oN
v
so% 30 suogmn3
puE ($
uoynqysIp
+ s)aZ/E_(o~vtrz)$v
aJaqM
Y. Onishi / Difjiiion-slip flow
38
surface after the interaction S through the wall, i.e.,
with it. n$
ofa binary gas mixture
is determined
by the no net mass flow condition
for gas
(13) We now assume 1973)
the solution
of the present
(PA= Q$(L
5,, 5,> v) -+-W$(L
cpB=cp~(s~~
5,, 5,, Y) + wF(L
I4A = U”(Y),
7AZZ. t
UB = zt”( Y),
pSO
>
problem
in the form (Sone,
1966; Onishi,
sy, Ez),
(14)
‘ty‘y,t,)3
05)
p=p“LyA& nB=pB=
1972,
(16) - ~~~/~~)YA~,
07)
The last expression of where y A is a given constant which is equal to $v’&ZA(l/PoA)(WA/aX). eq. (17) is a consequence of the total pressure of the mixture being kept uniform everywhere. Following the method of analysis in the previous papers (e.g., Onishi, 1973, 1986a), we obtain, after some laborious manipulations, the following set of integral equations for the macroscopic velocities for uA and uB: as \i;ruA = iWuABJ-r(
I Y -y,, I) dy, +
(1 -
aA)fom~ABJ-,(y
+Y,)
dya
(18)
where { = cuy, and J,(y)
is the Abramowitz
function
(Abramowitz
and Stegun,
1964) defined
by -r2-y’t
dt
In the above equations, we extract (Sone, 1966; Onishi, 1986a)
(n: integer). the Knudsen-layer
behavior
of the solutions
by putting
(2%21)
u
where kp* and k: are constants, and YPA and YPB are functions of y which are required to vanish exponentially rapidly as y -+ 00. Y,” and YPB may be called the Knudsen-layer correction functions in view of the nature of the corrections to k: and kp” for small y. It is noted that when the velocity gradient is imposed at infinity (y -+ co), the viscous slip part which has already been studied by Onishi (1986~) may also be derived. Substitution of eqs. (20) and (21) into eqs. (18) and (19) yields a set of coupled integral equations for (k,*, k:, YPA,Y,“), which is subject to the condition Y$
YpB+O sufficiently
rapidly
as y -+ 00.
(22)
Y. Onishi / Diffusion-slip flow of a binary gas mixture
Explicit evaluation of this set of equations in the limit which should be satisfied between kp” and k,*:
39
y ---*cc gives the following
k;=k;-1.
This relation equations:
reduces
relation
(23) the set of integral
equations
above mentioned
to the following
(qA- y,“) J-,(lS-Sol)
dlo
1
(qA-
-cxBk;JO({)
new set of
qB)]J-l(6+s,)
(%I
+ ct”J,({).
(25)
For a special case (YA = aB = 1 (perfect accommodation), these equations reduce exactly to those that have been studied by Onishi (1986a). The fairly accurate solutions to the above integral equations can also be obtained by the refined moment method (Sone and Onishi, 1973) for arbitrary values of aA and aB, except for the case aA = aB = 0 which is briefly discussed in Section 3. That is, YPAand YPBare obtained in series expansions of J,, of the following forms: b,,]J,(z)
withz=
n=O
i’=
a
y
9
zz:’
where a,‘s and b,‘s are constants to be found, and kp* is determined simultaneously constants. These solutions depend on m,/m,, NoB/NoA, K~B /K~ and K~~/K~ as cxA and (Ye. Some of the profiles of YPAand YPBare shown in Fig. l(a), and the values various sets of parameters are listed in Table 1. It may often be preferable to express the velocity field in terms of the mean velocity U = 1/-u, which is calculated from eqs. (20) and (21) as u=
I(mANoA PO
or in its dimensional
u A +mBNoBuB)
= - zyA[k:+
yr\l(z)]j
with these well as on of k: for mass flow
(27)
form,
(28) where k,M=kp*--
mBNOB
PO
ypM =
’
$
[ mANtYpA + msNoBYpB] T
(29)
and p. = m,Nt + mBNoB, p. being the reference density of the mixture (at X= 0). Here it is noted that P;‘(WA/aX) = No-‘(aNA/aX) holds. Some numerical values of kE are also listed in Table 1, and some profiles of YPM and the non-dimensional mass flow velocity, kg + YpM, are shown in Figs. l(b) and 2 respectively. It may be noted that, for aA = (yB = 1, the values of
Y. Omshi / Diffusion-slip flow of a binary gas mixture
40 .6 -
a
Fig. 1. Knudsen-layer correction functions for the diffusion-slip velocity..
k: and kg listed in Table 1 are identical to those given by Onishi (1986a) (the same symbols are used there). It is seen from Fig. 2 that for a special molecular interaction case (K~ = ~~~ = Key), (N~NoB/No2)U is in the direction of the diffusion velocity of the heavier component gas -in other words, in the opposite direction to the partial pressure (or concentration) gradient of
Table 1 The values of kp* and kg mE/mA
K?/%’
for aAA = kAB = kBa
aA=lO
k,*
,
2=1.0
kM I)
aA
=
1.0, aB = 0.8
k,*
kM D
aA= k,*
>cy’=l.O kM D
aA = aB= 0.8 k,*
kM D
0.1
0.1 1.0 10.0
0.06294 0.39792 0.86643
0.05304 0.30701 0.36643
0.05108 0.34477 0.83675
0.04118 0.25386 0.33675
0.06996 0.42666 0.88024
0.06006 0.33575 0.38024
0.05686 0.37203 0.85308
0.04696 0.?8112 0.35308
0.5
0.1 1.0 10.0
0.08005 0.46478 0.89653
0.03243 0.13144 0.06320
0.06594 0.41267 0.87487
0.01832 0.07934 0.04154
0.09388 0.50901 0.91208
0.04626 0.17567 0.07875
0.07754 0.45615 0.89326
0.02992 0.12282 0.05992
1.0
0.1 1.0 10.0
0.09091 0.50000 0.90909
0.0 0.0 0.0
0.07586 0.45086 0.89144
- 0.01504 - 0.04914 - 0.01766
0.10856 0.54914 0.92414
0.01766 0.04914 0.01504
0.09091 0.50000 0.90909
0.0 0.0 0.0
2.0
0.1 1.0 10.0
0.10347 0.53522 0.91995
- 0.06320 -0.13144 - 0.03243
0.08792 0.49099 0.90612
- 0.07875 - 0.17567 - 0.04626
0.12513 0.58733 0.93406
- 0.04154 - 0.07934 - 0.01832
0.10674 0.54384 0.92246
- 0.05992 - 0.12282 - 0.02992
10.0
0.1 1.0 10.0
0.13355 0.60207 0.93706
- 0.36645 - 0.30702 - 0.05304
0.11975 0.57333 0.93004
- 0.38025 - 0.33576
0.16323 0.65522 0.94892
- 0.33677 - 0.25387 - 0.04118
0.14691 0.62796 0.94314
-0.35309 - 0.28113 - 0.04696
Y. Onishi / Diffusion-slip flow of a binary gas mixture
0’
_P
C
\
\ \
C
13
*z
N
0:
Y. Onishi / Diffusion-slip
42
jlow of a binary gas mixture
the heavier component gas, although a slight directional reversion is observed near the wall for aA # aB. The effect of the concentration ratio of the component gases manifests itself mainly in the magnitude of the velocity. The coefficient - kzN:/( N$NeB), which we denote by a,&, i.e.,
* =-/g-
uAB
NOZ
(30)
N,AN,B ’
is the one commonly called the diffusion-slip coefficient for the mean mass flow velocity. Loyalka (1971a) has also obtained an expression for the corresponding coefficient (a,‘;) in his notation) on the basis of the linearized Boltzmann equation with the use of a variational technique, and for Maxwellian molecules (obeying the inverse fifth power force law), to which Hamel’s model equation corresponds (as far as the transport coefficients are concerned), his result is given as (in our notation) (8) -
U12
Noz
1
-
5mApo(N&+
NOB) [ (2 - CX^)N,j% + (2 - CX”)N,B] ( MaA - M1’2aB) N,AaA + NeaaBm
)>
(31)
with 2N, + 3A( M-‘N,A - N,B) + N,B( mA + ma)NoDAB/nB ‘=
2N,+3A(MN,B-N,A)+N;(mA+mB)NODAB/nA
(32)
’
where A = 0.517 (for Maxwellian molecules). It is noted that A and E which appear in his paper are identical to those in Chapman and Cowling (1970), and here E has been eliminated by the relation E = :No(mA + mB)DAB (see Eq. (9.81, 1) of Chapman and Cowling (1970) for the relation between E and DAB). Lang and Muller (1975), with the application of the modified Maxwell-method to the same equation, also obtained the coefficient. Their result, which should correspond to uzB, is given as (in our notation) N,2 s;, P: PO N,ANoB d,, ’
(33)
---
U LM
=
with -=_
s;,
1
61
2 (nA+qB)
na
A
f&
$B+
qB
P:
(M~/$-JL~B)
[(2- “A)(qA/vB)+ (2 -a”)] JMaA
+ (Pi/P;)
aB
I ’
(34) where l,, and Fiji are given in their paper, and here use has been made of the relation between (see Ferziger and Kaper, 1972). All the dir and DAB, i.e., DAB = (po/p~)2(NoANoB/No2)~11 accommodation coefficients for each component gas appearing there are taken to be equal, as they should be, to the present Maxwell’s type accommodation coefficient for the same component gas. The comparison among uAB, ~$28)and uLM are made in Table 2 for the following typical five cases, gas A-gas B = N,-CO,, N,-O,, N,-H,, Ne-Ar, and Ne-He. These diffusion-slip
1.0 0.8 LO 0.8 0.5 1.0
1.0 0.8 LO 0.8 0.5 1.0
1.0 0.8 1.0 0.8 0.5 1.0
1.0 0.8 1.0 0.8 0.5 1.0
1.0 I.0 0.8 0.8 0.5 1.0
1.0 1.0 0.8 0.8 0.5 1.0
1.0 1.0 0.8 0.8 0.5 1.0
1.0 1.0 0.8 0.8 0.5 1.0
0.5
1.0
2.0
IO.0
1.0 0.8 1.0 0.8 0.5 1.0
LO 1.0 0.8 0.8 0.5 1.0
0.1
aB +-TB
up OLM
N242 -* ~,=a
-1.793 0.218 0.068 - 2.592 -0.315 -1.168 0.386 0.284 - 1.886 -0.238 -2.027 -0.008 -0.190 -2.787 -0.390 -1.375 0.193 0.069 -2.059 -0.299 -0.749 0.153 0.070 -1.260 -0.274 -0.229 0.072
QLM
Nz-02
3.011 3.163 2.309 2.443 I.591
0.184 0.193 0.369 0.396 0.025 0.021 0.188 0.181 0.177 0.182 0.170
-0.070 0.01.5 0.138 0.248 -0.223 -0.169 -0.047 0.024 -0.010 0.038 0.061
0,778 0.008 0.028 0.954 0.216 0.257 0.49S -0.174 -0.17s 0.662 0.018 0.036 0.488 0.034 0.046 0.064 2.345 2.529 f.727 1.888 1.203
0.389 0.591 0.133 0.324 0.227 -5.562 -5.174 -5.704 -5.34% -4.930 -3.764
- 1.911 -1.639 -2.040 -1.770 -1.527 -1.005
-1.399 -0,989 -2.273 -1.874 -2,587
-3.384 -2.847 -3.95u -3.460 -3.574
0.299 0.266 0.470 0.450 0.143 0.106 0.285 0.257 0.263 0.244 0.221
0.459 0.338 0.637 0.538 0.262 0.144 0.426 0.325 0.312 0.306 0.273
-0.976 -0.760 -1.213 -0.981 -0,990 -0.906 -0.700 -1.148 -0.925 -0.955
3.358 3.491 2.612 2.730 1.790
0.999 1.158 0.702 0.854 0.636
0.659 0.847 0.404 0.591 0.489
0‘594 0.404 0.771 0.616 0.370 O.ISf 0.546 0.387 0.468 0.362 0.320
0.419 0.069 0.039 0.026 -1.124 -0.527 0.621 0.272 0.265 0.253 -0.923 -0.396 0.178 -0.132 -0.179 -0.191 -1.240 -0.687 0.379 0.069 0.044 0.034 -1.030 -0.537 0.319 0.068 0.053 0.046 -0.876 -0.552 -0.577 0.067 -0.825 -0.645 -0.970 -0.775 -0.701
-1.257 -0.672 -1.522 -0.904 -0.374
‘LM
0.376 0.615 0.129 0.377 0.378
-0.863 -0.679 -0.963 -0.765 -0.619
0.298 0.233 0.48110.445 0.10s 0.033 0.280 0.226 0.251 0.215 0.197
0.391 0.267 0.577 0.484 0.115 0.045 0.361 0.258 0.312 0.243 0.219
?&
op
-2.096 1.002 0.659 -1.669 1.154 0.901 -2.160 0.135 0.337 -1.723 0.924 0.629 - 1.162 0.789 0.584 0.508
flLM
Ne-Ar
0.147 0.488 0.917 0.712 0.501 0.231 0.684 0.467 0.582 0.435 0.381
-0.216 -0.157 -0.292 -0.219 -0.223
01’;’
-0.348 -0.257 -0.462 -0.353 -0.361
0.489 0.307 0.086 0.133 0.050 -0.356 -0.677 0.338 0.324 0.272 -0.077 -0.536 0.667 OS?0 0.256 0.059 -0.143 -0.083 -0.183 -0.557 -0.776 0.447 0.295 0.118 0.121 0.054 -0.271 -0.622 0.379 0.277 0.166 0.103 0.059 -0.143 -0.53s -0.375 0.069 0.246 -I
0.636 0.379 0.795 0.609 0.386 0.086 OS80 0.362 0.486 0.337 0.295
%?B
0p
N,-CO,
-2.112 -2.383 -2.882 -2.588 -2.364 -1.838
-1.369 -1.112 -1.534 -1.283 -1.138 -0,842
-0.914 -0.706 -1.070 -0.854 -0.757 -0.571
-0.612 -0.450 -0.754 -0.576 -0.519 -0.414 ^_-
-0.333 -0.227 -0.451 -0.324 -0.308 -0.281
%FB
Ne-He
-0.648 -0,423 -0.919 -0.687 -0.74s
-0.724 -0st9 -0.911 -0.693 -0.647
-0.869 -0.653 -1.010 -0.778 -0.641
-2.350 -1.851 -2.450 -1.936 -1.315
4LM
-1.713 0.775 -1.285 LOP4 -2.110 -0.044 -1.738 0.252 -1.775 -0.532
-0.770 -0.531 -1.029 -0.785 -0.806
-0.518 -0.344 -0.720 -0.529 -0.544
-0.374 -0.241 -0.536 -0.382 -0.394
-0.252 -0.158 -0.375 -0.258 -0.266
Up
ofthediffusion-slip coefficient calcuh&~I fromvarious expressions presented here. Theunderscored numerical vahwssrethose of a,'$) defined ineq.(36)
NOB/N: aA
The values
Table2
Y. Onishi / Diffiuion-slip Jlow of a binary gas mixture
44
coefficients have been evaluated all the pairs. In their evaluation, (ma/m,,
nA/nB,
at To = 20 o C and PO = 1 atm (thus N, = 2.504. 1019 cm-3) the values
D,,)
= (1.571,1.191,0.16
cm2/s)
for N, - CO,,
= (1.143,
cm2/s)
for N, - O,,
= (0.0714,
0.862,0.22
1.989, 0.76 cm2/s)
= (1.982,1.408,0.32
for
for N, - H,,
cm’/s)
for Ne - Ar,
= (0.198,
1.600, 1.07 cm2/s)
for Ne - He,
with 9* = 0.0175.
10e2 dyne.
s/cm2
mA = 4.649.
1O-23 g
for N,,
nA = 0.0314.
lop2 dyne.
s/cm21
mA = 3.351.
1O-23 g:
for Ne,
were used. The data for the viscosities were taken from the American Institute of Physics Handbook (1972) and those for the diffusion coefficients from the Handbook of Chemistry, Fundamentals II (edited by Chem. Sot. Japan, 1984). The values of ~~~~~~ and K~*/K~, which are needed for the evaluation of uzB, have been calculated from the relation (e.g., Onishi, 1986a) KAB -=
Cm*+
mB)9A
mAmBNoDAB
KAA
’
A
Key -=-
“I
KAA
qB
(35)
It is seen from Table 2 that, in general, the present result uzB and that of Loyalka ~$28)are in qualitative agreement; particularly when the amount of the heavier component gas is large, the two results may be said to agree quite well. However, Lang and Muller’s result gives a somewhat different aspect of the slip coefficient, especially when the amount of either of the component gases is large. In any case, the effects of the imperfect accommodation as well as of the molecular mass and the concentration ratios are not so simple as to be described in general terms, as the Tables listed here show. However, we note, in particular, that 1uzB 1 (and also 1ai!’ I) becomes large as the accommodation coefficient of the heavier component gas becomes poor, and also as its amount in the mixture decreases. Table 2 also contains, for comparison, the numerical values for the corresponding coefficient by Kramers and Kistemaker (1943) (see also Waldmann, 1961; Waldmann and Schmitt, 1966) (v) =
012
_
\/=crnA
-
mB)
(36) h/&)[hWo)
+
which was obtained on the basis accommodation (Y* = aB = 1.
,im,m,]
’
of the approximate
Maxwell’s
approach
under
the perfect
3. Discussion Here we examine some properties (i) For the case K~ = ~~~ = Key: kp*(M,
N,B/%“,
of the solutions to eqs. (24) and (25). Let these solutions be denoted by
aA, aB),
YpA(z; M, NOB/N;,
aA, aB),
YpB(z; M, NOB/No*, aA, aB),
(37)
Y. Onishi / Diffusion-slip flow of a binary gas mixture
where the arguments replacements
above
indicate
that
the
solutions
NoB/N,A + N,A/N,B )
M-M-‘,
in eqs. (24) and (25), we notice that the solutions given by the solutions (37) before the replacement, N,A/N$,
k;(M-I,
depend
on them.
(CIA, a”) + (2, to the resulting i.e.,
CY*,(Y“) = 1 - kp*(M,
45
the
a”) equations
N,B/N,A, a*,
N,A/N,B, a*, a”) = - yP*(z; M, N,B/N,A, a*, a*),
yP*(z;
N,A/N,B, a*, CY”) = - q*(z;
M, N,B/N,A,
(24) and (25) are
a*),
yp”(z; M-‘, M-i,
By making
(38)
aA, a*),
and N,A/N,B, a*, CY”) = -k;(M,
k,M(M-‘,
N,B/N,A,
aA, a*),
where the argument of space z is also given by eq. (26) with (Y= m. The last relation can be confirmed from Table 1. It may also be noted that for a particular case K~ = ICKY = K**, and aA = CY*, for which gas A and gas B are considered virtually identical in physical = mBy mA nature, we have k;
%JSB = -,
andhencekg=O,andYPM=O,
PO
(39)
as may be expected (see also Table 1). (ii) For the limiting case CX*, LX*-+ 0: When (Y* = CX*= 0 (complete specular reflection) in eqs. (24) and (25) then YF = YPB= 0, and k; becomes indeterminate. However, in the limiting case CX*,CX*-+ 0 with the ratio c = CX*/CX*being kept constant (c may be taken to be unity), we can obtain a definite value of kp* as k;
=
&i?~,Bc/(
N; + mN,Bc).
(40)
This is easily shown in the same way as that used by Sone (1970) for the thermal-creep flow case of a single component gas. That is, expand k;, Y* and YPBin the following power series of ff*, k;
=
K.
+
aAK1
+
. . . ,
Yp”= Q,” + CX”Q~ + . . . ,
YP”= Q,” + CY”Q,”+ . . . .
Substitution of these expansions into eqs. (24) and (25) yields a sequence of integral equations for Qf and QB. The first set of equations gives Q,^ = Q,” = 0, of course. The second set, which governs Qf and Q,“, involves K~. Integration of the second set of equations with respect to y from 0 to infinity (remember 5 = CYY)leads us to the definite value of ~~ [ = kp* (CX* -+ 0)] given in eq. (40).
References Abramowitz, M. and LA. Stegun (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series 55) p. 1001. American Institute of Physics Handbook (1972), D.E. Gray, ed. (McGraw-Hill, New York) pp. 2-232-2-248. Annis, B.K. and E.A. Mason (1970) Phys. of Fluids 13, 1452. Chapman, S. and T.G. Cowling (1970) The Mathematical Theory of Non-Uniform Gases (Cambridge Univ. Press, London) pp. 163, 171-172. Cercignani, C. (1975) Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh and London) pp. 118-122,212-215. Ferziger, J.H. and H.G. Kaper (1972) Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam) Section 6.5.
46
Y. Onishi / Diffusion-slip
flow of a binary gas mixture
Grad, H. (1949) Commun. Pure Appl. Math. 2, 331. Hamel, B.B. (1965) Phys. of Fluids 8, 418. Handbook of Chemistry, Fundamentals II (Kagaku Binran, Kiso-Hen II) in Japanese (1984), edited by Chem. Sot. Japan (Maruzen, Tokyo) pp. 11-68-H-69. Hidy, G.M. and J.R. Brock (1970) The Dynamics of Aerocolloidal Systems (Pergamon Press, Oxford) pp. 58-59. Khnc, T. and I. Kuscer (1972) Phys. of Fluids 15, 1018. Kramers, H.A. and J. Kistemaker (1943) Physica 10, 699. Kuscer, I. (1974) in: M. Becker and M. Fiebig, eds., Rarefied Gas Dynamics (DFVLR-Press, Porz-Wahn, Germany) Vol. II, pp.E.l-1-El-21. Lang, H. and S.K. Loyalka (1970) Phys. of Fluids 13, 1871. Lang, H. and W.J.C. Muller (1975) Z. Naturforsch. 3Oa, 855. Loyalka, S.K. (1971a) Phys. of Fluids 14, 2599. Loyalka, SK. (1971b) Phys. of Flui& 14, 2291. On&i, Y. (1972) Trans. Japan Sot. Aero. Space Sci. 15, 117. O&hi, Y. (1973) BUN. Uniu. Osaka Prefecture Ser. A 22, 91. Onishi, Y. (1984) in: H. Oguchi, ed., Rarefied Gas Dynamics (Univ. of Tokyo Press, Tokyo) Vol. II, pp. 875-884. O&hi, Y. (1986a) ZAMP 37, 573. O&hi, Y. (1986b) J. Fluid. Mech. 163, 171. Onishi, Y. (1986~) in: T. Matsui, ed., Proc. Third Asian Congress of Fluid Mechanics, pp. 523-526. Schmitt, K.H. and L. Waldmann (1960) Z. Naturforsch. IZa, 843. Sone, Y. (1966) J. Phys. Sot. Japan 21, 1836. Sone, Y. (1970) J. Phys. Sot. Japan 29, 1655. Sone, Y. and Y. Onishi (1973) J. Phys. Sot. Japan 35, 1773. Waldmann, L. (1961) in: L. Talbot, ed., Rarefied Gas Dynamics (Academic Press, New York) pp. 323-344. Waldmann, L. and K.H. Schmitt (1966) in: C.N. Davies, ed., Aerosol Science (Academic, New York) pp. 137-162. Yalamov, Yu.I., I.N. Ivchenko and B.V. Deryagin (1968) Sou. Pkys. Dokl. 13, 446. Zhdanov, V.M. (1967) Soo. Phys. Tech. Phys. 12, 134.