数理解析研究所講究録
1495 巻 2006 年 118-136

118

Cylindrical Couette flow of a vapor-gas mixture:
A ghost effect in the continuum limit
Hiroaki Yoshida

Department of Aeronautics and Astronautics,
Graduate School of Engineemng, Kyoto University, Kyoto 606-8501, Japan

Kazuo Aoki

artmenl of Mechanical Enginee rrng and Science,
Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan$De\mathrm{p}$Dedicated to Prof. Tai-Ping Liu on his 60th birthday
A binary mixture of gases is confined in a gap between two coaxial circular cylinders rotating atdifferent angular velocities. One of the component gases is the vapor of the substance that forms
the cylinders, so that evaporation or condensation (or sublimation) of the vapor may take place on
the surfaces of the cylinders. The other componcnt is a noncondcnsablc gas that neither evaporates
nor condenses on thc surfaces. Axisyrnmetric and axially uniform flows (the cylindrical Couetteflow) of such a mixture are investigated on the basis of kinetic theory with special interest in thecontinuum (or fluid-dynamic) limit in which the Knudsen number goes to zero. The fluid-dynamic
system that describes the behavior of the mixture is derived by a formal but systematic asymptotic
analysis of the Boltzmann system. The resulting system shows some nontrivial effects such as theghost effect and the flow bifurcation.

1.

INTRODUCTION

Axisyrnrnetric and axiallyflow between two rotating coaxial circular cylinders is a
textbook example known as cylindrical Couette flow in classical fluid dynamics. It is also one ofthc basic flows for a rarefied gas and has been investigated on the basis of kinetic theory (see,for example, [1-4] . The flow is a simple time-independent and spatially one-dimensional flow,
and it exhibits neither instability nor bifurcation unlcss thc axial uniformity is rclca. . However,if the gas is a vapor of the substance that forms the cylinders and it undergoes evaporation orcondensation (or sublimation) on the surfaces of the cylinders, the situation changes dramatically.
The flow exhibits bifurcation for relatively small Knudsen numbers even under the constraint ofaxial uniformity [5-8]. That is, there appear two or more different solutions for a certain rangeof parameters, such as the rotation speeds, the radii of the cylinders, and the Knudsen number.
The structure of this bifurcation was investigated in detail by a formal but systematic asymptotic
analysis of the Boltzmann equation with a delicate parameter setting in the case of slow rotationand small Knudsen numbers, and the essential features of the bifurcation were clarified [7]. Oneof the cases considered in Ref. [7] was then investigated mathematically in Ref. [9], and a rigorous
proof of the existence of the bifurcation in the asymptotic limit was given.
Now let us consider the continuum (or fluid-dynamic) limit where the Knudsen number goesto zero in the case with evaporation and condensation on the cylinders. When the inner cylinder
is rotating and the outer cylinder is at rest, there appear three different types of solution in awide parameter range. First type exhibits evaporation on the inner cylinder and condensationon the outer, the second type condensation on the inner cylinder and evaporation on the outer,and the last type no evaporation or condensation on the cylinders [5]. The first and second types
are described by the Euler equations with appropriate boundary conditions for cvaporation andcondensation [10]. For the third type, since there is no evaporation or condensation, it is naturalto think that the flow field between the cylinders is the same as that of the ordinary cylindrical
Couette flow. It is, however, not correct. The infinitesimal evaporation and condensation in thiscase have a significant effect on the flow field, such as the tangential velocity and temperaturefields [12]. This is an example of the ghost effect that was discovered in Ref. [11] and investigated, Refs. [12-18]). The fluid-dynamic system that describes
cxtcnsivcly in successive papers (the third-type solution was derived systematically from the Boltzmann equation in Ref. [12], andthe deformation of the tangential component of the flow velocity caused by the ghost effect wasclarified. Direct numerical analysis of the Boltzmann equation or its model equation for very small
Knudsen numbers was able to give numerical solutions that correspond to the first and the second$\mathrm{u}\mathrm{I}\dot{\mathrm{u}}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{I}\mathfrak{x}\mathrm{i}$$)$$\mathrm{S}\{^{\backslash \mathrm{d}}.$$\mathrm{s}\mathrm{c}\mathrm{e},$$\mathrm{c}.\mathrm{g}.$119type mentioned above [5, 6, 8]. However, it could not show the solution corresponding to the third
type $[5, 6]$ , most probably becausc of thc instability of thc solution. Whcn thc outer cylinder isrotating and the inner cylinder is at rest, there is no bifurcation of the flow in the continuum
limit, and thc solution without evaporation or condcnsation ( , the solution with thc gost cffect)appears for the rotation speed higher than a critical value. The solution corresponding to thissolution was obtained by the direct numerical analysis for sinall Knudsen numbers $[5, 12]$ .
In the present study, we consider the same problem, the cylindrical Couette flow of the vapor with
evaporation and condensation on the surfaces of the cylinders. The differonce from the previous
study is that we consider the case where another noncondensable gas, which neither evaporates
nor condenses on the surfaces of the cylinders, is also contained in the gap of the two cylinders.
According to the study of the plane Couette flow of the mixture of a vapor and a noncondensablegas [19], we can expect, in the cylindrical Couette flow, that evaporation and condensation areblocked by the noncondensable gas and vanish in the continuum limit in any parameter range.
Therefore, the solution that exhibits the ghost effect caused by the infinitesimal evaporation andcondensation is the natural and only possible solution when there is the noncondensable gas. We
investigate this problem and derive the fluid-dynamic system that describes the behavior of the
mixture in the continuum limit by a systematic asymptotic analysis of the Boltzmann system. As
expected, the resulting fluid-dynamic system shows the ghost effect. More specifically, althoughevaporation and condensation of the vapor vanish in the continuum limit, it changes the flow field,
such as thefile of the tangential component of the flow velocity, dramatically. In addition,such typc of solution cxhibits bifurcation in a ccrtain parametcr range. The glxost effect asas the bifurcation of solution is investigated in detail on the basis of the numerical solution of thefluid-dynamic systcrn. Furthcrmore, these results are compared with the Monte Carlo simulationof the Boltzmann equation for small Knudsen numbers.
$\mathrm{i}.\mathrm{c}.$$\mathrm{p}\mathrm{r}o$$\mathrm{w}\mathrm{e}\mathrm{U}$2.

FORMULATION OF THE PROBLEM
2.1.

Problem

Let us consider a binary mixture of a vapor ( -component) and a noncondensable gas (Bcomponent) confined in the gap between two coaxial circular cylinders made of the condensed
phase of the vapor, rotating at different angular velocities. The vapor may evaporate or condenseon the surfaces of the cylinders, whereas the noncondensable gas undergoes ordinary reflection
without evaporation or condensation there, The radius, temperature, and surface velocity of the, and , respectively, and the corresponding quantities of theinner cylinder are denoted by$A$$L_{I},$$T_{I}$$V_{I}$, aiidouter cylinder by.
Restricting ourselves to the case of axisymmetric and axially uniform flows, we investigate thesteady behavior of the mixture on the basis of the BoltzmaIm equation with special interest in thecontinuum limit where the Knudsen number vanishes.
$L_{II},$$T_{II}$$V_{II}$

2.2.

Basic equations

along the common
Let us introduce the cylindrical coordinatc systemwith thc, and components,, and its
axis of the cylinders. Let be the molecular velocity andthe velocity distribution function of the moleculos of the a-component (a $=A,$ ). In
follows,what
we will use the indices and to represent the components of the mixture, i.e.,$(r, \theta, z)$$\xi,.,$$\xi$$\xi_{\theta}$$z\mathrm{a}\mathrm{x}\dot{\iota}\mathrm{s}$$\xi_{z}$$r,$$\theta$$z$$B$$F^{\alpha}(r, \xi)$$\alpha$$\alpha,$$\beta=A,$ $B$$\beta$.

The Boltzmann equation in the present problem is written in the following form:$\xi_{f}\frac{\partial F^{\alpha}}{\partial r}+\frac{\xi_{\theta}^{2}}{r}\frac{\partial F^{\alpha}}{\partial\xi_{f}}-\frac{\xi_{\mathrm{r}}\xi_{\theta}}{r}\frac{\partial F^{\alpha}}{\partial\xi_{\theta}}=\sum_{\beta=A,B}J^{\beta\alpha}(F^{\beta}, F^{\alpha})$where$J^{\beta\alpha}$,(1)is the collision integral defined by:,$J^{\beta\alpha}(F, G)= \int[F(\xi_{*}’)G(\xi’)-F(\xi_{*})G(\xi)]B^{\beta\alpha}(|\mathrm{e}\cdot V|/V, V)\mathrm{d}\Omega(\mathrm{e})\mathrm{d}\xi_{*}$

(2)

120

with
$\xi’=\xi+\frac{/4^{\beta\alpha}}{m^{\alpha}}(\mathrm{e}\cdot V)\mathrm{e}$$V=\xi_{*}-\xi$,,$\xi_{*}’=\xi_{*}-\frac{/l^{\beta\alpha}}{m^{\beta}}(e\cdot V)\mathrm{e}$$V=|V|$ ,$\mu^{\beta\alpha}=\frac{2m^{\alpha}m^{\beta}}{m_{}^{\alpha}+m^{\beta}}$

,

(3)

.

(4)

Here,
the integration variable corresponding to , andis the unit
vector andis the so lid-angle element in its direction;is tho ma.ss of a molecule of the
a-component; the domain of integration with respect to is all direction, and that with respect$\epsilon*\mathrm{i}\mathrm{s}$$\xi$$\mathrm{d}\xi_{*}=\mathrm{d}\xi_{*f}\mathrm{d}\xi_{\mathrm{r}\theta}\mathrm{d}\xi_{\mathrm{s}z};\mathrm{e}$$m^{\alpha}$$\mathrm{d}\Omega(\mathrm{e})$$\mathrm{e}$tois its wholc spacc;
is a nonncgativc function ofand , whosefunctional form is determined by the intermolecular force.
In order to describe the boundary condition on the cylinders, we need to introduce the saturation
number densities
andof the vapor molecules at temperatures, respectively. For
andeach substance, the saturation vapor pressure (or number density) is a function of the temperature
only, determined by the Clausius-Clapeyron relation. Thus,andare determined by, respectively. However, we can regardandandas parameters independent ofand
if we keep the freedom of choosing substance of the vapor. For the purpose of describing theboundary conditions on the inner and outer cylinders in a unified fashion, we define the symbols$B^{\beta\alpha}(|\mathrm{e}\cdot V|/V, V)$$\xi_{*}$$|\mathrm{e}\cdot V|/V$$T_{II}$$T_{I}$$n_{II}$$n_{I}$$T_{I}$$n_{II}$$n_{I}$$T_{II}$$V$$T_{I}$$n_{II}$$n_{I}$$T_{II}$$\sigma_{w},$$n_{w},$$V_{w},$$\mathrm{a}\mathrm{I}\mathrm{l}\mathrm{d}T_{w}$by$\sigma_{w}=1$,$\sigma_{w}=-1$$n_{w}=n_{I}$,,$V_{w}=V_{I}$$n_{w}=n_{II}$,,$T_{w}=T_{I}$$V_{w}=V_{II}$The boundary conditions at $r=L_{I}$ and$L_{II}$,at $r=L_{I}$ ,,$T_{w}=T_{II}$(5)

at $r=L_{II}$ .

,are expressed in the following form [15, 20, 21]:$F^{\alpha}=g_{w}^{\alpha}( \xi)+\int_{\sigma_{w}\xi_{r}<0}.K_{w}^{\alpha}(\xi, \xi_{*})F^{\alpha}(r, \xi_{*})\mathrm{d}\xi_{*}$

,

(for

$\sigma_{w}\xi_{f}>0$

),

(6)

and the scattering kernel
where the source term
are given functions, the functional
forms of which arc determined by the nature of the boundary.
andneed tosatisfy the following conditions:$g_{w}^{\alpha}(\xi)$$K_{w}^{\alpha}(\xi, \xi_{*})$$K_{w}^{\alpha}(\xi,\xi.)$$\mathrm{T}1_{1}\mathrm{e}g_{w}^{\alpha}(\xi)$$g_{w}^{A}(\xi)\geq 0$,$(\sigma_{w}\xi_{f}>0)$

,

(7a)

$g_{w}^{B}(\xi)=0$,$(\sigma_{w}\xi_{f}>0)$

,

(7b)
(7c)

$K_{w}^{\alpha}(\xi, \xi_{*})\geq 0$,$(\sigma_{w}\xi_{r}>0, \sigma_{w}\xi_{\mathrm{t}f}<0)$$\int_{\sigma_{w}\xi_{\mathrm{v}}>0}|\xi_{\mathrm{r}}/\xi_{*r}|K_{w}^{B}(\xi,\xi_{*})\mathrm{d}\xi=1$,,$(\sigma_{w}\xi_{*r}<0)$

,

(7d)

whcre Eq. (7d) indicatcs that thcre is no net particle flux of the -componcnt across the surfaceof the cylinders, that is, the -component is noncondensable. In the present problem,
and
, and
, and
depend on
on , , and. Wc assume that Eq. (6) is satisfied
by the wall-state equilibrium distributions, i.e.,
$B$

$B$

$n_{w},$$V_{w},$$T_{w}$$m^{A}$$g_{w}^{A}$$K_{w}^{B}$$V_{w}$$T_{w}$$K_{w}^{A}$$m^{B}$$F_{w}^{\alpha}=g_{w}^{\alpha}( \epsilon)+\int_{\sigma_{u},\xi_{\mathrm{r}}<0}.K_{w}^{\alpha}(\xi, \xi_{*})F_{w}^{\alpha}(\xi_{*})\mathrm{d}\xi_{*}$

,

(8)

with
$F_{w}^{A}= \frac{n_{w}}{(2\pi k_{B}T_{w}/m^{A})^{3/2}}\exp(-\cdot\frac{\xi_{r}^{2}+(\xi_{\theta}-V_{w})^{2}+\xi_{z}^{2}}{2k_{B}T_{w}/m^{A}})$

,

(9a)

$F_{w}^{B}= \frac{\mathrm{c}}{(2\pi k_{B}T_{w}/m^{B})^{3/2}}\exp(-\frac{\xi_{f}^{2}+(\xi_{\theta}-V_{w})^{2}+\xi_{z}^{2}}{2k_{B}T_{w}/m^{B}})$

,

(9b)

where is an arbitrary constant, andis the Boltzmann constant. We also assume that Eqs. (9a)and (9b) are the only equilibrium distributions that satisfy Eq. (8).
$c$$k_{B}$

121

Let
and
be the molecular number density, the flow velocity, thepressure, and the ternpcraturc of thc a-component. These macroscopic quantities are expresscd interms of the velocity distribution functions as$n^{\alpha},$$v^{\alpha}=(v_{r}^{\alpha}, v_{\theta}^{\alpha}, 0),$$T^{\alpha}$$p^{\alpha})$$r \iota^{\alpha}=\int F^{\alpha}\mathrm{d}\xi$

,

(10a)

$v_{f}^{\alpha}= \frac{1}{n^{\alpha}}\int\xi_{r}F^{\alpha}\mathrm{d}\xi$,$v_{\theta}^{\alpha}= \frac{1}{n^{\alpha}}\int\xi_{\theta}F^{\alpha}\mathrm{d}\xi$

,

(10b)

,

(10c)

$p^{\alpha}=k_{B}n^{\alpha}T^{\alpha}= \frac{1}{3}\int m^{\alpha}[(\xi_{f}-v_{f}^{\alpha})^{2}+(\xi_{\theta}-v_{\theta}^{\alpha})^{2}+\xi_{z}^{2}]F^{\alpha}\mathrm{d}\xi$

where. In Eqs.

and in what follows, the domain of integration withrespect to is its wholc spacc unless the c.ontrary is statcd. Thc number density , the massdensity , the flow velocity $v=(v_{f}, v_{\theta}, 0)$ , the pressure , and the temperatureof the total
mixture are defined by$\mathrm{d}\xi=\mathrm{d}\xi_{r}\mathrm{d}\xi_{\theta}\mathrm{d}\xi_{z}$$(10\mathrm{a})-(10\mathrm{c})$

$\xi$

$n$

$T$

$p$

$\rho$$n= \int(F^{\Lambda}+F^{B})\mathrm{d}\xi$,$(11\mathrm{a})$$\rho=\int(m^{A}F^{A}+m^{B}F^{B})\mathrm{d}\xi$,$(11\mathrm{b})$$v_{f}= \frac{1}{\rho}\int\xi,.(m^{A}F^{A}+m^{B}F^{B})\mathrm{d}\xi$,$(11\mathrm{c})$$?) \theta=\frac{1}{\rho}\int\xi_{\theta}(m^{A}F^{A}+m^{B}F^{B})\mathrm{d}\epsilon$,$(11\mathrm{d})$$p=k_{B}nT= \frac{1}{3}\int[(\xi, -v_{\mathrm{r}})^{2}+(\xi_{\theta}-v_{\theta})^{2}+\xi_{z}^{2}](m^{A}F^{A}+m^{B}F^{B})\mathrm{d}\xi$.

$(11\mathrm{e})$Therefore, they are expressed in terms of the macroscopic quantities of each component as$n=n^{A}+n^{B}$ ,$(12\mathrm{a})$$\rho=m^{A}n^{A}+m^{B}n^{B}$

,

(12b)

$v_{r}=(m^{A}n^{A}v_{f}^{A}+m^{B}n^{B}v_{f}^{B})/\rho$

,

(12c)

$v_{\theta}=(m^{A}n^{A}v_{\theta}^{A}+m^{B}n^{B}v_{\theta}^{B})/\rho$

,

(12d)

$p= \sum_{\alpha=A,B}[\mathrm{p}^{\alpha}+m^{\alpha}n^{\alpha}(v_{f}^{\alpha}-v_{\gamma})^{2}/3+m^{\alpha}n^{\alpha}(v_{\theta}^{\alpha}-v_{\theta})^{2}/3]$.

$(12\mathrm{e})$To complctc thc physical setting of the problem, wc ncod to specify a quantity associated withthe amount of the noncondensable gas. As this parameter, we choose the average number densityof the noncondensable gas,.

$n_{av}^{B}= \frac{1}{\pi(L_{II}^{2}-L_{l}^{2})}\int_{-\pi}^{\pi}\int_{L_{I}}^{L_{II}}\int rF^{B}\mathrm{d}\xi \mathrm{d}r\mathrm{d}\theta$

By integrating both sides of Eq. (1) with respect to$\xi$$rn^{\alpha}v_{r}^{a}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$(13)over its whole space, we obtain

,

(14)

which expresses the mass conservation for each component gas. On the other hand, if we integrate
Eq. (6) (with $\alpha=B$ ) multiplied by with respect to over the space
and take Eq. (7d)into consideration, we obtain $n^{B}v_{r}^{B}=0$ at $r=L_{I}$ and. Thus we have$\xi_{r}$$\sigma_{w}\xi_{r}>0$$\xi$$L_{II}$$n^{B}v_{f}^{B}=0$,$(L_{\mathrm{J}}\leq r\leq L_{ll})$

.

(15)

122

2.3.

Dimensionless expressions

To derive the dimensionless expressions of the basic system, we introduce the following dimensionless variables:$\hat{r}=\frac{r}{L_{I}}$,$\hat{n}^{\alpha}=\frac{n^{\alpha}}{n_{I}}$$\hat{n}=\frac{n}{n_{I}}$$\zeta=(\zeta_{r},$,,$\zeta_{\theta},$$\hat{v}^{\alpha}=(\hat{v}_{r}^{a},\hat{v}_{\theta}^{\alpha}, 0)=\frac{v^{\alpha}}{V_{lh}}$$\hat{\rho}=\frac{\rho}{\rho_{I}})$,$f^{\alpha}= \frac{V_{th}^{3}}{n_{I}}F^{\alpha}$$(_{z})= \frac{\xi}{V_{th}},$,$\hat{p}^{\alpha}=\frac{p^{\alpha}}{p_{I}}$$\hat{v}=(\hat{v}_{r},\hat{v}_{\theta}, 0)=\frac{v}{V_{th}}$,$\hat{m}^{\alpha}=\frac{m^{\alpha}}{m^{A}}$,$\hat{T}^{\alpha}=\frac{T^{\alpha}}{T_{\mathit{1}}}$$\hat{p}=\frac{p}{p_{I}}$

,

(16)

,

$\hat{T}=\frac{T}{T_{I}}$is the most probable speed of the molecules of the -component in
where
the equilibrium state at rest at temperature $T_{I;}\rho_{I}=m^{A}n_{I}$ and $p_{I}=k_{Bn_{I}}T_{I}$ are, respectively,
the density and the pressure of the vapor in the saturated equilibrium state at rest at temperature$V_{th}=(2k_{B}T_{I}/\mathrm{m}^{A})^{1/2}$$T_{I}$$A$.

Then the Boltzmann equation (1) becomes,$Df^{\alpha}= \frac{2}{\sqrt{\pi}\mathrm{K}\mathrm{n}}\sum_{\beta=A,B}K^{\beta\alpha}\hat{J}^{\beta\alpha}(f^{\beta}, f^{\alpha})$$D= \zeta_{f}\frac{\partial}{\theta\hat{r}}+\frac{\zeta_{\theta}^{2}}{\hat{r}}\frac{\partial}{\partial\zeta_{r}}-\frac{\zeta_{r}\zeta_{\theta}}{\hat{r}}\frac{\partial}{\partial\zeta_{\theta}}$

(17)

,

(18)

$\hat{J}^{\beta\alpha}(f, g)=\int[f(\zeta_{*}’)g(\zeta’)-f(\zeta_{*})g(()]\hat{B}^{\beta\alpha}(|\mathrm{e}\cdot\hat{V}|/\hat{V},\hat{V})\mathrm{d}\Omega(\mathrm{e})\mathrm{d}\zeta_{*}$$\zeta’=\zeta+\frac{\hat{\mu}^{\beta\alpha}}{\hat{m}^{\alpha}}(\mathrm{e}\cdot\hat{V})\mathrm{e}$V$=\zeta_{*}-\zeta$,,$\zeta’.=\zeta_{*}-\frac{\hat{\mu}^{\beta\alpha}}{\hat{m}^{\beta}}(\mathrm{e}\cdot\hat{V})\mathrm{e}$$\hat{V}=|\hat{V}|$,$\hat{B}^{\beta\alpha}(|\mathrm{e}\cdot\hat{V}|/\hat{V},\hat{V})=\frac{B^{\beta\alpha}}{B_{0}^{\beta\alpha}}$

,

(19)

,

(20)

,

(21)

$\hat{\mu}^{\beta\alpha}=\frac{2\hat{m}^{\alpha}\hat{m}^{\beta}}{\hat{m}^{\alpha}+\hat{m}^{\beta}}$,$K^{\beta\alpha}= \frac{B_{0}^{\beta\alpha}}{B_{0}^{AA}}$

,

(22)

$B_{0}^{\beta\alpha}= \frac{1}{n_{I}^{2}}\int F_{0}^{\alpha}(\xi)F_{0}^{\beta}(\xi_{*})B^{\beta\alpha}(|\mathrm{e}\cdot V|/V, V)\mathrm{d}\Omega(\mathrm{e})\mathrm{d}\xi \mathrm{d}\xi_{*}$

,

(23)

.

$F_{0}^{\alpha}= \frac{n_{I}}{(2\pi k_{B}T_{I}/m^{\alpha})^{3/2}}\exp(-\frac{\xi_{r}^{2}+\xi_{\theta}^{2}+\xi_{z}^{2}}{2k_{B}T_{I}/m^{\alpha}})$

(24)

Here,
is the integration variable corresponding to , and; the domain of
integration is the whole space of and all directions of [ andare the same as in Eq. (2)];Kn on the right-hand side of Eq. (17) is thc rcfcrcncc Knudscn nurnbcr dcfined by Kn,
whereis the mean free path of the molecules of the vapor in the equilibrium
state at rest with temperatureand molccular numbcr dcnsity .
When the intermolecular potential extends to infinity, the so-called angular cutoff is introducedin Eq. (23), that is,is set to be zero for the doxnain( : a spccificd constant),
which corresponds to grazing collisions. It should be noted that the function
in general, wheredepends on the dimensionless paraineteris the characteristic size of theintermolecular potential for the interaction of a molecule of the -component with a molecule ofthe -component [15]. This parameter is not shown explicitly in the above equations. When bothcomponent gases consist of lnrd-sphere molecules,
does not depcnd on this pararnetcr, and, and are given by$\zeta_{\mathrm{r}}$$\mathrm{d}\zeta_{*}=\mathrm{d}\zeta_{f}.\mathrm{d}\zeta_{*\theta}\mathrm{d}\zeta_{*z}$$\zeta$$\zeta_{*}$$\mathrm{e}$$\mathrm{e}$$\mathrm{d}\Omega(\mathrm{e})$$=\ell_{0}/L_{I}$$\ell_{0}=2V_{\iota h}/\sqrt{\pi}n_{\mathit{1}}B_{0}^{AA}$$T_{I}$$n_{I}$$B^{\beta\alpha}$$|\mathrm{e}\cdot V|/V\leq\delta$$\delta$$\hat{B}^{\beta\alpha}$$U_{0}^{\beta\alpha}/k_{B}T_{I}$$U_{0}^{\beta\alpha}$$\alpha$$\beta$$\hat{B}^{\beta\alpha}$$\hat{B}^{\beta\alpha},$$K^{\beta\alpha}$$\ell_{0}$$\hat{B}^{\beta\alpha}=\frac{(\hat{\mu}^{\beta\alpha})^{1/2}}{4\sqrt{2\pi}}|\mathrm{e}\cdot\hat{V}|$,$K^{\beta\alpha}= \frac{1}{(\hat{\mu}^{\beta\alpha})^{1/2}}(\frac{d^{\beta}+d^{\alpha}}{2d^{A}})^{2}$,$\ell_{0}=\frac{1}{\sqrt{2}\pi(d^{\Lambda})^{2}n_{I}}$

,

(25)

123

where
is the diameter of a molecule of the a-component.
Corresponding to Eq. (5), we introduce the following symbols:$d^{\alpha}$$\sigma_{w}=1$,$\hat{n}_{w}=1$,$\hat{V}_{w}=\frac{V_{I}}{V_{th}}$,,$\hat{T}_{w}=1$

at$\hat{r}=1$,(26)$\sigma_{w}=-1$,$\hat{n}_{w}=\frac{n_{II}}{n_{I}}$,$\hat{V}_{w}=\frac{V_{II}}{V_{th}}$,$\hat{T}_{w}=\frac{T_{II}}{T_{I}}$,at

The dimensionless form of the boundary conditions is as follows: at$f^{\alpha}= \hat{g}_{w}^{\alpha}(\zeta)+\int_{\sigma_{w}\zeta_{r}<0}.\hat{K}_{w}^{\alpha}(\zeta$

,

$($

.

$)f^{\alpha}(\hat{r}, \zeta_{\mathrm{r}})\mathrm{d}\zeta_{*}$,$\hat{r}=\frac{L_{II}}{L_{I}}$and$\hat{r}=1$

(for

.

$L_{II}/L_{I}$$\sigma_{w}\zeta_{f}>0$

,
(27)

),

where
$\hat{g}_{w}^{\alpha}=\frac{V_{th}^{3}}{n_{I}}g_{w}^{\alpha}$

and$\hat{g}_{w}^{\alpha}(\zeta)$

and,$\hat{K}_{w}^{\alpha}=V_{th}^{3}K_{w}^{\alpha}$

,

(28)

satisfy$\hat{K}_{w}^{\alpha}(\zeta, \zeta_{\mathrm{r}})$$\hat{g}_{w}^{A}(\zeta)\geq 0$,$(\sigma_{w}\zeta_{f}>0)$$\hat{g}_{w}^{B}(\zeta)=0$,$(\sigma_{w}\zeta_{f}>0)$$\hat{K}_{w}^{\alpha}(\zeta, \zeta_{*})\geq 0$

,

,
,

(29a)
(29b)

,

$\int_{\sigma_{w}\zeta_{r}>0}|\zeta_{r}/\zeta_{*\mathrm{r}}|\hat{K}_{w}^{B}(\zeta, \zeta_{*})\mathrm{d}\zeta=1$

In addition, corresponding to Eqs. (8)(29c)$(\sigma_{w}\zeta_{r}>0, \sigma_{w}\zeta_{*r}<0))$$-(9\mathrm{b})$$(\sigma_{w}\zeta_{*f}<0)$

.

(29d)

, we havc$f_{w}^{\alpha}= \hat{g}_{w}^{\alpha}(\zeta)+\int_{\sigma_{w}\zeta_{\mathrm{r}}<0}.\hat{K}_{w}^{\alpha}(\zeta, \zeta_{*})f_{w}^{\alpha}(\zeta_{*})\mathrm{d}\zeta_{*}$$f_{w}^{A}= \frac{\hat{n}_{w}}{(\pi\hat{T}_{w})^{3/2}}\exp(-\frac{\zeta_{f}^{2}+(\zeta_{\theta}-\hat{V}_{w})^{2}+\zeta_{z}^{2}}{\hat{T}_{w}})$

,

(30)

,

$f_{w}^{B}= \frac{\hat{c}}{(\pi\hat{T}_{w}/\hat{m}^{B})^{3/2}}\exp(-\frac{\zeta_{f}^{2}+(\zeta_{\theta}-\hat{V}_{w})^{2}+\zeta_{z}^{2}}{\hat{T}_{w}/\hat{m}^{B}})$

(31a)

,

(31b)

where is an arbitrary (dimensionless) constant.
$V_{II}/V_{th)}T_{II}/T_{I}$ ,It is seen from Eqs. (26) and (30)that the parameters
and $m^{B}/m^{A}$ are contained in the boundary c,ondition. In general, however,andalso dcpcnd
on the reference temperature , the reference number density , and the reference molecular
, and
mass
also depends on
and. In the case of the so-called complete condensation
condition [for the vapor, and the diffuse reflection condition [15] for the noncondensable gas,andare given by$\hat{c}$$n_{Il}/n_{I},$$-(31\mathrm{b})$$V_{I}/V_{th},$$\hat{K}_{w}^{A}$$\hat{g}_{w}^{A}$$T_{I}$$m^{A}$$\hat{K}_{w}^{B}$$n_{I}$$T_{I}$$m^{A}$$15_{\mathrm{J}}^{\rceil}$$\hat{K}_{w}^{\alpha}$$\hat{g}_{w}^{\alpha}$$\hat{g}_{w}^{A}(\zeta)=\frac{\hat{n}_{w}}{(\pi\hat{T}_{w})^{3/2}}\exp(-\frac{\zeta_{r}^{2}+(\zeta_{\theta}-\hat{V}_{w})^{2}+\zeta_{z}^{2}}{\hat{T}_{w}})$$\hat{g}_{w}^{B}(\zeta)=0$

(32)

,

,

$\hat{K}_{w}^{A}(\zeta, \zeta_{*})=0$

(33)
(34)

,

$\hat{K}_{w}^{B}(\zeta, \zeta_{*})=-\frac{2}{\pi}(\frac{\hat{m}^{B}}{\hat{T}_{w}})^{2}\sigma_{w}\zeta_{*\tau}\exp(-\frac{\zeta_{r}^{2}+(\zeta_{\theta}-\hat{V}_{w})^{2}+\zeta_{z}^{2}}{\hat{T}_{w}/\hat{m}^{B}})$In this case,depend on$\hat{g}_{w}^{\alpha}$$T_{I},$

and, or$n_{I}$$\hat{K}_{w}^{a}$depend only on$m^{A}$.

$n_{II}/n_{I},$$V_{I}/V_{th},$$V_{II}/V_{th},$ $T_{II}/T_{I}$

.

(35)

, and $m^{B}/m^{A}$ and do not124

The dimensionless version of the relations between the macroscopic variables and the velocity
distribution functions, Eqs. (10) $-(12)$ are given by$\hat{n}^{\alpha}=\int f^{\alpha}\mathrm{d}\zeta$

,

(36a)

$l^{\wedge\alpha} \prime_{f}=\frac{1}{\hat{n}^{\alpha}}\int\zeta_{f}f^{\alpha}\mathrm{d}\zeta$,$\hat{v}_{\theta}^{\alpha}=\frac{1}{\hat{n}^{\alpha}}\int\zeta_{\theta}f^{\alpha}\mathrm{d}\zeta$

,

(36b)
(36c)

$\hat{p}^{\alpha}=\hat{n}^{\alpha}\hat{T}^{\alpha}=\frac{2}{3}\int\hat{m}^{\alpha}[(\zeta_{r}-\hat{v}_{r}^{\alpha})^{2}+(\zeta_{\theta}-\hat{v}_{\theta}^{\alpha})^{2}+\zeta_{z}^{2}]f^{\alpha}\mathrm{d}(,$$\hat{n}=\int(f^{A}+f^{B})\mathrm{d}\zeta=\hat{n}^{A}+\hat{n}^{B}$

,

(36d)

$\hat{\rho}=\int(\hat{m}^{A}f^{A}+\hat{m}^{B}f^{B})\mathrm{d}\zeta=\hat{m}^{A}\hat{n}^{A}+\hat{m}^{B}\hat{n}^{B}$

,

(36c)

$\hat{v}_{r}=\frac{1}{\hat{\rho}}\int\zeta_{r}(\hat{m}^{A}f^{A}+\hat{m}^{B}f^{B})\mathrm{d}\zeta=\frac{1}{\hat{\rho}}(\hat{m}^{A}\hat{n}^{A}\hat{v}_{r}^{A}+\hat{m}^{B}\hat{n}^{B}\hat{v}_{r}^{B})$

,

(36f)

,

(36g)

$\hat{v}_{\theta}=\frac{1}{\hat{\rho}}\int\zeta_{\theta}(\hat{m}^{A}f^{A}+\hat{m}^{B}f^{B})\mathrm{d}\zeta=\frac{1}{\hat{\rho}}(\hat{m}^{AA\wedge A}\hat{n}\uparrow)\theta+\hat{m}^{B}\hat{n}^{B}?^{\wedge B}\prime_{\theta})$$\hat{p}=\hat{n}\hat{T}=\frac{2}{3}\int[(\zeta_{r}-\hat{v}_{r})^{2}+(\zeta_{\theta}-\hat{v}_{\theta})^{2}+\zeta_{z}^{2}](\hat{m}^{A}f^{A}+\hat{m}^{B}f^{B})\mathrm{d}($$= \sum_{\alpha=A,B}[\hat{p}^{\alpha}+\frac{2}{3}\hat{m}^{\alpha}\hat{n}^{\alpha}(\hat{v}_{r}^{\alpha}-\hat{v}_{r})^{2}+\frac{2}{3}\hat{m}^{\alpha}n^{\alpha}(\hat{v}_{\theta}^{\alpha}-\hat{v}_{\theta})^{2}]$

,

(36h)

. Here and in what follows, the dornain of integration with respect to (iswhere
its whole space unless the contrary is stated.
The dimensionlcss forms of Eqs. $(113)-(15)$ are, respectively,$\mathrm{d}\zeta=\mathrm{d}(_{r}\mathrm{d}\zeta_{\theta}\mathrm{d}\zeta_{l}$$\frac{n_{av}^{B}}{n_{I}}=\frac{2}{(L_{II}/L_{I})^{2}-1}\int_{1}^{L_{\prime},/L,}\hat{\mathrm{r}}\hat{n}^{B}\mathrm{d}\hat{r}$$\hat{r}\hat{n}^{A}\hat{v}_{r}^{A}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$

3.

and

$\hat{n}^{B}\hat{v}_{r}^{B}=0$

,

for

,

(37)

$1\leq\hat{r}\leq L_{II}/L_{I}$

.

(38)

ASYMPTOTIC ANALYSIS FOR SMALL KNUDSEN NUMBERS

In this section, we consider the case where the Knudsen number Kn is small and carry out
a systematic asymptotic analysis of the boundary-value problem, Eqs. (17) and (27), following
the asymptotic theory developed by Sone (see, e.g., Rek. [11, 12, 15, 22-24]) as a guideline. The

analysis here can be carried out in a way parallel to Ref. [19]. For convenience, we use the following
parameter rather than Kn in this section:$\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{U}$$\epsilon$$\epsilon=(\sqrt{\pi}/2)\mathrm{K}\mathrm{n}\ll 1$

3.1.

.

(39)

Fluid-dynamic equations and boundary conditionsFirst, we leave aside the boundary condition (27) and look for a moderately varying solution, in the form of a power series of :, which satisfics$df_{H}^{\alpha}/\partial\hat{r}=O(f_{H}^{\alpha})$$f_{H}^{\alpha}$$\epsilon.$$f_{H}^{\alpha}=f_{H0}^{\alpha}+f_{H1}^{\alpha}\epsilon+\cdots$

.

(40)

is called
Hilbert solution or expansion. Substitution of Eq. (40) into Eqs.
gives the corresponding power series expansions of the macroscopic variables of each component
and of the total mixture:This$(36\mathrm{a})-(36\mathrm{h})$$\mathrm{t}\}_{1}\mathrm{e}$$f_{H}^{\alpha}$$h_{H}^{\alpha}=h_{H0}^{\alpha}+h_{H1}^{\alpha}\epsilon+\cdots$,$h_{H}=h_{H0+\prime 1,H1^{\zeta+}}.\cdots$

,

(41)
(42)

125

, etc., and the subscript $H$ indicates the quantities correwhere the letter stands forsponding to the Hilbert solution. The coefficient functions
and $h_{Hm}$ are cxpresscd in tcrms. Their explicit forms for $m=0$ and 1 are given in Appendix A. After substitutingof
Eq. (40) into Eq. (17) and arranging thc powcr of , we obtain a scrics of intcgral equations forthe coefficients:$\hat{n},\hat{v},\hat{T}$$h$$h_{Hm}^{\alpha}$$f_{Hn}^{\alpha}(n\leq m)$$\epsilon$$f_{Hm}^{\alpha}$

,

(43)

$\sum_{\beta=A,B}K^{\beta\alpha}\hat{J}^{\beta\alpha}(f_{H0}^{\beta}, f_{H0}^{\alpha})=0$$\sum_{\beta=A,B}K^{\beta\alpha}[\hat{J}^{\beta\alpha}(f_{H0}^{\beta}, f_{Hm}^{\alpha})+\hat{J}^{\beta\alpha}(f_{Hm}^{\beta}, f_{H0}^{\alpha})]$$=Df_{Hm-1}^{\alpha}- \sum_{l=1}^{m-1}\sum_{\beta=A,B}K^{\beta\alpha}\hat{J}^{\beta\alpha}(f_{Hl}^{\beta}, f_{Hm-i}^{\alpha})$,$(m=1,2, \ldots)$ ,(44)

is set to be zero when $m=1$ .
where
As is well known [25], the solution of Eq. (43) is a local Maxwellian distribution for each component with a common temperature and a common flow velocity, The solution is expressed in the, andof the Hilbert cxpansion
following form using the leading ordcr termsof the macroscopic quantities:$\sum_{l=1}^{m-1}$$T_{H0}$$\hat{n}_{H0}^{\alpha},2J_{fH0}\wedge,\hat{v}_{\theta H0}$

$f_{H0}^{\alpha}= \frac{\hat{n}_{H0}^{\alpha}}{(\pi\hat{T}_{H0}/\hat{m}^{\alpha})^{3/2}}\exp(-\frac{(\zeta_{r}-\hat{v}_{rH0})^{2}+(\zeta_{\theta}-\hat{v}_{\theta H0})^{2}+\zeta_{z}^{2}}{\hat{T}_{H0}/\hat{m}^{\alpha}})$

.

(45)

This gives the obvious relations$\hat{v}_{rH0}^{A}=\hat{v}_{rH0}^{B}=\hat{v}_{rH0}$,$\hat{v}_{\theta H0}^{A}=\hat{v}_{\theta H0}^{B}=\hat{v}_{\theta H0}$,$\hat{T}_{H0}^{A}=\hat{T}_{H0}^{B}=\hat{T}_{H0}$

.

(46)

On the other hand, frorn Eq. (38) we have$\hat{n}_{H0}^{B}\hat{v}_{rH0}^{B}=0$,$(1 \leq\hat{r}\leq L_{II}/L_{I})$

.

(47)

, i.e., there is no noncondensable gas in the leading order (of the Hilbert
The case where
solution), has been investigated for gencral gcomctry in Rcf. [26]. In this case, a small amountof the noncondensable gas is blown away by the vapor flow and pushed against the surface wherecondensation of the vapor is taking place. As a result, the noncondensable cornponent of a verysmall amount accumulates in a very thin layer adjacent to the condensing surface and may have asignificant effect on the global vapor flow [26-28]. In thc prcscnt study, we consider the oppositc
vanishes in a certain
case, i.e., the case where(the case whereinvanishesinterval of can be handlcd in thc kamework of the present analysis). Thcn,
identically, so that Eq. (46) gives$\hat{n}_{H0}^{B}\equiv 0$$1\leq\hat{r}\leq L_{II}/L_{I}$$\hat{n}_{H0}^{B}>0$$\hat{n}_{H0}^{B}$$\hat{v}_{\tau H0}^{B}$$\hat{r}$$\hat{v}_{rH0}^{A}=\hat{v}_{rH0}^{B}=\hat{v}_{rH0}\equiv 0$

Therefore, the leading-order solution$f_{Il0}^{\alpha}$

.

(48)

reduces to$f_{H0}^{\alpha}= \frac{\hat{n}_{H0}^{\alpha}}{(\pi\hat{T}_{H0}/\hat{m}^{\alpha})^{3/2}}\exp(-\frac{\zeta_{r}^{2}+(\zeta_{\theta}-\hat{v}_{\theta H0})^{2}+\zeta_{z}^{2}}{\hat{T}_{H0}/\tau\hat{n}^{\alpha}})$

.

(49)

. BecauseThe-order equation, Eq. (44), is an inhomogeneous linear integral equation forof thc form of Eq. (49), the left-hand side of Eq. (44) is esscntially thc linearized collision term for abinary mixture of gases. Therefore, the corresponding homogeneous equation has the independent.
nontrivial solutions:, whereare arbitrary constants, andIn consequence, in order that Eq. (44) has a solution, its inhomogeneous term should satisfy acertain solvability condition, which is reduced to the following form:$m\mathrm{t}\mathrm{h}$$f_{Hm}^{\alpha}$$(a^{\alpha},\hat{m}^{\alpha}\zeta,\hat{m}^{\alpha}\zeta^{2})f_{H0}^{\alpha}$$\zeta^{2}=\zeta_{f}^{2}+\zeta_{\theta}^{2}+\zeta_{z}^{2}$$a^{\alpha}$$\int Df_{Hm-1}^{\alpha}\mathrm{d}\zeta=0$

(50)

,

$\int\sum_{\alpha=A,B}\hat{m}^{\alpha}\zeta Df_{Hm-1}^{\alpha}\mathrm{d}\zeta=0$

(51)

,

$\int\sum_{\alpha=A,B}\hat{m}^{\alpha}\zeta^{2}Df_{Hm-1}^{\alpha}\mathrm{d}\zeta=0$

.

(52)

126

If we insert Eq. (49) in Eqs. (50) $-(52)$ with $m=1$ , the -component of Eq. (51) gives a constraint
shown in Eq. (54) bclow, but othcr conditions are satisficd autoaatically. With this constraint,
Eq. (44) with $m=1$ can be solved. The explicit form of the solution
is given in Appendix B.
If we substitute this
into Eqs. (50)with $m=2$ , we obtain some constraints for the. With these constraints, we can proceed to the higher order.
macroscopic variables contained in
In this way, the functional forxn ofwith respect to is determined successivcly $(n=0,1, \ldots)$ ,and at the same time the constraints on the (unknown) macroscopic variables
andareobtained in the form of ordinary differential equations. The latter equations are the so-callcd
fluid-dynamic equations. The set of equations necessary to determine the leading-order solution
is csscntially obtaincd from Eq. (50) with $m=2$ , thc -component of Eq. (51) with $m=1$ ,the -component of Eq. (51) with $m=2$ , and Eq. (52) with $m=2$ , and can be summarized asfollows:$\hat{r}$$f_{H1}^{\alpha}$$-(_{\iota}^{r_{)}}2)$$f_{H1}^{\alpha}$$f_{H1}^{\alpha}$$f_{Hn}^{\alpha}$$\zeta$$h_{Hn}^{\alpha}$$h_{Hn}$$\hat{r}$$f_{H0}^{\alpha}$$\theta$,(53)$\frac{\mathrm{d}}{\mathrm{d}\hat{r}}(\hat{r}\hat{\rho}_{H0}\hat{v}_{\mathrm{r}H1})=0$$\frac{1}{2}\frac{\mathrm{d}\hat{p}_{H0}}{\mathrm{d}\hat{r}}-\frac{\hat{\rho}_{H0}\hat{v}_{\theta H0}^{2}}{\hat{r}}=0$

,

(54)

,

(55)

$\hat{r}\hat{\rho}_{H0}\hat{v}_{rH1}\frac{\mathrm{d}}{\mathrm{d}\hat{r}}(\hat{r}\hat{v}_{\theta H0})=\frac{\mathrm{d}}{\mathrm{d}\hat{r}}[\hat{r}^{2}\hat{\mu}\frac{\hat{T}_{H0}^{1/2}}{2}(\frac{\mathrm{d}\dagger^{\wedge}\rangle\theta H0}{\mathrm{d}\hat{r}}-\frac{||\theta H0\wedge}{\hat{r}})]$$\hat{r}\hat{\rho}_{H0}\hat{v}_{rH1^{\frac{\mathrm{d}}{\mathrm{d}\hat{r}}}}(\hat{v}_{\theta H0}^{2}+\frac{\backslash r_{)}}{2}\hat{T}_{H0})=\frac{\mathrm{d}}{\mathrm{d}\hat{r}}[\hat{r}\hat{\mu}\hat{T}_{H0}^{1/2}\hat{v}_{\theta H0}(\frac{\mathrm{d}_{\mathrm{t}^{1\theta H0}}^{\wedge}}{\mathrm{d}\hat{r}}-\frac{\hat{v}_{\theta H0}}{\hat{f}})$$+ \hat{r}\hat{\lambda}\hat{T}_{H0}^{1/2}\frac{\mathrm{d}\hat{T}_{H0}}{\mathrm{d}\hat{r}}-k_{?},\frac{\hat{r}\hat{\rho}_{H0}\hat{v}_{rH1}\hat{T}_{H0}}{\chi_{H0}^{A}}]$

,

(56)

where
$\hat{r}\hat{\rho}_{H0}\hat{v}_{rH1}=-\hat{r}\frac{\hat{T}_{H0}^{1/2}}{\chi_{H0}^{B}}(\hat{D}_{T}\frac{\mathrm{d}\mathrm{h}\hat{T}_{H0}}{\mathrm{d}\hat{r}}+\hat{D}_{AB}\frac{\mathrm{d}\chi_{H0}^{\Lambda}}{\mathrm{d}\hat{r}})$

,

(57)

$-2( \hat{m}^{B}-\hat{m}^{\Lambda})\hat{D}_{AB}\frac{\chi_{H0}^{\Lambda}}{\hat{T}_{H0}^{1/2}}\hat{v}_{\theta H0}^{2}$,$\hat{\rho}_{H}0=\hat{m}^{A}\hat{n}_{H0}^{A}+\hat{m}^{B}\hat{n}_{H0}^{B}$$\chi_{H0}^{\alpha}=\frac{\hat{n}_{H0}^{\alpha}}{\hat{n}_{H0}}$,$\hat{p}_{H0}=\hat{n}_{H0}\hat{T}_{H0}$$\hat{n}_{H0}=\hat{n}_{H0}^{\Lambda}+\hat{n}_{H0}^{B}$

,

(58a)

.

(58b)

, and
in Eqs. (55)-(57), which are functions of
(or
) and
and
depend on the model of molecular interaction, correspond to the viscosity, thermal conductivity,
thermal-diffusion ratio, thermal diffusion coefficient, and mutual diffusion coefficient, respectively.
The definition of these transport coefficients is given in Appendix B. With Eqs. (58a) and (58b),, andEqs. (53) $-(57)$ fora a closed set of equations for. Equations (53)(56) are, respectively, the conservation equations for the total mass, the -component of the total
momentum, its -compouent, and the total cncrgy. Equation (57) expresses the diffusion in theradial direction.
In the derivation of thc Hilbcrt solution, we have used the fact that thc radial component of theflow velocity of the noncondensable gas is identically zero, which is a consequence of the boundary
condition (27) on the cylinders. Exc.ept this, howcver, the boundary c,ondition has not been takcn
into account so far. We now consider the boundary condition. Let us assume that the macroscopic, andin
takc thc following values on the surfaces of the cylinders; atvariablcs,$\mathrm{T}\mathrm{h}\mathrm{e}\hat{/4},\hat{\lambda},$$k_{T},\hat{D}_{T}$$\hat{D}_{AB}$$\hat{T}_{H0}$$\chi_{H0}^{A}$$\chi_{H0}^{B}$$\hat{T}_{H0}$$\hat{n}_{H0}^{A},\hat{n}_{II0}^{B},\hat{v}_{rH1},\hat{v}_{\theta H0}$$\hat{r}$$\theta$$\hat{n}_{H0}^{\alpha},\hat{v}_{\theta H0}$$\hat{T}_{H0}$$f_{H0}^{\alpha}$$\hat{r}=1$$\hat{n}_{H0}^{A}=1$,$\hat{v}_{\theta II0}=\frac{V_{I}}{V_{th}}$,$\hat{T}_{H0}=1$

,

(59)

127

and at$’\hat{r}=L_{l\mathit{1}}/L_{I}$,$\hat{n}_{H0}^{A}=\frac{n_{II}}{n_{I}}$,$\hat{v}_{\theta H0}=\frac{V_{II}}{V_{th}}$,$\hat{T}_{l\mathit{1}0}=\frac{T_{II}}{T_{I}}$

.

(60)

Then, because of the relation (30), it is seen thatsatisfies the boundary condition (27) at theleading order (thc zcroth ordcr in ). Thc rcstrictions (59) and (60) give the consistent boundary
conditions on the cylinders for the fluid-dynamic equations (53)-(58). Note that the boundary
conditions (59) and (60) contain only the dimensionlcss paramctcrs, andand do not depend on the reference quantityor .
The fluid-dynamic system, Eqs. (53)-(58) and Eqs. (59) and (60), gives thc solution,, and, with which the leading-order termis determined [Eq. (49)]. In this way,
we obtain the leading-order solution of the boundary-value problem, Eqs. (17) and (27), by theHilbert expansion.
$f_{H0}^{\alpha}$$\epsilon$$\gamma|,II/n_{I},$$V_{II}/V_{\mathrm{t}h}$$T_{II}/T_{I},$$V_{I}/V_{th}$$T_{I}$$n_{I}$$\hat{n}_{H0}^{A},\hat{n}_{H0}^{B}$$\hat{v}_{rH1},\hat{v}_{\theta H0}$$\hat{T}_{H0}$$f_{H0}^{\alpha}$

Continuum limit and the ghost effect

3.2.

Here, we give a brief remark on the behavior of the mixture in the continuum limit where Kn or
vanishes. In this limit, the solution is given by Eq. (49), and thc macroscopic variables reduceto their respective leading order terms, i.e.,[cf.
Eq. (48) . That is, in this limit, evaporation and condensation of the vapor stop, and the radial
component of the flow velocity
vanishes. Therefore, one might expect that the flow field inthis lixnit is givcn by thc flow ficld in the sarnc limit of thc ordinary cylindrical Coucttc flow of abinary mixture of noncondensable gases (or between impermeable cylinders). This is not true. The
leading-order terms, andare determined together with the first-order termof the radial component of the flow velocity. This means that, in spite of the fact that there isno evaporation or condensation, the flow field in the continuum limit is affected by the infinitesimalradial flow (or infinitesimal evaporation and condensation) [note that, butis of $O(1)]$ . This is an example of the ghost effect that was pointed out in Ref. [11] andinvestigated extensively in subsequent works (see, Refs. [12-18] and the references in Refs. $[15, 18]$ ;for the case of a gas rnixturc, we refer to Rcfs. [19, 29-31] . In particular, thc situation in thepresent problem is essentially the same as that of the plane Couette flow of the mixture of a vapor
and a noncondensable gas [19]. Some numerical examples of the continuum limit obtained fromthe fluid-dynamic system, Eqs. (53)-(58) and Eqs. (59) and (60), will be given in Sec. 4.
In the case of the ordinary cylindrical Couette flow of a binary mixture of noncondensable
gases, i.e., in the case where the -component is also a noncondensable gas, the fluid-dynamic
systcm can be derived readily by a slight modification of the analysis in the prescnt scction. The
resulting fluid-dynamic equations are Eqs. (53) $-(58)$ with $\hat{v}_{rH1}=0$ , and the boundary conditions
are Eqs. (59) and (60) with thc conditions fordiscarded. In this system, , which appears in
the nondimensionalization (16), should be regarded as an appropriate reference molecular number
density, anddisappears. In addition, we have to specify a parameter associated with theamount of the -component, saybeing the average molecular number density ofwiththe -component. Some numerical solution of this system (ordinary Couette flow, for short) will
also be shown in the next section.
$\epsilon$$(\hat{n}^{A},\hat{n}^{B},\hat{T},\hat{v}_{\theta},\hat{v}_{r\prime})arrow(\hat{n}_{H0}^{A},\hat{n}_{H0}^{B},\hat{T}_{H0},\hat{v}_{\theta H0},0)$$]$$\hat{v}_{f}$$\hat{n}_{H0}^{A},\hat{n}_{H0}^{B},\hat{T}_{H0}$$\hat{v}_{\theta H0}$$\hat{v}_{rH1}$$\hat{v}_{r}=\hat{v}_{rH1}\epsilon+\cdotsarrow 0$$\hat{v}_{fH1}$

$)$

$A$

$\hat{n}_{H0}^{A}$$n_{I}$$n_{II}$$A$$n_{av}^{A}/n_{I}$$n_{av}^{A}$

$A$

4.

RESULTS AND DISCUSSIONSIn this sec.tion, we show somc numerical results
ed on the fluid-dynamic system derived inSec. 3 and discuss the behavior of the vapor and noncondensable gas in the continuum limit.
We a.ssrune that both components consist of hard-spherc molecules. Therefore, since the fluiddynamic boundary conditions do not depend onor , the problem is characterized by the$m^{B}/m^{A}$ , anddimensionless paramcters.
We let $T_{II}/T_{I}=1,$ $V_{II}/V_{th}=0$ , and $L_{II}/L_{I}=2$ throughout this section. The relations between
the terxns, etc. occurring in the Huid-dynaIuic systcrn and the physical quantities$\mathrm{b}\mathrm{a}‘ \mathrm{s}$$n_{I}$$r\iota_{II}/n_{I},$$\hat{n}_{lJ0}^{\alpha},\hat{v}_{\theta H0},\hat{T}_{H0}$$T_{II}/T_{I},$$V_{I}/V_{th},$$T_{I}$$V_{II}/V_{th},$ $n_{av}^{B}/n_{I},$$L_{II}/L_{I},$$d^{B}/d^{A}$

128

in the continuum limit are as follows:$\hat{n}_{H0}^{\alpha}=\frac{n^{\alpha}}{n_{l}}$,$\hat{\rho}_{H0}=\frac{\rho}{m^{A}n_{J}}$,$\hat{T}_{H0}=\frac{T}{T_{l}}$,$\hat{v}_{rH1}=\mathrm{l}\mathrm{i}\mathrm{r}\mathrm{n}\frac{v_{r}}{V_{th}}\frac{1}{\epsilon}\epsilonarrow 0=\lim_{\mathrm{K}\mathrm{n}arrow 0}\frac{v_{r}}{V_{th}}\frac{2}{\sqrt{\pi}\mathrm{K}\mathrm{n}}$$?^{\wedge}) \theta H0=\frac{v_{\theta}}{V_{th}}$

(61)

,

.

$\frac{2}{(L_{II}/L_{I})^{2}-1}\int_{1}^{L_{\mathrm{f}J}/L,}\hat{r}\hat{n}_{H0}^{B}\mathrm{d}\hat{r}=\frac{n_{av}^{B}}{n_{I}}$4.1.

Ghost effect, ft,
In Figs. 1 and 2, we show the profiles of the macroscopic quantities$m^{B}/m^{A}=0.2$$d^{B}/d^{A}=1$$V_{I}/V_{th}=0.5,$,
in the case:andfor various values ofand, and Fig. 2 for $n_{II}/n_{l}=0.5$ . In Fig. 1, infinitesimal evaporation takes place
Fig. 1 is foron the outer cylindcr and infinitesimal condensation on the inner, so that there is an infinitcsimal
inward flow $(\hat{v}_{tII1}<0)$ . In Fig. 2, the situation is opposite $(\hat{v}_{rII1}>0)$ . The magnitude of theincreases as the amount of the noncondcnsable gas decreases,
infinitcsimal radial flowi.e., asdecreases. Correspondingly, the profile of the tangential velocity component. In Figs. 1 and 2,
is pushcd inward (Fig. 1) or outward (Fig. 2) with thc decrcase of
the results for the ordinary cylindrical Couette flow (i.e., the case where the -component is also
noncondensable) arc also shown bylines for the corrcsponding parameters, $V_{I}/V_{th}=0.5$ ,$m^{B}/m^{A}=0.2$ , and $d^{B}/d^{A}=1$ . Here, the reference molecular number density
has been, takes thedetermined in such a way that the total number of the -component,same value as the corresponding case where the -component is the vapor (see the last paragraphfor the ordinary cylindrical Couette flow is practically independent ofof Sec. 3.2). The, i.e., the concentration of the components.
We should now recall that there is no evaporation or condensation (i.e., the radial flow is infinitesimal) in the case of the mixture of a vapor and a noncondensable gas. However, the behavior
is quite different from that of the ordinary cylindrical Couette flow. For example, the taiigendecreases. This
tial velocity profile deviates significantly, and the deviation increases asdeviation is due to the ghost cffcct causcd by thc infinitcsimal cvaporation and condcnsation.
We have also performed a Monte Carlo simulation for the original Boltzmann system, Eqs. (17)and (27), using thc dircct simulation Montc Carlo (DSMC) method $[32, 33]$ . We have assumed
the complete condensation condition for the vapor and the diffuse reflection condition for the, i.e., Eq. (27) with Eqs. (32) $-(35)$ and changed the Knudsen number Kn fromnoncondensable
0.05 to 0.005. The results for the tangential and radial components of the flow velocity are shownof Fig. 1 with $n_{av}^{B}/n_{I}=0.2$ , and Fig. 4 to thein Figs. 3 and 4. Figure 3 corrcsponds to thc$n_{av}^{B}/n_{I}=0.2$. As Kn is reduced, the magnitude of the radial component
case of Fig. 2 with. This is consistent with thedec,reases, showing the tendency that it vanishes in the limit
fact that $\hat{v}_{rH0}=0$ in the as ymptotic analysis. On the other hand, the tangential component
approaches the solid line that indicates the solution of the fluid-dynamic system. By the way, thedashed line indicates the solution of the fluid-dynamic system for the ordinary cylindrical Couette$\hat{n}_{H0}^{A}$$\hat{T}_{H0}$$BH0’\hat{v}_{\theta H0},\hat{v}_{rH1},\hat{\rho}_{H0}$$n_{av}^{B}/n_{I}$$n_{l\mathit{1}}/n_{l}=2$$(|\hat{v}_{\mathrm{r}H1}\epsilon|)$$n_{av}^{B}/n_{I}$$(\hat{v}_{\theta H0})$$n_{av}^{B}/r\iota_{I}$$A$$\mathrm{d}_{r}\backslash \mathrm{s}\mathrm{h}\mathrm{e}\mathrm{d}$$n_{I}$$A$$n_{I} \int_{1}^{2}\hat{r}\hat{n}_{H0}^{A}\mathrm{d}\hat{r}$$A$$\hat{v}_{\theta H0}$$n_{av}^{B}/n_{I}$$n_{av}^{B}/n_{I}$$\mathrm{g}\mathrm{a}_{\iota}\mathrm{s}$$\mathrm{c}\mathrm{a}_{\iota}\mathrm{s}\mathrm{c}$$v_{f}$$\mathrm{K}\mathrm{n}arrow \mathrm{O}$$v_{\theta}$

flow.
4.2.

Bifurcation

thc InasI; flow of the vapor in the radial dircction per unit timc and per
Let us denotc byunit length of the axial coordinate , which is infinitesimal in the continuum limit. Then, it is’asandrelated to$M_{r}$$z$$\hat{\rho}_{H0}\tau_{tH1}^{\wedge}$$\hat{m}^{A}\hat{n}_{H0^{\mathrm{t}^{\wedge}}rH1}^{AA}\rangle$,$\hat{r}\hat{\rho}_{H0}\hat{v}_{rH1}=\hat{\mathrm{r}}\hat{m}^{A}\hat{n}_{H0}^{A}\hat{v}_{fH1}^{A}=\lim_{\mathrm{K}\mathrm{n}arrow 0}\overline{M}$where$\overline{M}$values of$\overline{M}=\frac{M_{r}}{2\pi L_{l}m^{A}n_{I}V_{lh}}\frac{2}{\Gamma\pi \mathrm{K}\mathrm{n}}$versusis introduced for the later convenience. Figure 5 showsin the case $n_{Il}/n_{I}=1.5,$ $m^{B}/m^{A}=1$ , and $d^{B}/d^{A}=1$ . When$\hat{r}\hat{\rho}_{H}0\hat{v}_{rH1}$$n_{av}^{B}/n_{\mathit{1}}$

,

(62)

for various
is decreased$V_{I}/V_{th}$$n_{av}^{B}/n_{\mathit{1}}$129from 1 to 0.1, the curve becomes steep near $V_{I}/V_{th}=1$ , and the gradient of the curve there becomes
negative with thc further dccrcasc of. For examplc, for $n_{av}^{B}/n_{I}=0.05$ , there are three valuesofat the samein a certain interval. In other words, the solution is not unique,and the bifurcation of thc solution takes placc. As
is dccrcascd further, the curve tends
to approach the dot-dashed curve that indicates the pure-vapor case [12]. In this special case of$m^{B}/m^{A}=d^{B}/d^{A}=1$ , all the curves pass the point. The prohlesof the tangential componentof the three solutions at $V_{I}/V_{th}=$ 1.0053 for $n_{av}^{B}/n_{I}=0.08$ ,0.05, and 0.02 arc shown in Fig. 6. The solution withis c,ommon to all
andcorresponds to the ordinary cylindrical Couette flow.
We have also performed a Monte Carlo simulation (DSMC) of the original Boltzmann system.
Figure 7 shows the results for the case corresponding to Fig. 5. More specifically,in Eq. (62)versusis shown at Kn $=0.005$ for,, and 0.02 in the case$m^{B}/m^{A}=1$ , and $d^{B}/d^{A}=1$ The corresponding curve in the continuum
based on the fluiddynamic system is also shown for comparison. The DSMC result demonstrates the nonuniquenessof the solution. However we were not able to obtain the solution that roughly corresponds tothe part of the curve with the negative gradient in Fig. 5. The solution on this part might beunstable. The profiles of the tangential velocity of thc DSMC rcsult corrcsponding to thc points$(V_{I}/V_{th},\overline{M})=(1.0053, -3.36)$ and (1.0053, 4.46) for $n_{av}^{B}/n_{I}=0.05$ are shown in Fig. 8 together
with the corresponding result in the continuum limit, i.e., $(V_{I}/V_{th},\hat{r}\hat{\rho}_{H0}\hat{v}_{rH1})=(1.0053, - 4.643)$ ,1.0053, , and (1.0053, 12.17).
In Figs. 7 and 8, the DSMC results at Kn
0.005 still deviate significantly from the corresponding continuum limit. The discrepancy should reduce if we are able to carry out the DSMC
computation for smaller Knudsen numbers. Although such a computation is seemingly easy in spatially one-dimensional problems such as the present one, in reality it is not an easy task because of
the structure of the flow. When the Knudsen
is small, the-order tangential velocity
is determined together with the Kn-order radial velocity. Therefore, in the DSMC
computation, we need to obtain the srnall radial vclocity precisely in ordcr to describe thc tangcntial velocity correctly. As is well known, however, one of the drawbacks of the DSMC method isthe difficulty in obtaining sznall quantities because they are buried in the statistical fluctuations
inherent to the method. Since the radial velocity in the present problem decreases in proportion
to the Knudsen number, the computation becomos increasingly difficult with the dec.rease of theKnudsen number.
$n_{av}^{B}/n_{I}$$V_{l}/V_{t’\iota}$$\hat{r}\hat{\rho}_{H0}\hat{v}_{\mathrm{r}H1}$$n_{av}^{B}/n_{I}$$(V_{I}/V_{th},\hat{r}\hat{\rho}_{H0^{\mathrm{t})}rH1})\wedge=(1,0053,0)$$\hat{v}_{\theta H0}$$n_{av}^{B}/n_{I}$$\hat{r}\hat{\rho}_{H0}\hat{v}_{rH1}=0$$\overline{\mathrm{J}f}$$V_{I}/V_{\iota h}$$n_{av}^{B}/n_{\mathit{1}}=0.5,0.05$.

$n_{\mathit{1}\mathit{1}}/n_{I}=1.5$$\mathrm{l}\mathrm{i}\iota \mathrm{I}\mathrm{l}\mathrm{i}\mathrm{t}$$i$$v_{\theta}$

$0)$

$($

$=$

$\mathrm{K}\mathrm{n}^{0}$$\mathrm{n}\mathrm{u}\iota \mathrm{n}\mathrm{b}\mathrm{e}\mathrm{r}$$\hat{v}_{rH1}\epsilon$$\hat{v}_{\mathit{0}H0}$

Acknowledgment$\mathrm{s}$The authors tharik Prof.
Takata for his valuable discussions. This work is supportedby the Grants-in-Aid for Scientific Research (Nos. 17360041 and 16001161) from JSPS and by theCenter of Excellence for Research and Education on Cornplex Rnctional Mechanical Systems.
$\mathrm{S}\mathrm{l}\dot{\mathrm{u}}\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{u}$APPENDIX$\mathrm{A}$: MACROSCOPIC QUANTITIES OF THE HILBERT EXPANSION

The explic,it expressions of the macroscopic quantities of the Hilbert expansion bysummarized. The coefficients of each gas
are$f_{Hm}^{\alpha}$are$h_{Hm}^{\alpha}$$\hat{n}_{H0}^{\alpha}=\int f_{H0}^{\alpha}\mathrm{d}\zeta$

(A1)

,

$\hat{v}_{rH0}^{\alpha}=\frac{1}{\hat{n}_{H0}^{\alpha}}\int\zeta_{r}f_{H\mathit{0}}^{\alpha}\mathrm{d}\zeta$$\hat{v}_{\theta H0}^{\alpha}=\frac{1}{\hat{n}_{H0}^{\alpha}}\int\zeta_{\theta}f_{H0}^{\alpha}\mathrm{d}\zeta$

,

(A2)

,

(A3)

$\hat{p}_{H0}^{\alpha}=\hat{n}_{H0}^{\alpha}\hat{T}_{H0}^{\alpha}$,$= \frac{2}{3}\int[(\zeta_{r}-\hat{v}_{rH0}^{a})^{2}+(\zeta_{\theta}-\hat{v}_{\theta H0}^{\alpha})^{2}+\zeta_{z}^{2}]\hat{m}^{\alpha}f_{H0}^{\alpha}\mathrm{d}\zeta$

(A4)

130

and
$\hat{n}_{H1}^{\alpha}=\int f_{H1}^{\alpha}\mathrm{d}\zeta$

,

(A5)

,

(A6)

”

(A7)

$\hat{v}_{rH1}^{\alpha}=\frac{1}{\hat{n}_{II0}^{\alpha}}\int\zeta_{r}f_{H1}^{\alpha}\mathrm{d}\zeta-\frac{\hat{n}_{H1}^{\alpha}}{\hat{n}_{H0}^{\alpha}}\hat{v}_{\mathrm{r}H0}^{\alpha}$$\hat{v}_{\theta H1}^{\alpha}=\frac{1}{\hat{n}_{H0}^{\alpha}}\int\zeta_{\theta}f_{H1}^{\alpha}\mathrm{d}\zeta-\frac{\hat{n}_{H1}^{\alpha}}{\hat{n}_{H0}^{\alpha}}\tau_{\theta H0}^{\wedge\alpha}$$\beta_{H1}^{\alpha}=\hat{n}_{H0}^{\alpha}\hat{T}_{H1}^{\alpha}+\hat{n}_{H1}^{\alpha}\hat{T}_{H0}^{\alpha}$

.

(A8)

$= \frac{2}{3}\int[(\zeta_{f}-\hat{v}_{rH0}^{\alpha})^{2}+(\zeta_{\theta}-\iota^{\wedge\alpha}\prime_{\theta H0})^{2}+\zeta_{z}^{2}]\hat{m}^{\alpha}f_{H1}^{\alpha}\mathrm{d}\zeta$

The coefficients of the total mixture $h_{Hm}$ are expressed in terms of those of each gas:,,$\hat{n}_{H0}=\hat{n}_{H0}^{\Lambda}+\hat{n}_{H0}^{B}$$\hat{\rho}_{H0}=\hat{m}^{A}\hat{n}_{H0}^{A}+\hat{m}^{B}\hat{n}_{H0}^{B}$$\hat{v}_{rH0}=\frac{1}{\hat{\rho}_{H0}}(\hat{m}^{A}\hat{n}_{H0}^{A}\hat{v}_{rH0}^{A}+\hat{m}^{B}\hat{n}_{H0}^{B}\hat{v}_{rH0}^{B})$$\hat{v}_{\theta l\mathit{1}0}=\frac{1}{\hat{\rho}_{H0}}(\hat{m}^{A}\hat{n}_{H0}^{A}\hat{v}_{\theta H0}^{A}+\hat{m}^{B}\hat{n}_{H0}^{B}\hat{v}_{\theta H0}^{B})$

(A9)
(A10)

,

(All)

,

(A12)

$\hat{p}_{H0}=\hat{n}_{H}0\hat{T}_{H0}=\sum_{\alpha=A,B}[\hat{p}_{H0}^{\alpha}+\frac{2}{3}\hat{m}^{\alpha}\hat{n}_{H0}^{\alpha}(V_{rH0}^{\alpha})^{2}+\frac{2}{3}\hat{m}^{\alpha}\hat{n}_{H\mathit{0}}^{a}(V_{\theta H0}^{\alpha})^{2}]$

,

(A13)

and
$\hat{n}_{H1}=\hat{n}_{H1}^{A}+\hat{n}_{H1}^{B}$

,

(A14)

,

(A15)

$\hat{\rho}_{H1}=\hat{m}^{A}\hat{n}_{H1}^{A}+\hat{m}^{B}\hat{n}_{H1}^{B}$$\hat{v}_{fH1}=\frac{1}{\hat{\rho}_{H0}}[\sum_{\alpha=A,B}(\hat{m}^{\alpha}\hat{n}_{H0}^{\alpha}\hat{v}_{rH1}^{\alpha}+\hat{m}^{\alpha}\hat{n}_{H1}^{\alpha}\hat{v}_{rH0}^{\alpha})-\hat{\rho}_{H1}\hat{v}_{rH0}]$$\hat{v}_{\theta H1}=\frac{1}{\hat{\rho}_{H0}}[\sum_{\alpha=A,B}(\hat{m}^{\alpha}\hat{n}_{H0}^{\alpha}\hat{v}_{\theta H1}^{\alpha}+\hat{m}^{\alpha}\hat{n}_{H1}^{\alpha}\hat{v}_{\theta H0}^{\alpha})-\hat{\rho}_{H1}\hat{v}_{\theta H0}]$

,

(A16)

,

(A17)

$\hat{p}_{H1}=\hat{n}_{H0}\hat{T}_{H1}+\hat{n}_{H1}\hat{T}_{H0}$$= \sum_{\alpha=A,B}\{\hat{p}_{ll1}^{\alpha}+\frac{2}{3}\hat{m}^{\alpha}\hat{n}_{lI1}^{\alpha}[(V_{rH0}^{\alpha})^{2}+(V_{\theta IJ0}^{\alpha})^{2}]+\frac{4}{3}\hat{m}^{\alpha}\hat{n}_{ll0}^{\alpha}(V_{\Gamma lJ0}^{\alpha}V_{rll1}^{\alpha}+V_{\theta \mathit{1}J0}^{\alpha}V_{\theta l\mathit{1}1}^{\alpha})\}$

,
(A18)

where
$V_{rHm}^{\alpha}=\hat{v}_{rHm}^{\alpha}-\hat{v}_{rHm}$

APPENDIX$\mathrm{B}$,$V_{\theta Hm}^{\alpha}=\hat{v}_{\theta Hm}^{\alpha}-\hat{v}_{\theta Hm}$

: HILBERT SOLUTION

.

(A19)

$f_{H1}^{\alpha}$

In this appendix the index is used to represent the labels
andof the gas species, inaddition to a and . The flrst-order Hilbert solution
is given in the following form:$A$$\gamma$$\beta$$B$$f_{H1}^{\alpha}$$f_{II1}^{\alpha}=f_{lI0}^{\alpha}[ \frac{\hat{p}_{H1}^{\alpha}}{\hat{p}_{H0}^{\alpha}}+2\hat{m}^{\alpha}\frac{\hat{v}_{rH1}\tilde{\zeta}_{r}+\hat{v}_{\theta H1}\tilde{\zeta}_{\theta}}{\hat{T}_{H0}^{1/2}}+\frac{\hat{T}_{H1}}{\hat{T}_{H0}}(\hat{m}^{\alpha}\tilde{\zeta}^{2}-\frac{5}{2})$$- \tilde{\zeta}_{r}A^{\alpha}(\tilde{\zeta})\frac{1}{\hat{p}_{H0}}\frac{\mathrm{d}\hat{T}_{H0}}{\mathrm{d}\hat{r}}-\tilde{\zeta}_{r}\tilde{\zeta}_{\theta}B^{\alpha}(\tilde{\zeta})\frac{\hat{T}_{H0}^{1/2}}{\hat{p}_{H0}}(\frac{\mathrm{d}\hat{v}_{\theta H0}}{\mathrm{d}\hat{r}}-\frac{\hat{v}_{\theta H0}}{\hat{r}})$$- \tilde{\zeta}_{r}\sum_{\beta=A,B}D^{(\beta)\alpha}(\tilde{\zeta})\frac{1}{\hat{n}_{H0}\hat{p}_{H0}}(\frac{\mathrm{d}\hat{p}_{H0}^{\beta}}{\mathrm{d}\hat{r}}-\frac{2\hat{m}^{\beta}\hat{n}_{H0}^{\beta}\hat{v}_{\theta II0}^{2}}{\hat{r}})]$

,

(B1)

131

where
$\tilde{\zeta}_{r}=\frac{\zeta_{r}}{\hat{T}_{H0}^{1/2}}$

The functions,$\tilde{\zeta}_{\theta}=\frac{\zeta_{\theta}-\hat{v}_{\theta H0}}{\hat{T}_{H0}^{1/2}}$. and$A^{\alpha}((), B^{\alpha}(\zeta)\text{ノ}$$D^{(\beta)\alpha}$,$\tilde{\zeta}_{z}=\frac{\zeta_{r}}{\hat{T}_{H0}^{1/2}}$.

,

are the solutions of the following integral equations:,$\sum_{\beta=A,B}K^{\beta\alpha}\chi_{H0}^{\beta}\tilde{L}_{T}^{\beta\alpha}(\zeta_{r}A^{\beta}, (_{r}A^{\alpha})=-\zeta_{r}(\hat{m}^{\alpha}\zeta^{2}-\frac{\backslash r_{\mathrm{J}}}{2})$subsidiary condition:

(B3)

,

$\sum_{\alpha=\Lambda,B}\hat{m}^{\alpha}\chi_{H0}^{\alpha}I_{4}^{\alpha}(A^{\alpha})=0$,$\sum_{\beta=A,B}K^{\beta\alpha}\chi_{H0}^{\beta}\tilde{L}_{T}^{\beta\alpha}(\zeta_{f}\zeta_{\theta}B^{\beta}, \zeta_{f}\zeta_{\theta}B^{\alpha})=-2\hat{m}^{\alpha}\zeta_{r}\zeta_{\theta}$(B4)$\sum_{\beta=A,B}K^{\beta\alpha}\chi_{H0}^{\alpha}\chi_{H0}^{\beta}\tilde{L}_{T}^{\beta\alpha}(\zeta_{r}D^{(\gamma)\beta}, \zeta_{r}D^{(\gamma)\alpha})=-\zeta_{r}(\delta_{\alpha\gamma}-\frac{\hat{m}^{\alpha}n_{H0}^{\alpha}}{\hat{\rho}_{H0}})$subsidiary condition:

where$\zeta=(\zeta_{r}^{2}+\zeta_{\theta}^{2}+\zeta_{z}^{2})^{1/2}$(B2)$\tilde{\zeta}^{2}=\tilde{\zeta}_{r}^{2}+\tilde{\zeta}_{\theta}^{2}+\tilde{\zeta}_{z}^{2}$$\sum_{\alpha=A,B}\hat{m}^{\alpha}\chi_{H0}^{\alpha}I_{4}^{\alpha}(D^{(\beta)\alpha})=0$. The operators$\tilde{L}_{T}(f, g)$

,
(B5)

,

and $I_{n}(F)$ is defincd as follows:$\tilde{L}_{\Gamma}^{\beta\alpha},(f, g)=\int[f(\zeta_{*}’)+g(\zeta’)-f(\zeta_{*})-g(\zeta)]E^{\beta}(\zeta_{*})\hat{B}_{T}^{\beta\alpha}\mathrm{d}\Omega(\mathrm{e})\mathrm{d}\zeta_{*}$$\hat{B}_{\mathit{1}’}^{\beta\alpha},=\hat{B}_{\mathit{1}’}^{\beta\alpha},(|\mathrm{e}\cdot\hat{V}|/\hat{V},\hat{V})=\hat{B}^{\beta\alpha}(|\mathrm{e}\cdot\hat{V}|/\hat{V})\hat{V}\hat{T}_{H0}^{1/2})/\hat{T}_{H0}^{1/2}$

,

,

,

(B6)
(B7)
(B8)

$I_{\mathfrak{n}}^{\alpha}(F)= \frac{8\pi}{15}(\frac{\hat{m}^{\alpha}}{\pi})^{3/2}\int_{0}^{\infty}\zeta^{n}F(\zeta)\exp(-\hat{\mathrm{m}}^{\alpha}\zeta^{2})\mathrm{d}\zeta$

where$E^{\beta}( \zeta_{*})=(\frac{\hat{m}^{\beta}}{\pi})^{3/\mathrm{z}}\exp(-\hat{m}^{\beta}\zeta_{*}^{2})$$(_{*}=(\zeta_{*f}^{2}+\zeta_{*\theta}^{2}+\zeta_{*z}^{2})^{1/2}$

The transport coefficients
, and
, and
functionsas follows:$\hat{\mu},\hat{\lambda},$$A^{\alpha},$$B^{\alpha}$$k_{\mathit{1}’},,\hat{D}_{\mathit{1}’}’$$\hat{D}_{AB}$

,

(B9)

.

(B10)

in Eqs. (55)-(57) are defined by means of the$D^{(\gamma)\alpha}$$\hat{D}_{AB}=\chi_{H0}^{A}\chi_{H0}^{B}(\hat{\Delta}_{AA}+\hat{\Delta}_{BB}-\hat{\Delta}_{AB}-\hat{\Delta}_{BA})$$\hat{D}_{T}=\chi_{H0}^{A}\chi_{H0}^{B}(\hat{D}_{TA}-\hat{D}_{TB})$

,

,
(Bll)

$\hat{\mu}=\sum_{\alpha=A,B}\hat{m}^{\alpha}\chi_{H0}^{\alpha}\hat{\mu}^{\alpha}$$k_{T}= \frac{\hat{D}_{T}}{\hat{D}_{AB}}$,,$\hat{\lambda}=\hat{\lambda}’-k_{T^{\frac{\hat{D}_{T}}{\chi_{H0}^{A}\chi_{H0}^{B}}}}$

,

wherc
$\hat{\Delta}_{\alpha\beta}=\frac{5}{2}I_{4}^{\alpha}(D^{(\beta)\alpha})$$\hat{D}_{T\alpha}=\frac{5}{2}I_{4}^{\alpha}(A^{\alpha}\rangle$

,

,

(B12)
$\hat{/}r^{\alpha},=\hat{m}^{\alpha}I_{6}^{\alpha}(B^{\alpha})$,$\hat{\lambda}^{\prime\alpha}=\frac{5}{2}I_{4}^{\alpha}([\hat{m}^{\alpha}\zeta^{2}-\frac{5}{2}]A^{\alpha})$

.

132

From the subsidiary conditions and appropr\’iate integration of the integral equations in Eqs (B3),(B5), the following relations are derived, with the help of which thc form of Eqs. (55)-(57) is

obtained:$\hat{\Delta}_{\alpha\beta}=\hat{\Delta}_{\beta\alpha}$,$\sum_{\beta=A,B}\hat{m}^{\beta}\chi_{H0}^{\beta}\hat{\Delta}_{\alpha\beta}=0$$\sum$

(B13)

,

(B14)

.

(B15)

$\hat{m}^{\beta}\chi_{H0}^{\beta}\hat{D}_{T\beta}=0$$\beta=A,B$, and
A numerical database for the coefficientsfor hard-sphere molecules has
been constructed in Rpf. [34], and it
been used in obtaining the numerical solution presented
in Sec. 4.
$\hat{\mu},\hat{\lambda},$$k_{\Gamma},,\hat{D}_{\mathit{1}^{\tau}},$$\hat{D}_{AB}$$\mathrm{h}_{\mathfrak{c}}\backslash \mathrm{s}$.Fluids

10, 1200 (1967).
”[(1996).
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[4] K. Aoki, H. Yoshida, T. Nakanishi, and A. L. Garcia, Phys. Rev.
, 016302 (2003).
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[6] Y. Sone, T. Ohwada, and Y. Makihara, in Rarefied Gas Dynamics, editcd by R. Brun, R. Campargue,
R. Gatignol, and J.-C. Lengrand (C\’epadu\‘es-Editions, Toulouse, 1999), Vol. 1, p. 511.
[7] Y. Sonc and T. Doi, Phys. Fluids 12, 2639 (2000).
[8] Y. Sone, M. Handa, and H. Sugimoto, }. Theory Stat. Phys. 31, 299 (2002).
[9] L. Arkeryd and A. Nouri, J. Stat. Phys. 118, 849 (2005).
[10] K. Aoki and Y. Sone, in Advances in Kinetic Theory and Continuum Mechanics, edited by R. Gatignol
and Soubbaramayer (Springer-Verlag, Berlin, 1991), p. 43.
[11] Y. Sone, K. Aoki, S. Takata, H. Sugimoto, and A. V. Bobylev, Phy\S . Fluids 8, 628 (1996); Erratium:
ibid 8, 841 (1996).
[12] Y. Sone, S. Tahta, and H. Sugimoto, Phys. Fluids 8, 3403 (1996); Erratum: ibid 10, 1239 (1998).
[13] Y. Sone, in Rarefied Gas Dynamics, edited by Ching Shen (Peking University Press, Beijing, 1997),
p. 3.
[14] Y. Sone, Annu. Rev. Fluid Mech. 32, 779 (2000).
, Boston, 2002).
[15] Y. Sone, Kinetic Theory and Fluid Dynamics ([16] Y. Sone, M. Handa, aud T. Doi, Phys. Fluids 15, 2903 (2003).
[17] Y. Sone and T. Doi, Phys. Fluids 16, 952 (2004).
[18] Y. Sone, Molec ular Gas Dynamics: Theory, Techniques, and Applications (Birkh\"auser, Boston, 2006).
[19] S. Takata and K. Aoki, Phys. Fluids 11, 2743 (1999).
[20] C. Cercignani, Rarefied Gas Dynamics (Cambridge University Press, 2000).
[21] C. Cercignani and M. Lampis, Transp. Theory Stat. Phys 1, 101 (1971).
[22] Y. Sone, in Rarefied Gas Dynamics, edited by L. Trilling and H. Y. Wachman (Academic Press, New
York, 1969), p. 243.
[23] Y. Sone, in Rarefied Gas Dynamics, edited by D. Dini (Editrice Tecnico Scientifica, Pisa, Italy, 1971),
Vol. II, p. 737.
[24] Y. Sone, in Advances in Kinelic Theory and Continuum Mechanics, edited by R. Gatignol and Soubbaramayer (Springcr, Bcrlin, 1991), p. 19.
[25] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd Ed. (Cambridge
University Press, Cambridge, 1995).
, Eur. J. Mech. B/Flui&22, 51 (2003).
[26] K. Aoki, S. Takata, and S.
[27] K. Aoki, S. Takata, and S. Kosuge, Phys. Fluids 10, 1519 (1998).
[28] S. Taguchi, K. Aoki, and S. Takata, Phys. Fluids 16, 4105 (2004).
[29] S. Takata, K. Aoki, and T. Muraki, in Rarefied Gas $Dynarr\iota ics$ , editcd by R. Brun, R. Carnpargue, R.
, Vol. 1, p.479.
Gatignol, and J.-C. Lengrand[30] S. Takata and K. Aoki, Transp. Theory Stat. Phys. 30, 205 (2001); 31, 289(E) (2002).
[31] S. Takata, Phys. Fluids 16, 2182 (2004).
[32] G. A. Bird, Molecular Gas Dynamics (Oxford University Press, Oxford, 1976).
[33] G. A. Bird, Molec ular Gas Dynamics and the Direct Simulation of Gas Flows (Oxford University
Press, Oxford, 1994).
[34] S. Takata, S. Yasuda, K. Aoki, and T. Shibata, in Rarefied Gas Dynamics, edited by A. D. Ketsdeveraiid E. P. Muntz (Melville, Ncw York, 2003), p.106.
$2[1|\mathrm{K}\mathrm{C}..$$\mathrm{C}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{F}.\mathrm{S}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{N}\mathrm{a}\mathrm{n}\mathrm{b}\mathrm{u},\mathrm{P}\mathrm{h}\mathrm{y}\mathrm{s}.\mathrm{F}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{i}\mathrm{d}\mathrm{s}27,2632$$(\mathrm{l}984)\mathrm{P}\mathrm{h}\mathrm{y}\mathrm{s}.$$\mathrm{B}/\mathrm{F}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}\mathrm{s}16,493$$\mathrm{b}^{\backslash }68$$?$$\mathrm{a}\mathrm{o}\mathrm{p}$$\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{k}\mathrm{h}\ddot{\mathrm{a}}\iota \mathrm{l}\mathrm{s}\mathrm{e}\mathrm{r}$$r_{1\mathrm{a}\mathrm{g}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{i}}’$$(\mathrm{C}\text{\’{e}} \mathrm{p}\mathrm{a}\mathrm{d}\mathrm{u}\text{\‘{e}} \mathrm{s}-\acute{\mathrm{b}}^{\backslash }\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}o\mathrm{o},\mathrm{f}\prime \mathrm{o}\mathrm{u}\mathrm{l}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{e}, 1999)$

133(hard-sphere
FIG. 1: Profiles of the macroscopic quantities in the continuum limit for various$m^{B}/m^{A}=0.2$ , and $d^{B}/d^{A}=1(L_{II}/L_{I}=2$ ,molecules) in the case of $n_{II}/n_{I}=2,$$T_{II}/T_{I}=1$ , and $V_{II}=0$ ). The dashed line indicates the corresponding results for ordinary cylindricalin this case).
Couette flow (see the main text for the meaning of$n_{av}^{B}/n_{l}$$V_{\mathit{1}}/V_{\mathrm{t}h}=0.5,$$n_{I}$

134

FIG. 2: Profiles of the macroscopic quantities in the continuum limit for various(hard-spheremolecules) in the case of $n\iota I/n_{I}=0.5,$ $V\mathrm{r}/V_{th}=0.5,$ $m^{B}/\mathrm{m}^{A}=0.2$ , and $d^{B}/d^{A}=1(L_{II}/L_{I}=2$ ,$n_{av}^{B}/n_{I}$$T_{II}/T_{\mathrm{f}}=1$, and$V_{\mathrm{f}I}=0$). See the caption of Fig. 1.

135

$0$

.

$-0.05.:.:.:.:.:.\cdot:.:\cdot.\mathfrak{h}.::0:.\mathrm{i}::.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:0_{:}005\mathit{0}.02^{\cdot}$$....\dot{\mathrm{K}}.\mathrm{n}..=..0^{\cdot}:_{0_{\theta}^{r}}\ldots.\ldots\ldots\ldots\ldots\ldots.$.

$\backslash \mathrm{s}_{\partial}^{\mathrm{S}}\llcorner$$-0.1.\cdot.\cdot..$

1.5$-0.1\mathrm{s}_{1}$2$r/L$ ;

FIG. 3: Profiles of the flow velocity for small Kn (DSMC result for hard-sphere molecules) in the case of$n_{av}^{B}/n_{I}=0.2,$ $n_{II}/n_{I}=2,$$V_{II}=\mathit{0})$$V_{l}/V_{th}=0.5,$ $m^{B}/m^{A}=0.2$, and $d^{B}/d^{A}=1(L_{II}/L_{I}=2,$ $T_{II}/T_{I}=1$ , and. The solid line indicates the profile of the continuum limit, and the dashed line that of the same

limit for the ordinary cylindrical Couette flow.

0.15$......$

0.1.

.
$....\mathrm{K}..\mathrm{n}..=..0.\cdot.0.5\ldots\ldots\ldots\ldots$.

$.......\ldots\ldots\ldots\ldots.0\ldots 0.2\ldots\ldots\ldots\ldots.$$\backslash \mathrm{s}_{\tilde{\check{\aleph}}}^{\mathrm{s}}$0.05$\ldots\ldots\ldots\ldots\ldots\ldots 0.\cdot.0!\ldots\ldots\ldots\ldots$.

.

$\ldots\ldots\ldots\ldots\ldots\ldots.\mathrm{A}\cdot 99?\ldots\ldots\ldots\ldots$$01$

FIG. 4:

1.5$r/L_{I}$2of the lflflow velocity for small Kn (DSMC restt for hard-8phere molecules) in the case of$0.5$ , $V_{I}/V_{th}=0.5,$ $m^{B}/m^{A}=0.2$ , and $d^{\hslash}/d^{A}=1(L_{l/}/Lr=2,$ $T_{ll}/T_{l}=1$ , and$VII=0)$ . See the caption of Fig. 4,$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{R}\mathrm{l}\mathrm{e}\mathrm{s}$$n_{av}^{B}/I=0.2,$$nII/n\mathrm{I}$$–$FIG. 5: The
versusin the continuum limit for various
in the case of $n_{lI}/n_{I}=1.5,$ $m^{B}/m^{A}=1$ , and $d^{B}/d^{A}=1$ (dot-dashed line indicatcs the purc vapor case [12].
$\hat{r}\hat{\rho}_{H0}\hat{v}_{fH1}$$V_{l}/V_{ch}$$L_{\mathit{1}I}/L_{\mathit{1}}=2,$$n_{a\tau}^{B},/n_{I}$(hard-sphere molecules), and $V_{Il}=0$ ). The$T_{l\mathit{1}}/T_{l}=1$

136

FIG. 6: Profiles of$\hat{v}_{\theta H0}$of the three different solutions in the continuum limit at $V_{I}/V_{th}=1.0053$ (hard-

sphere molecules) for $n_{av}^{B}/I=0.08,0.05$ , and 0.02 in thc caqe of $n_{II}/n_{I}=1.5,$ $m^{B}/m^{A}=1$ , and$d^{B}/d^{A}=1$ ( $L_{II}/L_{I}=2,$ $T_{II}/T_{I}=1$ , and $V_{Il}=0$ ).

at Kn $=0.005$ (DSMC result for hard-sphere molecules)
FIG. 7: Magnified mass flow rate versus
for $n_{av}^{B}/n_{I}=0.5,0.05$ , and 0.02 in the case of $n_{II}/n_{I}=1.5,$ $m^{B}/m^{A}=1$ , and $d^{B}/d^{A}=1(L_{II}/L_{I}=2$ ,$T_{II}/T_{I}=1$ , and $V_{II}=0$ ). The solid line indicates the results in the continuum limit.
$\overline{M}$$V_{l}/V_{\mathrm{t}h}$(DSMC result for hard-sphere molecules) correat Kn
and (1.0053, 4.46) in the case of $n_{a}^{B},,/n_{l}=0.05$ , $n_{Il}/n_{I}=1.5$ ,, and). The corresponding re\S lllt\epsilon in the con$(V_{I}/V_{th,\hat{r}\hat{\rho}_{H0}\hat{v}_{r}H1})=(1.005\delta, - 4.643)$ ,1.0053, , and (1.0053, 12.17),

FIG. 8: Profiles of the tangential velocity
sponding to$m^{B}/m^{A}=1$ , and $d^{B}/d^{A}=1$ (tiuuum limit, i.e., the profiles atare also shown by the solid line.

$v_{\theta}$$=0.0\mathit{0}5$$(V\iota/V_{\mathrm{t}h)}\overline{M}\rangle=(1.0053, -3.36)$$L_{II}/L_{\mathrm{J}}=2,$ $T_{l\mathit{1}}/T_{l}=1$$V_{\mathit{1}l}=0$$($$0)$