数理解析研究所講究録
第 824 巻 1993 年 224-254

224

Kinetic theory analysis of steady evaporating flows
from a spherical condensed phase into a vacuum
Yoshio Sone and Hiroshi Sugimoto
宏
良夫
曽根
杉本
Department of Aeronautical Engineering, Kyoto University, Kyoto 606-01 Japan
京大・工Abstract
Steady evaporating flows from a spherical condensed phase into a vacuum are considered. On the basis of the Boltzmann-Krook-Welander equation, the behavior of the gas(the velocity distribution function as well as the density, velocity, and temperature) fromthe sphere to downstream infinity is analyzed numericaUy in detail for the whole range ofthe Knudsen number. An analytic solution is also given for small Knudsen numbers. The
discontinuity of the velocity distribution function in the gas, a typical behavior of the gas
around a convex body, is analyzed curately. lt extends, with appreciable size, to downstream infinity not only for intermediate and large Knudsen numbers but also even for rather$smaU$ Knudsen numbers. The flow is highly in nonequilibrium over the whole field except
for very small Knudsen numbers. The velocity and temperature at downstream iniinity takefinite values, determined by the Knudsen number, except the temperature at zero Knudsen
numb er.
$ac$

I.

INTRODUCTION

Considerable kinetic theory studies on gas flows around a condensed phase, where evaporation or condensation is taking place, have been carried out in the last twenty years (e.g.,
Refs. 1-34). Most of them are concerned with one-dimensional problems, but various importantand interesting results are presented. The boundary conditions for hydrodynamic equation in
the continuum hmit on a interface between a gas and its condensed phase are derived from theanalysis of a half-space problem of evaporation or condensation, where a semi-infinite expanse ofa gas is considered.10,11,22,29,32 A steady evaporating flow of the half-space problem, determinedby a parameter at downstream infinity in addition to the condition of the condensed phase,
can neither be supersonic nor take a downstream pressure lower than some positive value Pmin
(Ref. 22). In the case where a plane condensed phase is adjacent to a vacuum, a time-dependent
process occurs, and finally the whole field is filled with the gas and approaches a steady evaporating flow that is sonic and takes the pressure Pmin at downstream infinity. The initial vacuum
state disappears.
When a cylindrical or spherical condensed phase is set in an infinite expanse of a vacuum,its vapor gas expands into a wider region, and finally a steady evaporating flow into a vacuum
will be established. The flow behavior depends on the Knudsen number of the condition on thecondensed phase (the initial Knudsen number, for short). In the flow into a vacuum, as thegas going downstream, the mean free path increases indefinitely since the density decreases tozero, but the characteristic length becomes longer indefinitely corresponding to vanishing of thevariation of the variables. According to the es timate by the spherically expanding isentropic flowin the continuum limit, the effective Knudsen number (the local mean free path divided by the
local characteristic length), which characterizes the variation of the flow, increases indefinitely
with the flow. Thus, various features are expected to be seen along the flow. Unfortunately

225
few are known about this interesting flow into a vacuum; some discussions were made about
the deviation from local equilibrium well downstream of an isentropic expanding flow in thecontinuum limit.35-38 Recent engineering problems such as isotope separation by a laser beam39
and vacuum vapor deposition require detailed information, at the level of the velocity distributionfunction, of the flow. In the present paper, therefore, we investigate a steady evaporating flowfrom a spherical condensed phase into a vacuum, mainly numerically, for the entire range of theinitial Knudsen number to obtain the comprehensive information about the flow.
In the numerical analysis of the problem, there are two points to be treated carefully. First,
as discussed in Refs. 34 and 40, the velocity distribution function has discontinuities in a gas
around a convex body. Fortunately, a hybrid difference system (an ordinary difference system
supplemented with a characteristic method) capable of describing the discontinuity accurately
is devised in Ref. 34. This is applied to the present problem. Second, practical numericalcomputation has to be carried out over a finite domain. Unfortunately, in the present problem
the flow field approaches a vacuum state at infinity fairly slowly. In order to obtain reliable
results, therefore, we will analyze the problem for many cases of various domain sizes andexamine the results carefully.
II.

PROBLEM AND BASIC EQUATION

Consider a steady evaporating flow from a spherical condensed phase (radius: , temper) into an infinite expanse of a
, the saturation gas pressure at temperature:
ature:
vacuum. We investigate the behavior of the flow around the sphere for a wide range of theKnudsen number $Kn_{w}$ (the mean free path at the equilibrium state at rest with temperatureand pressure
divided by the radius of the sphere) under the following assumptions:(i) the behavior of the gas is described by the Boltzmann-Krook-Welander (BKW or BGK)
$equation$ ;(ii) the gas molecules leaving the condensed phase constitute the corresponding(the conand temperaturepart of the stationary Maxwellian distribution with pressure
ventional boundary condition for evaporation and condensation). The extension of the result toa more general boundary condition will be given in Sec. VI.
The Boltzmann-Krook-Welander equation for a spherically symmetric steady flow is given$L$$T_{w}$$T_{w}$$T_{w}$$p_{w}$$p_{w}$$T_{w}$$p_{w}$$as^{44}$,$\xi_{r}\frac{\partial f}{\partial r}+\frac{\xi_{t}^{2}}{r}\frac{\partial f}{\partial\xi_{r}}-\frac{\xi_{r}\xi_{t}}{r}\frac{\partial f}{\partial\xi_{t}}=A_{cd}\rho(f_{e}-f)$,$f_{e}= \frac{\rho}{(2\pi RT)^{3/2}}\exp(-\frac{(\xi_{r}-u)^{2}+\xi_{t}^{2}}{2RT}I$$\rho=2\pi\int\int f\xi_{t}d\xi_{r}d\xi_{t}$,$\rho u=2\pi\iint f\xi_{f}\xi_{t}d\xi_{r}d\xi_{t}$

(2)

(3a)

,

(3b)

$3 \rho RT=2rr\iint f[(\xi_{r}-u)^{2}+\xi_{t}^{2}]\xi_{t}d\xi_{r}d\xi_{t}$$p=\rho RT$

(1)

,

,

(3c)
(3d)

where is the radial distance from the center of the sphere;is the radial component of themolecular velocity;is the magnitude of the molecular velocity normal to the radial direction,$r$$\xi_{r}$$\xi_{t}$226
is the molecular speed, which will be denoted by for short; $f$ is thethat is,, andis the density of the gas;
velocity distribution function, which is a function ofcomponent; $T$
velocity,
only
of
which
is
the
nonvanishingcomponent
radialtheflow
is the$R$
is a constant; is the specificis the temperature of the gas; is the pressure of the gas;
gas constant. The integration in Eqs. $(3a)-(3c)$ , and in what follows unless otherwise stated, isis the colhsion frequency
carried out over the range $(-\infty<\xi_{r}<\infty, 0\leq\xi_{t}<\infty)$ . The
of a gas molecule, which is independent of molecular velocity for BKW equation. The parallel, to the flow, are given by
andand normal temperatures$(\xi_{r}^{2}+\xi_{t}^{2})^{1/2}$

$\xi$

$r,$

$\xi_{t};\rho$$\xi_{r}$$u$$A_{cd}$$p$$A_{col}\rho$$T_{||}$$\tau_{\perp}$$\rho RT_{||}=2\pi\iint f(\xi_{r}-u)^{2}\xi_{t}d\xi_{r}d\xi_{t}$$\rho RT_{\perp}=\pi\int\int f\xi_{t}^{3}d\xi_{r}d\xi_{t}$

(4a)

,

.

(4b)

Thus,

.

$T=(T_{N}+2T_{\perp})/3$(5)

The boundary condition on the condensed phase (at $r=L$ ) is,

$f= \frac{\rho_{w}}{(2\pi RT_{w})^{3/2}}\exp(-\frac{\xi_{f}^{2}+\xi_{t}^{2}}{2RT_{w}})$$\rho_{w}=p_{w}/RT_{w}$$(\xi_{r}>0)$

.

,

(6)
(7)

The condition at infinity $(rarrow\infty)$ is$f=0$ ,$(\xi_{r}<0)$.

(8)

For the convenience of the following analysis, we introduce the following nondimensionalvariables:$r=Lr\wedge$

,

(9a)

$\xi_{r}=(2RT_{w})^{1/2}\zeta\cos\theta_{\zeta}$$\xi_{t}=(2RT_{w})^{1/2}\zeta\sin\theta_{\zeta}$$f= \frac{\rho_{w}}{2\pi(2RT_{w})^{3/2}}\hat{f}$

,

(9b)

,

,

$\rho=\rho_{w}\rho\wedge$,$u=(2RT_{w})^{1/2\wedge}u$$p=p_{wP}^{\wedge}$,$T=T_{w}\hat{T}$

(9c)

,

(9d)

.

With these new variables, the BKW equation (1) is reduced to$D \hat{f}=\frac{2}{\sqrt{\pi}Kn_{w}}\rho\wedge(\hat{f_{e}}-\hat{f})$

,

(10)

where
$D=( \cos\theta_{\zeta}\frac{\partial}{\partial r\wedge}-\frac{\zeta\sin\theta_{\zeta}\partial}{r\wedge\partial\theta_{\zeta}}$

,

(lla)

227

$Kn_{w}= \frac{l_{w}}{L}$,$l_{w}= \frac{(8RT_{w}/\pi)^{1/2}}{A_{col}\rho_{w}}$

,

(llb)

$\hat{f_{e}}=\frac{2\rho\wedge}{\pi^{1/2}\hat{T}^{3/2}}\exp(-\frac{\zeta^{2}+\wedge u^{2}-2(u\wedge\cos\theta_{(}}{\hat{T}}I$

.

(llc)

and temperatureis the mean free path at the equilibrium state at rest with pressure
The
,
, and
is the Knudsen number at the state. In the new independent variables ,the BKW equation contains only the derivatives with respect to and . The nondimensional, and are related to as
macroscopic variables$\ell_{w}$$p_{w}$$Kn_{w}$$T_{w}$$(r\wedge$$(, \theta_{\zeta})$$\wedge r$$\theta_{\zeta}$$\rho\wedge,$$u\wedge,\hat{T}$$\hat{f}$$p\wedge$$\rho\wedge=\iint\hat{f}\zeta^{2}\sin\theta_{\zeta}d\zeta d\theta_{(}$

,

(12a)

$\overline{\rho u}=\iint\hat{f}\zeta^{3}\cos\theta_{(}\sin\theta_{\zeta}d\zeta d\theta_{\zeta}$

,

(12b)
(12c)

$\frac{3}{2}\hat{\rho}\hat{T}=\iint\hat{f}(4\sin\theta_{\zeta}d\zeta d\theta_{\zeta}-\overline{\rho u}^{2},$$\hat{p}=\hat{\rho}\hat{T}$

.

(12d)

in Eqs. $(12a)-(12c)$ , and in the following
The double integration with respect to (andequations unless otherwise stated, is carried out over the domain $(0\leq\zeta<\infty, 0\leq\theta_{\zeta}\leq\pi)$.
,
The nondimensional forms of the boundary conditions are, at$\theta_{\zeta}$$\wedge r=1$$\hat{f}=\frac{2}{\sqrt{\pi}}\exp(-\zeta^{2})$,$(0\leq\theta_{\zeta}<\pi/2)$

,

(13)

and, at infinity,$\hat{f}=0$,$(\pi/2<\theta_{\zeta}\leq\pi)$

.

(14)

The boundary-value problem [Eqs. (10), (13), and (14)] contains only a parameter, the(Ref. 45). We will investigate the problem numerically for a wide range
Knudsen numberof the Knudsen number and clarify the comprehensive behavior of the evaporating flow from aspherical condensed phase into a vacuum.
, orover the domain $(0\leq$Integrating Eq. (10) multiplied by$0\leq\zeta<\infty$ ), we obtain the following conservation equations:$Kn_{w}$$\zeta^{2}\sin\theta_{\zeta},$$(^{3}\cos\theta_{\zeta}\sin\theta_{\zeta}$$\zeta^{4}\sin\theta_{\zeta}$$\theta_{\zeta}\leq\pi,$$\overline{\rho u}r^{2}\wedge=Q/4\pi\rho_{w}(2RT_{w})^{1/2}L^{2}$$\frac{d}{dr\wedge}[(u^{2}\wedge+\hat{T}_{||}/2)\overline{\rho r}^{2}]=\rho r’\hat{\tau}_{\perp^{\wedge}}$

,

(15)

,

$\hat{w}r^{2}\wedge=W/4\pi p_{w}(2RT_{w})^{1/2}L^{2}$

(16)

,

$\hat{w}=\iint\hat{f}\zeta^{5}\cos\theta_{\zeta}\sin\theta_{\zeta}d\zeta d\theta_{\zeta}$

(17)

,

(18)

where
and $W$ are the mass flux and energy flux from the sphere respectively. Equations (15)and (17) will be used in an accuracy test of our computation.
The saturation gas pressure
is a function of
given by the Clausius-Clapeyron relation.46
is never used, and thus
In the following analysis, however, the relation between
andandcan be chosen freely in the results. Thus the result has wider application. Sec. VI is oneof the examples.
$Q$$p_{w}$$T_{w}$$p_{w}$$T_{w}$$T_{w}$$p_{w}$228
III.

METHOD OF NUMERICAL ANALYSISA.

Preliminary remarks

In the numerical analysis of the boundary-value problem, Eqs. (10), (13), and (14), we haveto hmit the domain ( $1\leq r\wedge<\infty,$ $0\leq(<\infty, 0\leq\theta_{\zeta}\leq\pi)$ of our interest to a finite domain, $0\leq\zeta\leq(_{D}, 0\leq\theta_{(}\leq\pi)$ . Since is seen to decay rapidly with (from our(numerical tests, accurate computation of the problem can be carried out with a reasonable size ofis rather slow, and therefore. On the other hand, approach to the state at infinity as
and confirm the asymptotic behavior for
we will carry out detailed tests for various large
large . This requires rather large computation.
Another difficulty in the numerical computation is the discontinuity of the velocity distribution function in the gas, which is pointed out and discussed in Refs. 34 and 40. The velocity(between
distribution function on the condensed phase $(r\wedge=1)$ is discontinuous at
the incoming and leaving molecules), and this discontinuity propagates into the gas along the. In the following numerical computation by a finite
characteristic of Eq. (10), i.e.,
difference method, we have to take this discontinuity into account. Fortunately, a finite difference scheme capable of describing the discontinuity is developed in Ref. 34, which will be applied
to the present computation.
$\hat{f}$$1\leq r\wedge\leq\wedge r_{D}$$r\wedgearrow\infty$$\zeta_{D}$$\wedge r_{D}$$\wedge r$$\theta_{\zeta}=\pi/2$$r\wedge\sin\theta_{\zeta}=1$

Difference scheme and process of solution

B.

The boundary-value problem, Eqs. (10), (13), and (14), is considered over a finite domain$0\leq\zeta\leq(D, 0\leq\theta_{(}\leq\pi)$ , whereand ( are chosen properly depending
be the lattice points in the domain, where $i=0,1,$
on situations. Let(1$\leq r\wedge\leq\wedge r_{D},$$\wedge r_{D}$$(^{\triangleleft_{r}:)},$

$D$

$I$

$(^{(j)}, \theta_{\zeta}^{(k)})$$\ldots,$$(^{\triangleleft_{r}0)}=1, \wedge r^{\langle I)}=\wedge r_{D}),$$j=0,1,$$\ldots,$$J(\zeta^{(0)}=0, \zeta^{(J)}=\zeta_{D})$, and $k=0,1,$$\ldots,\overline{K},$$\ldots,$, and). For the convenience of analysis, the points$(s)$(of a continuous variable, andas the values of smooth functions
integer points $s=i,$ , and :$\theta_{(}^{(\hat{K})}=\pi/2,$$\theta_{(}^{(K)}=\pi$$\wedge r^{(t)},$$r\wedge(s),$$\wedge(:)\wedge r=r(i)$$\hat{f},$$\rho\wedge$$\theta_{\zeta}(s)$,are taken
evaluated at$\theta_{\zeta}^{(k)}$

$s$

$k$

$j$

The variables
lattice point:$\zeta^{(g)}$$K(\theta_{(}^{(0)}=0$,$(^{(j)}=\zeta(j), \theta_{\zeta}^{(k)}=\theta_{\zeta}(k)$

.

(19)

, etc. at a lattice point are denoted by the superscripts corresponding to the$\hat{f}^{(i,j,k)\triangleleft i)}=\hat{f}(r, \zeta^{()}J\theta_{\zeta}^{(k)})$,$\hat{\rho}^{\langle i)}=\rho\wedge(r^{i)}\triangleleft)$

.

(20)

depends on
Note that, and[see Eq. (llc)].
only throughWe construct the discrete solutionof Eqs. (10), (13), and (14) as the limit of the sequenceobtained by the iteration process described below. Corresponding$\hat{f_{e}}^{(\cdot,j,k)}$$\acute{r}^{\langle\cdot)}$$\hat{\rho}^{\langle t)},$$u^{(i)}\wedge$$\hat{T}^{(i)}$$\hat{f}^{(\iota,j,k)}$$\hat{f}_{(n’)}^{(\dot{t}}\dot{J}^{k)}(n=0,1,2, \ldots)$

to Eq. (10), the following finite difference equation for$\hat{f}_{(n)}^{(i_{\dot{J}},k)}$is adopted:$\zeta^{(j)}\cos\theta_{\zeta}^{(k)}\frac{\Delta_{1}^{(i,j,k)}\hat{f_{(n)}}\zeta^{(j)}\sin\theta_{(}^{(k)}\Delta_{2}^{(\cdot,j,k)}\hat{f_{(n)}}}{(dr\wedge/di)r^{\langle i)}\wedge(d\theta_{\zeta}/dk)}$$= \frac{2}{\sqrt{\pi}Kn_{w}}\rho_{(n-1)}(\hat{f_{e(n-1)}}-\acute{f}_{(n)}^{\langle i,j,k)})\triangleleft\cdot)(i,j,k)$

where

and
, respectively.

$\triangle_{1}^{(ij,k)}\hat{f_{(n)}}/(dr\wedge/di)$$\partial\hat{f_{(n)}}/\partial r\wedge$

and$\partial\hat{f_{(n)}}/\partial\theta_{\zeta}$$\triangle_{2}^{(\cdot,j,k)}\hat{f_{(n)}}/(d\theta_{(}/dk)$(21), given explicitly below, correspond to229
We use the following formulae for the operators$\Delta_{1}$and$\Delta_{2}$

:

(22d)

$\triangle_{2^{1}}^{(,j,k)}\hat{f_{(n)}}=-\frac{3}{2}\hat{f}_{(n)}^{(\cdot,j,k)}+2\hat{f}_{(n)}^{(i,j,k+1)}-\frac{1}{2}\hat{f}_{(n)}^{(,j,k+2)}|$$(dr\wedge/di)=(dr\wedge/ds)_{s=:}$,$(d\theta_{(}/dk)=(d\theta_{\zeta}/ds)_{s=k}$$\hat{f}_{(n)}^{(i,j,K+1)}=\hat{f}_{(n)}^{(i,j,K-1)}$

(22e)

,

(22f)

, and
, and
is defined by Eq. (llc) whereare substituted forrespectively. Equation (22f) corresponds to the symmetry relation with respect toaround. The variable
outside the domain is specified as a part of boundary conditions

and$\hat{T}$,$\hat{f_{e(n-1)}}^{(1j,k)}$$\hat{\rho}_{(n-1)}^{(1)},$$u\wedge\wedge$$\rho,$$\hat{T}_{(n-1)}^{(i)}$$u_{(n-1)}^{(\cdot)}\wedge$$\theta_{(}$$f_{(n)}^{XI+1,j,k)}$$\theta_{\zeta}=\pi$[Eq. (27b) below]. The$\hat{f}_{(n)}^{(i,j,K+2)}$

does not appear since$\sin\theta_{\zeta}^{(K)}=0$.

As explained in Sec. III
is discontinuous along $r\wedge\sin\theta_{\zeta}=1(0<\theta_{\zeta}\leq\pi/2)$ . The$(22a)-(22d)$difference formulae
should be modified when they consist of the data on both sidesof the discontinuity. That is,$A,\hat{f}$$\triangle_{1}^{(l_{J},k)}\hat{f_{(n)}}=\{\begin{array}{l}A_{1}\hat{f}_{(n)}^{\langle i,j,k)}-A_{2}\hat{f}_{(n)}^{(|-1,j,k)}+A_{3}\hat{f}_{(n)}^{(i(k),j,k)+}(\hat{r}^{\langle i-1)}>1/sin\theta_{\zeta}^{(k)}>r^{i-2)}\triangleleft)A_{4}(f_{(n)}^{j,k)}X\cdot,-\hat{f}_{(n)}^{(|(k),j,k)+}),(\acute{r}^{\{i)}>1/sin\theta_{\zeta}^{(k)}>r^{\langle\iota-1)}\wedge)\end{array}$$(23b)(23a)$$\triangle_{2}^{(i,j,k)}\hat{f_{(n)}}=\{\begin{array}{l}B_{1}\acute{f}_{(n)}^{(\cdot,j,k)}-B_{2}\hat{f}_{(n)}^{(t,j,k+1)}+B_{3}\hat{f}_{(n)}^{(i,j,k(i))-}[\theta_{\zeta}^{(k+1)}<Arcsin(1/\acute{r}^{\langle i)})<\theta_{\zeta}^{(k+2)}]B_{4}(\acute{f}_{(n)}^{\langle i,j,k)}-\hat{f}_{(n)}^{\langle i,j,k(i))-}),[\theta_{(}^{(k)}<Arcsin(1/r^{i)}\triangleleft)<\theta_{(}^{(k+1)}]\end{array}$$(23d)(23c)$

where
,

and the constants$A_{1},$$\acute{f}_{(n)}^{\langle i(k)_{J},k)\pm}=\hat{f_{(n)}}(1/\sin\theta_{\zeta}^{(k)}\pm 0, \zeta^{(j)}, \theta_{\zeta}^{(k)})$(23e)$\hat{f}_{(n)}^{(l,j,k(i))\pm}=\hat{f_{(n)}}(r-()(, Arc\sin(1/r)\pm 0),$(23f)

$A_{2}$, and$A_{3}$(

curacy for$\partial\hat{f_{(n)}}/\partial_{\Gamma}^{\wedge}(\partial\hat{f_{(n)}}/\partial\theta_{\zeta})$

at

, and
) are chosen in such a way thatis the difference expression of the second order acand
give corresponding first order$B_{1},$$\triangle_{1}^{(i,j,k)}\hat{f_{(n)}}/(dr\wedge/di)[\Delta_{2}^{(i,j,k)}\hat{f_{(n)}}/(d\theta_{\zeta}/dk)]$$(^{\triangleleft_{r}i)},$$B_{2}$$(^{(j)}, \theta_{\zeta}^{(k)});A_{4}$$B_{3}$$B_{4}$

230
depend only on and . It is noted that $(i(k), j, k)$ andand$(i,j, k(i))$ are not regular lattice points. The $i(k)$ and $k(i)$ represent the intersection of therespectively. That is,andwith the lattice hnes
characteristic

formulae.

$A_{1},$$\ldots,$$B_{1},$$A_{4}$$k$$i$$B_{4}$$\ldots,$$r\wedge=\acute{r}^{\langle i)}$$\theta_{(}=\theta_{\zeta}^{(k)}$$r\wedge\sin\theta_{(}=1$$\triangleleft_{r}i(k))=1/\sin\theta_{\zeta}^{(k)}$

,

(24)

$\theta_{\zeta}^{(k(:))}=Arc\sin(1/\acute{r}^{\prec i)})$.

For a discontinuous function on the characteristic, the two hmiting values as in Eqs. (23e) and, i.e.,
on eitheror(23f) should be considered. The formulae contain$\hat{f}_{(n)}^{(i,j,k(t))-}$$\hat{f_{(n)}}^{(i(k),j,k)+}$corresponding toside of the discontinuity. The sequence
the discontinuity, is constructed by the following difference scheme:$\hat{f}_{(n)}^{(i,j,k(i))\pm}$$\hat{f_{(n)}}$at$\hat{f_{(n)}}$on each side of$r^{\langle i)}\wedge$$\zeta^{(j)}\cos\theta_{(}^{(k(i))}\Delta_{1}^{(\cdot,j)\pm}\hat{f_{(n)}}=\frac{2}{\sqrt{\pi}Kn_{w}}\rho_{(n-1)}^{(i)}\wedge(\hat{f_{e(n-1)}}^{(i,j,k(i))}-\acute{f}_{(n)}^{\langle i,j,k(i))\pm})$

where$\triangle_{1}^{(i,j)\pm}\hat{f_{(n)}}$corresponds to$\Delta_{1^{t}}^{(,j,k)}\hat{f_{(n)}}(0\leq k\leq\overline{K}-1)$

,

(25)

in Eqs. (22a) and (22b) with$\hat{f}_{(n)}^{(l,j,k)}$. The $\pm sign$ is absent insince it is continuous.
replaced by
Corresponding to the boundary condition (13), we impose the condition:$\hat{f_{e(n-1)}}(i,j,k(|))$$\hat{f}_{(n)}^{\langle i,j,k(t))\pm}$$\hat{f}_{(n)}^{(0_{J)}k)}=\frac{2}{\sqrt{\pi}}\exp(-(\zeta^{(j)})^{2})$which is independent of . As the condition at$k$$\hat{f}_{(n)}^{(I,j,k)}=0$,,$(k=0,1, \ldots, \overline{K}-1)$$r\wedge=r_{D}\wedge$, we assume$(k=\overline{K}, \ldots, K)$$\hat{f}_{(n)}^{(I+1_{d},k)}=0$

,

(26)

,

,

$(k=\overline{K}, \ldots, K)$

(27a)

.

(27b)

Two conditions are required since the second order difference scheme, Eq. (22c), is used in
Eq. (21).
With these preparations of difference formulae, we construct the sequence$\hat{f}_{(n)}^{(i,j,k)}(n=$$0,1,2,$(i) For$\ldots$) by the following process. Let$\overline{K}\leq k\leq K$, starting fromand (27b) in descending order ofLet$\hat{f}_{(n)}^{(i’,j,k)}(i’>i)$$\hat{f}_{(n-1)}^{(i,j,k)}$$\hat{f}_{(n)}^{(I-1,j,k)}$$i$, thus, compute

down tobe given. Starting from$\hat{f}_{(n)}^{(0,j,k)}$$\hat{f}_{(n)}^{(\cdot,j,K)}$$\rho^{i)}A_{(n-1)}\hat{u}_{(n-1)}^{(l)}$$\hat{f}_{(n)}^{(i,j,k)}$, and$\hat{T}_{(n-1)}^{(l)}$, be known.47

using Eqs. (21), (22c), (22d), (27a),. The step $i=i+1$ to $i=i$ is as follows.

, compute$\hat{f}_{(n^{t})}^{(,j,k)}$using Eqs. (21), (22c), and(22d) [and Eqs. (27a) and (27b) for $i=I-1,$ $I-2$ ] in descending order of down to.
Carry out this step for every .
(ii) Computealong the discontinuity using Eq. (25):is constructed from$k$$\hat{f}_{(n)}^{\langle\cdot,j,\hat{K})}$$j$$\hat{f}_{(n)}^{(i,j,k(\cdot))\pm}$$\hat{f}_{(n)}^{(i,j,k(i))+}$

the initial dataobtained in the preceding stepfrom the initial data
determined by Eq. (13). The computation is continued until the discontinuity becomes negligibly, obtainsmall. Then, from a set of
by interpolation. [The data on the
discontinuity in Eqs. $(23a)-(23d)$ are now prepared.](\"ui) For $0\leq k\leq K-1$ , the computation is carried out in ascending order of , independently
in the two regions
andseparated by the discontinuity. The step$\hat{f}_{(n)}^{(0j,\hat{K})}$$(i);^{48}\hat{f}_{(n)}^{(l,jk(i))-})$$\hat{f}_{(n)}^{(i_{\theta},k(i))+}$$\hat{f}_{(n)}^{(i(k),j,k)+}$$i$$\wedge r>1/\sin\theta_{(}$$r\wedge<1/\sin\theta_{\zeta}$

231$i=i-1$ to $i=i$ is as follows. Let$\hat{f}_{(n)}^{(i,j,\hat{K}-1)}$, compute$\hat{f}_{(n)}^{(c,j,k)}$$\hat{f}_{(n’)}^{(\cdot,j,k)}(i‘ <i)$be given. (a) For$\wedge r>1/\sin\theta_{\zeta}$, starting fromusing Eqs. (21), (22a), (22d), (23a), (23b), and the data obtained insteps (i) and (ii) in descending order of$k$down to$\hat{f}_{(n)}^{(l,j,\overline{k})}$where$\theta_{\zeta}^{(\overline{k}-1)}<Arc\sin(1/r^{t}\triangleleft))<\theta_{(}^{(\overline{k})}$, starting from, compute(Ref. 49). (b) For
using Eqs. (21), (22a),(22b), (22d), (23c), (23d), (26), and the data obtained in step (ii) in descending order of down
. Carry out these steps for every . For $i>i_{c}$ , where the discontinuity is negligibly
to$r\wedge<1/\sin\theta_{(}$$f_{(n)}^{\triangleleft\cdot j,\overline{k}-1)}$$\hat{f}_{(n)}^{\langle\cdot,j,k)}$$k$$\cdot$$j$$\hat{f}_{(n)}^{(\cdot,j,0)}$small, neglecting the discontinuity, compute.
from $k=\overline{K}-1$ toformulae (22a), (22b), and (22d) as in step (i).
(iv) Applying the Simpson formula to Eqs. $(12a)-(12c)$ , compute)$\hat{f}_{(n)}^{(\cdot,jk)}$$\rho_{(n)}^{(l)}\wedge,$(v) Repeat the process of $(i)-(iv)$ with shift of the subscript ($0$

only with the standard$u_{(n)}^{(i)}\wedge$, andto $n+1$ ) until$n$$\hat{T}_{(n)}^{(i)}$from$\hat{f}_{(n’)}^{(i_{\dot{J}},k)}$$\hat{f}_{(n)}^{\langle i,j,k)}$.

converges.

We take the hmit as the solutibn.
The order of computation in the preceding process is consistent with the natural course of
integration of Eq. (10) along its characteristics in the direction of molecular velocity.
$\hat{f}^{(;,j,k)}$IV.

SOLUTIONS IN THE FREE MOLECULAR AND CONTINUUM LIMITS

Before presenting the result of the numerical analysis, we give the solutions of two extreme
cases: the free molecular and the continuum limits. The general solution of a free molecularflow around a convex body is well-known.44,45,50 Thus, the solution of the present problem for$Kn_{w}=\infty$ is easily obtained as$\hat{f}(r\wedge, \zeta, \theta_{(})=\{\begin{array}{l}\frac{2}{\sqrt{\pi}}exp(-\zeta^{2}),[0\leq\theta_{(}<Arcsin(1/r\wedge)]0,[Arcsin(1/\wedge r)<\theta_{\zeta}\leq\pi]\end{array}$$(28b)(28a)$

At a given point in the gas, only the molecules whose velocities are inside the circular cone$\theta_{(}<Arc\sin(1/\wedge r)$ are present. The cone becomes more slender as the distance from the sphere
increases. The height of is invariant with respect to the distance. From Eqs. $(3a)-(3c),$ $(4a)$ ,
$T,$
, and
are given byand (4b), the macroscopic variables$\hat{f}$$\rho,$$u,$$T_{||}$$\rho/\rho_{w}=\frac{1}{2}(1-\sqrt{1-(L/r)^{2}})$$\tau_{\perp}$,$u/(2RT_{w})^{1/2}= \frac{1}{\sqrt{\pi}}(1+\sqrt{1-(L/r)^{2}})$

(29a)
,

$T/T_{w}=1- \frac{2}{3\pi}(1+\sqrt{1-(L/r)^{2}})^{2}$

(29b)

,

(29c)

$T_{||}/T_{w}=(1- \frac{2}{\pi})(1+\sqrt{1-(L/r)^{2}})^{2}-\sqrt{1-(L/r)^{2}}$$T \perp/T_{w}=\frac{1}{2}(1+(L/r)^{2}-\sqrt{1-(L/r)^{2}})$The mass and energy fluxes

$Q$

.

,

(29d)

(29e)

and $W$ are, from Eqs. (15) and (17),$Q/4 \pi\rho_{w}(2RT_{w})^{1/2}L^{2}=\frac{1}{2\sqrt{\pi}}$

,

(30a)

232

$W/4 \pi p_{w}(2RT_{w})^{1/2}L^{2}=\frac{1}{\sqrt{\pi}}$

.

(30b)

The general behavior of steady flows of a shghtly rarefied gas past its condensed phase, where(strong) evaporation or condensation is taking place, is studied in Ref. 26. In the limit of theKnudsen number being zero, the flow field is described by the solution of the Euler system of
equations for an ideal gas under appropriate boundary conditions, with local corrections such
as Knudsen $1ayer^{22,25,32}$ or shock $1ayer^{51,52}$ . According to Ref. 26 the boundary condition to besatisfied on the condensed phase at rest where evaporation is taking place is given as follows:$\frac{p}{p_{w}}=h_{1}(M)$,$\frac{T}{T_{w}}=h_{2}(M)$,,$(M\leq 1)$(31)

where $M$ is Mach number $(3u^{2}/5RT)^{1/2}$ and the numerical data of the functions of $M,$ $h_{1}(M)$and $h_{2}(M)$ , for the BKW equation are shown in Table I. In addition, the tangential velocity
should be zero. Thus, in the present problem with the spherical condensed phase, the conditions, and
Eq. (31) among $T,$ $M,$
are apphed at $r=L$ .
The solution of the Euler system for the evaporating flow from a sphere into a vacuum isgiven by the following parametric expression in $M$ :$p,$$T_{w}$$p_{w}$$\rho/\rho_{*}=8/(3+M^{2})^{3/2}$

,

(32a)

$u/(5RT_{*}/3)^{1/2}=2M/(3+M^{2})^{1/2}$ ,(32b)$T/T_{*}=4/(3+M^{2})$ ,(32c)$r/L=(3+M^{2})/4M^{1/2}$ ,

(32d)

where
$\rho_{*}=h_{1}(1)\rho_{w}/h_{2}(1)$$T_{*}=h_{2}(1)T_{w}$

,

(32e)

.

(32f)

Incidentally,$T_{||}=T_{\perp}=T$

,

(32g)

since the Euler system corresponds to the local Maxwellian. For the BKW equation under theconventional boundary condition,$h_{1}(1)/h_{2}(1)=0.3225$ ,The mass and energy fluxes$Q$$h_{2}(1)=0.6434$

.

(33)

and $W$ from the sphere are given by

,

(34a)

.

(34b)

$Q/4\pi\rho_{w}(2RT_{w})^{1/2}L^{2}=(5/6)^{1/2}h_{1}(1)/h_{2}(1)^{1/2}$$W/4\pi p_{w}(2RT_{w})^{1/2}L^{2}=4(5/6)^{3/2}h_{1}(1)h_{2}(1)^{1/2}$

Thus, for the BKW equation,$Q/4\pi\rho_{w}(2RT_{w})^{1/2}L^{2}=0.2361$

,

(35a)

$W/4\pi p_{w}(2RT_{w})^{1/2}L^{2}=0.5065$

.

(35b)

233
For large $r/L$ , Eqs. $(32a)-(32d)$ are reduced to
,

(36a)

$u/(5RT_{*}/3)^{1/2}=2-3(4r/L)^{-4/3}$ ,(36b)$\rho/\rho_{*}=\frac{1}{2}(r/L)^{-2}$$T/T_{*}=4^{-1/3}(r/L)^{-4/3}$$M=(4r/L)^{2/3}$

,

(36c)

.

(36d)

There are two unclear points in this isentropic solution. The first point is that $d\rho/dr,$ $du/dr$ ,etc. are infinite at $r=L$ . This violates the assumption imposed in the derivation of the Euler
system from the Boltzmann or BKW equation. The second point is the behavior of the solution
for large $r/L$ . The mean free path at the equihibrium at rest with and is$l$$l=(8RT/\pi)^{1/2}/A_{cd}\rho$

and the length scale$L_{r}$

of variation$T$$\rho$$\rho,$

$u$

,

(37)

, etc. is $O[\rho(d\rho/dr)^{-1}]$ . That is,$L_{r}=O(r)$

.

(38)

The local Knudsen number $Kn_{r}(=l/L_{r})$ for large $r/L$ increases with
$Kn_{r}=O[Kn_{w}(r/L)^{1/3}]$

$r$

as

.

(39)

, which characterizes the variation of the flow,53 is not uniformly
The local Knudsen number
small for small. It ranges from $O(Kn_{w})$ to .
When $Kn_{w}$ is very $smaU$ , there is a region whereis small but $M$ is large. Then, theisentropic solution is valid up to this region, with the reservation of the first unclear point
raised in the preceding paragraph. The behavior downstream of this region can be studied by
the hypersonic approximation, where a simplification is made under the assumption that thewidth of the velocity distribution function is much smaller than the flow speed (Refs. 35-38).
According to Refs. 35 and 36, the solution of this approximation of a spherically expanding flow
is expressed by the confluent hypergeometric functions as follows:$Kn_{r}$$Kn_{w}$$\infty$$Kn_{r}$$\rho=c_{0}r^{-2}\wedge\wedge$$u=c_{1}\wedge$

,

(40a)
(40b)

,

$\hat{T}=\frac{1}{\wedge,r^{2}}[c_{2}U(2/3,3, \alpha/Kn_{w}r\wedge)+c_{3}M(2/3,3, \alpha/Kn_{w}r\wedge)]$$\alpha=2c_{0}/\sqrt{\pi}c_{1}$

,

(40c)

,

, and are undetermined constants, and $U(a, b, x)$ and $M(a, b, x)$ are Kummer’s
wherefunctions (of the same notation as in Ref. 54; don’t confuse with Mach number $M$ ). The innerexpansion forof the solution $(40a)-(40c)$ is$c_{0},$$c_{1},$$($$c_{2}$$c_{3}$$Kn_{w^{7^{\tau}}}^{\wedge}\ll 1)^{55,56}$$\rho=c_{0}r^{-2}\wedge\wedge$$u=c_{1}\wedge$

,

(41a)

(41b)

,

$\hat{T}=c_{2}(\frac{Kn_{w}}{\alpha})^{2/3}\frac{1}{\wedge,r^{4/3}}+c_{3}\frac{2}{\Gamma(2/3)}(\frac{Kn_{w}}{\alpha})^{7/3}r^{1/3}\exp\wedge(\frac{\alpha}{Kn_{w}r\wedge})$

,

(41c)

234
is the gamma function54. Matching Eqs. $(4la)-(4lc)$ with the outer expansion,where$(36a)-(36c)$, of the isentropic solution, we have57
Eqs.
$\Gamma(x)$$c_{0}=h_{1}(1)/2h_{2}(1)$,$c_{1}=[10h_{2}(1)/3]^{1/2}$$c_{2}=[3h_{1}(1)^{2}/40\pi]^{1/3}Kn_{w}^{-2/3}$

From Eqs. (40c) and (42), we have, for$rarrow\infty$$T/T_{w}arrow A_{0}Kn_{w}^{4/3}$,$c_{3}=0$

,

(42)

.

,

,
(43)

,

$A_{0}=( \frac{5\pi}{3})^{2/3}\frac{h_{2}(1)^{3}}{\Gamma(2/3)h_{1}(1)^{4/3}}=4.828$

which is the frozen temperature for small
can be shown to decay as$Kn_{w}$. In this approximation the normal temperature$\tau_{\perp}$$\tau_{\perp}/\tau_{w}arrow A_{1}Kn_{w}^{1/3}L/r$,

as$A_{1}=( \frac{5\pi}{24})^{1/6}\frac{h_{2}(1)^{3/2}}{\Gamma(2/3)h_{1}(1)^{1/3}}=0.5999$

V.

RESULT AND DISCUSSIONA.

Macroscopic variables$Kn_{w}(r/L)arrow\infty$

.

(44)

,

,
The distributions of the density, flow velocity, temperature, and Mach number, i.e.,in, and $M[=u/(5RT/3)^{1/2}]$ , are shown for various Knudsen numbers
Figs. 1, 2, and Tables II-IV. Figure 1 shows the behavior near the sphere, and Fig. 2 shows
the long range behavior. The data for $Kn_{w}=0$ are the solution of the Euler system given by
Eqs. $(32a)-(32d)$ . Their Knudsen-layer corrections, which flatten on $r/L=1$ , are also shownin the figures. In Fig. 2(c), the temperature profiles of the hypersonic approximation for small$Kn_{w}$ are also shown by dashed lines. The reason of their deviation from the numerical solutions
is given in connection with the behavior of the velocity distribution function. The data for$Kn_{w}=\infty$ are the free molecular solution given by Eqs. $(29a)-(29c)$ . The behavior of the gas
varies very sharply near the sphere (Fig. 1) but approaches a vacuum state at infinity slowly(Fig. 2). The density vanishes as fast as, where $\alpha=2.00$ for all Knudsen numbers. The flow
velocity and temperature approach nonzero values (except for
at $Kn_{w}=0$ ), which depend$u_{\infty}/(2RT_{w})^{1/2}$
on the Knudsen number. The hmiting values
andof $u/(2RT_{w})^{1/2}$ and$T/T_{w}$ as $rarrow\infty$ , the latter of which is called frozen temperature, are shown in Fig. 3 andTable V. These data were determined by close examination of the behavior of variation of$\ln du/dr$ , and $\ln dT/dr$ versus, andfor large . The data
are found to behnear, with high accuracy, for large , from which the asymptotic forms of
, and
are
determined. The $u_{\infty}/(2RT_{w})^{1/2}$ andin Fig. 3 and Table V are the extrapolated values$hom$ the asymptotic forms. In Fig. 3, the frozen temperature (43) for small
obtained by
the hypersonic approximation is also shown.
$\rho/\rho_{w}$$u/(2RT_{w})^{1/2},$ $T/T_{w}$$Kn_{w}$$r^{-a}$$T$$\tau_{\infty}/T_{w}$$\rho,$

$u$

$T$

$r$

$\ln\rho,$$r$$\ln r$$\rho,$$u$$\tau_{\infty}/T_{w}$$Kn_{w}$

$T$

235
and
[cf. Eqs. (4a) and $(4b)$ ] andThe profiles of parallel and normal temperatures, which are often referred to in experimental studies of nonequihibrium flows, aretheir ratio
for moderate and large
first decreases sharply
shown in Fig. 4. The parallel temperatureand increases slowly to the value at infinity as increases. This behavior is easily understoodfor the free molecular flow $(Kn_{w}=\infty)$ by kinematic consideration. For the free molecular flow
[Eqs. (28a), $(28b)$ ] the velocities of the molecules on the sphere range $0\leq\xi_{r}<\infty(0\leq\theta_{\zeta}<\pi/2)$
and their speeds are practically hmited to a region of the order of $(2RT_{w})^{1/2}$ . With the distance, where $\xi_{r}/(2RT_{w})^{1/2}$ is small forfrom the sphere the molecules for Arcsin(L/r),
deviation$(r/L-1)$
disappears.
theof with respect to , which
small
Therefore
standard, decreases for $smaU(r/L-1)$ . With the decrease of the molecules of thecorresponds toincreases in this region.
central part of for moderate and large $(r/L-1)$ , the temperature, with, on the other hand, the shrink of with respect to , or the decrease of
For smallowing to frequent molecular collisions.
the distance from the sphere induces the decrease of,
From Figs. 4(b) and 4(c), as$T_{1}arrow 0$ ,(45)
$T_{||}$$\tau_{\perp}$$\tau_{\perp}/T_{||}$$Kn_{w}$$T_{||}$$r$$\leq\theta_{\zeta}\leq\pi/2$$f$$\xi_{r}$$T_{||}$$T_{||}$$\xi_{r}$$Kn_{w}$$f$$\tau_{\perp}$$\xi_{t}$$T_{||}$$rarrow\infty$( $Kn_{w}\neq 0$ or ) in Figs. 4(b) and 4(c), corresponding to theand the slopes of the curves
speed of decay, approach $-1$ as $rarrow\infty$ (Ref. 58), but the speed of convergence is much slower
approachthan that of the case of in Fig. 2. The corresponding slopes for $Kn_{w}=0$ and$-4/3$ and $-2$ respectively [Eqs. (29e), (32g), and $(36c)$ ]. From Eqs. (5) and (45),$\tau_{\perp}$$\infty$$\infty$$\rho$$T_{||}arrow 3T$,(as $rarrow\infty$ ).

(46)

is a measure of anisotropy of the velocity distribution function around the flow
The ratio
velocity (when it is isotropic). Except for very small Knudsen numbers, the velocity
distribution function shows strong anisotropy over the whole flow field (Fig. 4(d); see Sec. III
for more details).
$\tau_{\perp}/T_{||}$$\tau_{\perp}/T_{||}=1$

$C$

B.

Mass and energy fluxes

The nondimensional mass and energy fluxes $Q/4\pi\rho_{w}(2RT_{w})^{1/2}L^{2}$ and $W/4\pi p_{w}(2RT_{w})^{1/2}L^{2}$are shown in Fig. 5 and Table V. They( andfor short) versus the Knudsen number$Kn_{w}=0$[Eqs. (35a) and $(35b)$ ] to those at $Kn_{w}=\infty$
increase monotonically from the values atand $W$ of the molecules leaving the sphere are[Eqs. (30a) $and,(\sim 30b)$ ]. The contributions toequal to and $W$ at $Kn_{w}=\infty$ , respectively. Therefore, the differences from the free molecularvalues, which are negative, are the contributions of the molecules arriving on the sphere. For
example, at $Kn_{w}=0$ , where $\hat{Q}/\hat{Q}(Kn_{w}=\infty)=0.8370$ and $\overline{W}/\overline{W}(Kn_{w}=\infty)=0.8977$ , the16.30% of the molecules leaving the sphere return to the sphere, but only 10.23% of the energy
leaving the sphere return. This is because the returning molecules, on the average, have lessenergy than the leaving molecules.
The rate (16.30%) of the returning molecules at $Kn_{w}=0$ is equal to that in the evaporating
flow from a plane condensed phase that reaches sonic speed at infinity. This is obvious from thefollowing structure of the isentropic solution [Eqs. $(32a)-(32d)$ ] of the evaporating flow from thesphere at $Kn_{w}=0$ . The boundary condition (31) for the Euler system corresponds to connectionof an isentropic flow to an evaporating flow from a plane condensed phase.26 The latter cannot
be supersonic, but an isentropic expanding flow into a vacuum cannot be subsonic. Therefore,
the connection of the two flows, which makes up the evaporating flow [Eqs. $(32a)-(32d)$ ] fromthe sphere, is made at sonic speed.
$\hat{Q}$$\overline{W}$$Kn_{w}$$\hat{Q}$$\hat{Q}$236
C.

Velocity distribution function

The velocity distribution function $\hat{f}[=2\pi\rho_{w}^{-1}(2RT_{w})^{3/2}f]$ at several points in the gas, obtained by the numerical method in Sec. IV, are shown for $Kn_{w}=0.01,0.1,1$ , and 10 in Figs. 611. The free molecular flow solution, Eqs. (28a) and (28b), is given in Fig. 12 for reference.
At $Kn_{w}=\infty$ , the molecular velocity is locahzed. That is, at a point $(r)$ , the molecules with
Arcsin(L/r) come directly from the sphere and there are no molecules withvelocity
the other velocity. The height of the velocity distribution function remains unchanged along theflow. At $Kn_{w}=10$ (a large Knudsen number, Fig. 11), these features are well preserved andspace is seen. The effect of coUision is more eminent for smaller
only local correction inmolecular speed because of smaller free path. The height of the distribution decreases only very
slowly.
At $Kn_{w}=0.01$ (Figs. 6 and 7), the behavior is quite different. The discontinuity of thevelocity distribution function on the sphere decays in a very short distance (much shorter than) [Fig. $6(a)arrow Fig$ . $6(c)$ ]. The distribution is transformed into a distrithe mean free pathbution fairly close to the Maxwelhan distribution with the corresponding density, velocity, andtemperature (the local MaxweUian ) in a distance of the several mean free paths [Fig. 6(a)$arrow Fig$ . $6(d)arrow Fig$ . $7(a)$ ; see Figs. 13(a) and 14(a) for comparison with the local Maxwellian].
This is a kinetic transition process to a continuum region [Fig. $7(a)$]. The transition region is
the Knudsen layer and the region with discontinuity at the bottom of the Knudsen layer is thelayer, discussed in Refs. 34, 40, and 59. Along the flow, the density of the gas decreases andthe collision effect becomes less important. Therefore the velocity distribution function begins
to deviate from the Maxwellian, i.e., its width normal to the flow shrinks [Fig. $7(a)arrow Fig$ . $7(b)$$arrow Fig$ . $7(c)$ ]. In this stage, the collision is still appreciable and the height of the distribution
decreases considerably. Further away from the sphere, the density of the gas is so small thatmolecular collisions are rare. Thus, the flow is nearly free molecular and the height of the distribution decreases only very slowly [Fig. $7(c)arrow Fig$ . $7(d)$ ]. If the Mach number in the continuum
region is sufficiently large, the downstream flow can be well expressed by the hypersonic approximation. In the present case $(Kn_{w}=0.01),$ $M\sim 1.8$ in this region. This is not sufficiently large
for the solution of the hypersonic approximation to be very good approximation at $Kn_{w}=0.01$
or larger values as is seen in Fig. 2(c).
At $Kn_{w}=0.1$ , the discontinuity decays in the several mean free paths from the sphere[Fig. 8(c);[Fig. $8(a)arrow Fig$ . $8(c)$ ]. The distribution function there is of a shape with
see Figs. 13(b) and 14(b) for comparison with the local MaxweUian], and this ratio decreases
with the flow [Fig. $8(c)arrow Fig$ . $9(a)$ ]. That is, it is in a transition region that corresponds tothe transition region in the downstream of the continuum region in the case $Kn_{w}=0.01$ . The
behavior of further downstream is similar to the case $Kn_{w}=0.01$ , i.e., a transition region isfollowed by a free molecular region [Fig. $9(a)arrow Fig$ . $9(b)arrow Fig$ . $9(c)$ ]. There is no continuum
region in the flow. At $Kn_{w}=1$ , a transition region with large discontinuity [Fig. $10(a)arrow$
Fig. $10(c)$ ] is followed by a free molecular region with large discontinuity [Fig. $10(c)arrow Fig$ . $10(d)$ ].
$0\leq\theta_{\zeta}\leq$$(\xi_{r}, \xi_{t})$$1_{w}$$f_{e}$$S$$\tau_{\perp}/T_{||}<1$

In Fig. 15,. If $f=f_{e},$ $\delta=0$ ,versus $r/L$ is shown for various
and if andare disjoint, $\delta=2$ . The is a measure of deviation from the corresponding
equilibrium distribution . Figures 13(a) and 13(b), where comparison with the Maxwellian ismade, are, respectively, the distributions marked withand $0$ in Fig. 15. They are the points
with nearly the smallest in the flows at $Kn_{w}=0.01$ and 0.1. From Fig. 15, the flow is seen tobe highly nonequilibrium everywhere except for very small Knudsen numbers.
The transition to a free molecular flow region can clearly be seen from the profiles of atsome (versus
for various , which are shown in Fig. 16 for $Kn_{w}=0.01,0.2,1$ , and 10.
$Kn_{w}$$\delta=\rho^{-1}\iint|f-f_{e}|\xi_{t}d\xi_{r}d\xi_{t}$$f$$\delta$$f_{e}$$f_{e}$$\bullet$$\delta$$\hat{f}$$\wedge r\sin\theta_{(}$$\wedge r$237
Since the characteristics of Eq. (10) are given by
at a given is expressed bya single curve if the flow is free molecular. The flow approaches a free molecular condition forlarge $r/L$ even for $Kn_{w}=0.01$ . From Fig. 16 it is also seen that the discontinuity of the velocity
distribution function remains appreciable at $r/L=2000$ even for $Kn_{w}=0.2$ . By extrapolationis the collision frequency of a
it is found to be appreciable at downstream infinity. Since
molecule, the integral
may be considered as the number of collision of a molecule$\wedge r\sin\theta_{\zeta}=const,\hat{f}$$\zeta$$A_{cot}\rho$$\int_{r_{0}}^{\infty}A_{col}\rho\xi_{r}^{-1}dr$with radial velocity
to infinity. $6$ The integral takes a finite value
while it proceeds fromexperiences only a finite number of coUision
for large . A molecule
becausea molecule has rare chance of collision
before it reaches infinity. In other words, for large
while proceeding fromto infinity. This corresponds to our numerical results showing the freemolecular behavior (resulting in the persistence of discontinuity) in the far field.
According to the free molecular feature of (Fig. 16), the flow velocity and temperature
are nearly constant (or their leading terms are constants that depend on $Kn_{w}$ ) in the far field
but tois proportional to (nearly)
for $Kn_{w}\neq 0$ or(large ). On the other hand,$(29e)]^{61}$$Kn_{w}=\infty$
forthere Figs. 4(b), 4(c), Eq. (44), and Eq.
This difference due tomolecular collisions is a microstructure, and therefore more detailed consideration is required tounderstand this behavior. For large , as we have seen, the width of the distribution $f$ in theof the collision
direction normal to the flow (or in for short) is narrow. The gain termThus, the contribution of the gain term to theterm in Eq. (1) has the same width in as in-tail part of is of the same order as that to the central part. The effect of molecular collisions
relative to the local magnitude of in is much stronger on the tail. The contribution of thetail part to the normal temperature
is not negligible and plays an important role because
in Eq. (4b) significantly reduces the contribution of the central part of . Thus,
the factor
the effect of coUisions is amplified and the difference appears.
$\xi_{r}$$r_{0}$$\rho\sim r^{-2}$$(\xi_{r}\neq 0)$$r$$r_{0}$$r_{0}$$f$$r$$r^{-1}$$\tau_{\perp}$$r^{-2}$$\infty$$[$$r$$A_{co\mathfrak{l}}\rho f_{e}$$\xi_{t}$$\xi_{r}^{62}$$\xi_{t}$$f$$\xi_{t}$$f$$\xi_{t}$$\tau_{\perp}$$f$$\xi_{t}^{3}$D.

Lattice system and accuracy tests

lattice system [the lattice
Since the behavior of the gas depends considerably on
$K$
$I,$
, and
, and
and ] is chosen approfunctions
defined in Eq. (19);. Various tests have been carried out before the final computation.
priately depending on
Here, the data of the lattice system that was used to obtain the results in Sec. V $A,$ , andis given by
are summarized. (i) The lattice function$Kn_{w}\underline{t},he$$r\wedge(s),$$\zeta(s)$$J,$$\theta_{\zeta}(s)$$K;r_{D}\wedge$$\zeta_{D}$$Kn_{w}$$B$$C$$\wedge r(s)$$\frac{dr\wedge}{ds}=f_{1}(r\wedge)f_{2}(r\wedge)$,$\wedge r(0)=1$,$fi=d_{2}+(d_{1}-d_{2})\exp(-d_{3}(r\wedge-1))$

(47)

,

$f_{2}=1+(d_{4}-1)\exp(-d_{5}(r\wedge-1))$ ,where
are constants chosen properly in each case. The fundamental lattice moderately widening from the sphere is expressed by ; a finer lattice in the neighborhood of thesphere is expressed by . Examples of the lattice data are given in Table VI. The sizeof thedomain is chosen as:
for $Kn_{w}=10,$
for $0.5\leq Kn_{w}<10$ , andlattice,( : a constant), ($D=8$ , and $J=192$ for
for $Kn_{w}<0.5$ . (ii) For$Kn_{w}<0.1$ and $J=96$ for $Kn_{w}\geq 0.1$ . (iii) A fine lattice ofis requiredsince islocahzed there for large . The number of the lattice points are chosen as: $(K, K)=(2560, 2464)$
for $Kn_{w}<0.05$ and $(5760, 5568)$ for $Kn_{w}\geq 0.05$ . Typical data ofare shown in Table VII.
$d_{1},$$d_{2},$$d_{5}$$\ldots,$$f_{1}$$f_{2}$$\wedge r$$\wedge r_{D}$$r_{D}\wedge\sim 100$$\zeta^{(j)}$$r_{D}\wedge\sim 1000$$\zeta(s)=\alpha s^{3}$$r_{D}\wedge\sim 2000$$\alpha$$\hat{f}$$\theta_{(}$$nea\underline{r}\theta_{\zeta}=0$$r\wedge$$\theta_{\zeta}^{(k)}$238
Most of the lattice points are in $0<\theta_{\zeta}<\pi/2$ . (iv) The criterion of convergence of the sequence, andover all latticeis that the variations of the macroscopic variables
in an iteration are less than $10^{-12}$ .
points on
Here we summarize the accuracy test of our computation.
at ( $\geq 5.36(\zeta_{D}=8)$ is bounded by $4x10^{-12}$ . Further, the(i) The distribution function, and for ($D=8$ and $\zeta_{D}=10.2$ at $Kn_{w}=0.1$ aremacroscopic
thevariables
differences ofadopted in our main computationover. Thus, the size
bounded by
may be considered sufficiently large.
, and over the(ii) Typical data of the maximum variations of the macroscopic variables
for different choices ofare given in Table VIII. The
whole lattice points onvariation of is much smaller than the others in some cases. These data are also shown in theandare extrapolated from the data oftable. The $u_{\infty}/(2RT_{w})^{1/2}$ andare shown in Table IX.
for large . These data for different values, and over the whole lattice points on(i\"u) Typical data of the maximum variations ofare given in Table X. The maximum
for different numbers of the lattice points
thanthat of the others.
separately
of
is
shown
when
it
is
much
smaJlervariationover the whole lattice points on(iv) Typical data of the maximum variation ofare shown in Fig. 17.
of the lattice points offor different numbers and$W$,on(v) The mass and energy fluxes andare computed at all the lattice points
and the accuracies of the conservation relations (15) and (17) are examined. Their maximum
latticevariations $(Q_{\max}-Q_{\min})/Q_{\min}$ and $(W_{\max}-W_{\min})/W_{\min}$ are shown in Table XI. The
dependence of these variations is shown in Fig. 17.
(vi) For the conservation relation (15) to hold with a given accuracy, and should be computedwith the same relative accur acy for all. Then, the absolute error of for large should be very
. The test (v) in Table XI suggests that the computation
small since tends to zero asof is very accurate for large . Typical examples of the variation of at large in the tests
corresponding to (ii), (iii), and (iv) are shown in Table XII for further evidence of the accuracy.
The height of is almost invariant for large , and the domain where is appreciable shrinks as
according as shrink of the view angle of the sphere. Thanks to this behavior of , with, we can compute without losing its relative
sufficient lattice points
in this region
accuracy much. Thus accurate for large is obtained.
, the hmiting solution(vii) For some ofof the sequenceis examined to be$\rho^{i)}A_{(n)}$$\hat{f}_{(n)}^{(l,j,k)}$$u_{(n)}^{(i)}\wedge$$\hat{T}_{(n)}^{(i)}$$1\leq r\wedge\leq\wedge r_{D}$$\hat{f}$$\hat{T}$$\rho\wedge,\hat{u}$$2\cross 10^{-14}$$\zeta_{D}=8$$1\leq r\wedge\leq\wedge r_{D}$$\rho\wedge,$$1\leq\wedge r\leq r_{D}\wedge$$\hat{T}$$u\wedge$$r_{D}\wedge$$\rho\wedge$$|du\wedge/dr\wedge|$$T_{\infty}/T_{w}$$r\wedge$$|d\hat{T}/dr\wedge|$$r_{D}\wedge$$\hat{T}$$\rho\wedge,\hat{u}$$(^{(j)}$$J$$1\leq r\wedge\leq r_{D}\wedge$$p\wedge$$(\rho\wedge, \wedge u,\hat{T})$$I$$\wedge r_{D}$$(K,\overline{K})$$1\leq\wedge r\leq$$(r\wedge, \theta_{\zeta})$$r^{\langle)}\wedge$$Q$$1\leq r\wedge\leq r_{D}\wedge$$r\wedge\theta_{\zeta}$$u\wedge$$\rho\wedge$$r^{\langle i)}\wedge$$r\wedge$$\rho\wedge$$r\wedgearrow\infty$$\rho\wedge$$r\wedge$$\rho\wedge$$\rho\wedge$$\hat{f}$$r\wedge$$\hat{f}$$r\wedge$$\wedge rarrow\infty$$\hat{f}$$\theta_{\zeta}^{(k)}$$(\theta_{\zeta}\sim 0)$$\rho\wedge$$Kn_{w}$$\hat{f}$$\wedge r$$\hat{f}^{(i,j,k)}$$\hat{f}_{(n’)}^{(1j,k)}$independent of various choices of the initial functionof the iteration.
The computation was carried out by IIP 9000730 and MIPS RS 3230 computers at ourlaboratory.
$\hat{f}_{(0)}^{(i,j,k)}$VI.

EFFECT OF CONDENSATION FACTOR IN KINETIC BOUNDARY CONDITION

As in most works on a gas flow with evaporation and condensation, we considered the problem under the conventional boundary condition [the assumption (ii) in Sec. II]. A generalizationof the conventional boundary condition is reported in Ref. 12, where the effect of condensation factor is introduced and the conversion formula of the solution of the hnearized half-space
problem under the conventional condition to that under the generalized one is also given. The
conversion relation is generalized to the two surface, the non hnear half-space, and the cylindrical problems in Refs. 22, 26, 29, and 34. The generahzed boundary condition introduced in239
Ref. 12 is obtained by simply replacing the$\rho_{w}$in Eq. (7) by the following quantity:$\alpha_{c}\frac{p_{w}}{RT_{w}}-(1-\alpha_{c})(2\pi)^{3/2}(RT_{w})^{-1/2}\int\int_{\xi_{\Gamma}<0}\xi_{r}\xi_{t}fd\xi_{r}d\xi_{t}$

,

(48)

is a constant $(0\leq\alpha_{c}\leq 1)$ called the condensation factor of the boundary. The
corresponds to the conventional condition. The two problems differ only in theboundary condition (6) by a factor. We can, therefore, derive a simple conversion formula
between the two classes of solutions. Only the result is given here. Let the nondimensional
where

case

$\alpha_{c}$$\alpha_{c}=1$solution $(\hat{f}, u\wedge, \rho\wedge,p\wedge,\hat{T},\hat{Q}\{=Q/[4\pi\rho_{w}(2RT_{w})^{1/2}L^{2}]\},\overline{W}\{=W/[4\pi p_{w}(2RT_{w})^{1/2}L^{2}]\})$ under theconventional boundary condition for $Kn_{w}=K_{C}$ be denoted by the subscript , and let thesolution under the generahzed boundary condition for $Kn_{w}=K_{G}$ be denoted by the subscript. The following one-to-one correspondence holds between the two classes of solutions:
$C$

$G$

$(\hat{f}_{G}, K_{G})=\Gamma(\hat{f}_{C}, K_{C})$

,

(49)

where
$\Gamma=[1+2\sqrt{\pi}(1-\alpha_{c})\alpha_{c}^{-1}\hat{Q}_{C}]^{-1}$

.

(50)

Then,
$(p_{G}\wedge,p\wedge c,\hat{Q}_{G},\overline{W}_{G})=\Gamma(\rho\wedge c,\hat{p}_{C},\hat{Q}_{C},\overline{W}_{C})$

,

(51a)

.

(51b)

$(u_{G}\wedge,\hat{T}_{G}, M_{G})=(u_{C}\wedge,\hat{T}_{C}, M_{C})$It is noted that all thein Eqs. (9c), (9d), (llb), etc. [except in Eq. (7)] is the original$\rho_{w}=p_{w}/RT_{w}$defined by
but not the quantity (48).
$\rho_{w}$VII.

$\rho_{w}$CONCLUDING REMARKS

We have considered evaporating flows of a rarefied gas from its spherical condensed phase
into a vacuum and analyzed the problem on the basis of the BKW equation numerically with a
specially devised difference scheme capable of describing the discontinuity of the velocity distribution function. The velocity distribution function together with various important macroscopic
variables is obtained accurately over the whole flow field for the entire range of the Knudsen
number. The result is supplemented with an analytic solution for $smaU$ Knudsen numbers,
which is given by the solution of the Euler system of equations under the boundary condition(31) matched with the solution of hypersonic approximation for the far field. The distributionfunction shows strong anisotropy over the entire flow field even for small Knudsen numbers, andits discontinuity extends to downstream infinity for intermediate and large Knudsen numbers(and even for rather small Knudsen numbers). Even for $Kn_{w}=0.1$ , the velocity distributionfunction deviates considerably from the local Maxwelhan distribution over the whole flow field.
In most cases the velocity distribution function is far from an ellipsoidal distribution. The layer
at the bottom of the Knudsen layer are shown clearly. The velocity and temperature at downstream infinity take finite values, determined by the Knudsen number, except the temperature
at zero Knudsen number. A simple formula deriving the solution for an arbitrary condensation
factor in the kinetic boundary condition from that under the conventional boundary conditionis also presented. In order to confirm the accuracy of computation, various tests are carriedout. Especially, since the approach to the state at infinity is fairly slow, extensive tests are doneby changing the size of the domain of and examining not only quantities under consideration
themselves but also the rate of their variations.
$S$$\wedge r$

240
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17. A. A. Abramov and M. N. Kogan, “ Conditions for supersonic condensation of gas,” Sov.
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20. Y. Sone and H. Sugimoto, “ Strong evaporation from a plane condensed phase,” J. Vac. Soc.
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(

$p$

$r$

$p$

241
Soc. Jpn. 32, 214 (1989) (in Japanese).
22. Y. Sone and H. Sugimoto, “ Strong evaporation from a plane condensed phase,” in Adiabatic
Waves in Liquid-Vapor Systems, IUTAM Symposium, G\"ottingen, 1989, edited by G. E. A.
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23. Y. Sone, K. Aoki, H. Sugimoto, and T. Yamada, “ Steady evaporation and condensation ona plane condensed phase,“ Theor. Appl. Mech. (Bulgaria) 19, No. 3, 89 (1988).
24. Y. Sone, T. Ohwada, and K. Aoki, “ Evaporation and condensation on a plane condensed
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Phys. Fluids A 1, 1398 (1989).
25. K. Aoki, Y. Sone, and T. Yamada, “ Numerical analysis of gas flows condensing on its planecondensed phase on the basis of kinetic theory,” Phys. Fluids A 2, 1867 (1990).
26. K. Aoki and Y. Sone, “ Gas flows around the condensed phase with strong evaporationor condensation-Fluid dynamic equation and its boundary condition on the interface andtheir application-,” in Advances in Kinetic Theory and Continuum Mechanics, edited by R.
Gatignol and Soubbaramayer (Springer, Berlin, 1991), p. 43.
27. Y. Sone, “ Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen
numbers,” in Advances in Kinetic Theory and Continuum Mechanics, edited by R. Gatignol
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28. Y. Sone, “ Analytical and numerical studies of rarefied gas flows on the basis of the Boltzmann equation for hard-sphere molecules,” in Rarefied Gas Dynamics, edited by A. E. Beyhch(VCH, Weinheim, 1991), p. 489.
29. Y. Sone, T. Ohwada, and K. Aoki, “ Evaporation and condensation of a rarefied gas between its two parallel plane condensed phases with different temperatures and negative
temperature-gradient phenomenon -Numerical analysis of the Boltzmann equation-,” inMathematical Aspects of Fluid and Plasma Dynamics, Proceedings, edited by G. Toscani,
V. Boffi, and S. Rionero, Lecture Notes in Mathematics Vol. 1460 (Springer, Berhn, 1991),
p. 186.
30. M. N. Kogan and A. A. Abramov, “ Direct simulation solution of the strong evaporationand condensation problem,” in Ref.28, p. 1251.
31. A. P. Kryukov, “ Strong subsonic and supersonic condensation on a plane surface,” inRef.28, p. 1279.
32. K. Aoki, K. Nishino, Y. Sone, and H. Sugimoto, “ Numerical analysis of steady flows ofa gas condensing on or evaporating from its plane condensed phase on the basis of kinetic
theory: Effect of gas motion along the condensed phase,” Phys. Fluids A 3, 2260 (1991).
33. D. Sibold and H. M. Urbassek, “ Kinetic study of evaporation flows from cyhndrical jets,”
Phys. Fluids A 3, 870 (1991).
34. H. Sugimoto and Y. Sone, “ Numerical analysis of steady flows of a gas evaporating fromits cyhndrical condensed phase on the basis of kinetic theory,” Phys. Fluids A 4, 419 (1992).
35. B. B. Hamel and D. R. Wilhs, “ Kinetic theory of source flow expansion with apphcation
to free jet,” Phys. Fluids 9, 829 (1966).
36. R. H. Edwards and H. K. Cheng, “ Steady expansion of a gas into a vacuum,” AIAA J. 4,
558 (1966).
37. N. C. Freeman, “ Solution of the Boltzmann equation for expanding flows,” AIAA J. 5,
1696 (1967).
38. R. E. Grundy, “ Axially symmetric expansion of a monatomic gas from an orifice into a
vacuum,” Phys. Fluids 12, 2011 (1969).
39. Soubbaramayer, Lecture Notes on Uranium Enrichment (Tsinghua University, Peking, 1984)
Fascicle 2.

242
40. Y. Sone and S. Takata, “Discontinuity of the velocity distribution function in a rarefied gas
around a convex body and the layer at the bottom of the Knudsen layer,” Transp. Theory
Stat. Phys. (to be published).
41. P. L. Bhatnagar, E. P. Gross, and M. Krook, “ A model for coUision processes in gases, I,”
Phys. Rev. 94, 511 (1954).
42. P. Welander, “ On the temperature jump in a rarefied gas,” Ark. Fys. 7, 507 (1954).
43. In the BKW equation the gain term $G(f)$ in the collision term $(G(f)-A_{col}pf)$ of the. According to Kogan
Boltzmann equation for pseudo Maxwell molecules is replaced by(On the equations of motion of a rarefied gas,” J. Appl. Math. Mech. 22, 597 (1958)) thevelocity distribution function of the molecules just coUided, $G(f)$ , is fairly close toeven for a strongly nonisotropic . Accordingly in the coUision term $G(f)-A_{col}\rho f=[G(f)-$$A_{col}pf_{e}]+A_{col}\rho(f_{e}-f)$ , the second term on the righthand side, where
differs greatly fromin highly nonequihibrium flows, is much larger than the first. Thus, the essential feature of
nonequilibIium flow is well described by the BKW equation. Further, comparisons between
Knudsen layer results for Boltzmann and BKW equations are given in Refs. 24 and 28.
44. M. N. Kogan, Rarefied Gas Dynamics (Plenum, New York, 1969), p. 401, Appendix 1.
45. This is a special feature of the BKW equation and the Boltzmann equation for hard-spheremolecules. For other molecular models, the problem contains another parameter $U_{0}/mRT_{w}$ ,whereis the characteristic magnitude of the intermolecular potential and $m$ is the mass ofa molecule (Y. Sone, Lecture Notes on Molecular Gas Dynamics, Department of AeronauticalEngineering, Kyoto University, 1987).
46. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965),
p. 304.
, and
47. Practically, only the data
are used to obtainin the present
scheme for the BKW equation.
$S$$A_{col}pf_{e}$$A_{col}pf_{e}$$f$$f_{e}$$f$$U_{0}$$\hat{\rho}_{(n-1)}^{(i)},$$u_{(n-1)}^{(l)}\wedge$$\hat{T}_{(n-1)}^{(i)}$$f_{(n)}^{X^{i,j,k)}}$48. This means that
. The othershould be taken as the hmiting value ashmiting value is determined by Eq. (13).
49. In our lattice system, the characteristic
does not pass any lattice point.
50. C. Cercignani, The Boltzmann Equation and Its Application (Springer, Berhn, 1987) p. 262.
51. H. Grad, “ Singular and nonuniform limits of solutions of the Boltzmann equation,” inTransport Theory, edited by R. Bellman, G. Birkhoff, and I. Abu-Shumays (American Mathematical Society, Providence, RI, 1969), p. 269.
52. C. Cercignani and C. Lancellotti, “ Velocity shp for curved shock waves,” in Rarefied Gas
Dynamics, edited by A. E. Beylich (VCH, Weinheim, 1991), p. 153.
53. In a hypersonic region, the mem free path based on the flow speed instead of the thermal
speed characterizes the variation of the flow in its direction. With this mean free path, theeffective Knudsen number is given by $Kn_{r}=O[Kn_{w}(r/L)]$ . IncidentaUy, in a cyhndrically
expanding flow, $Kn_{r}=O(Kn_{w})$ for the latter mean free path.
54. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York,1968), p. 255, p. 504.
55. J. D. Cole, Perturbation Method in Applied Mathematics (Blaisdell Pub. Co., Mass., 1968)
Chap. 1.
56. M. D. Van Dyke, Perturbation Method in Fluid Mechanics (Academic Press, New York,1964) Chap. V.
57. From Eq. (42), is $O(1)$ and positive, which is implicitly assumed in deriving Eq. (41c).
58. For $Kn_{w}=10$ , the data for larger than shown in Fig. 4(c) are required.
59. Y. Sone, “ New kind of boundary layer over a convex solid boundary in a rarefied gas,”
Phys. Fluids 16, 1422 (1973).
$\hat{f}_{(n)}^{\langle 0,j,\hat{K})}$$\theta_{\zeta}arrow\pi/2+0$$r\wedge\sin\theta_{\zeta}=1$$\alpha$$r$$(\hat{r}^{\{i)}, \theta_{\zeta}^{(k)})$243
60. We are not tracing a particular particle, but are simply considering the sum of the local
expected number of coUisions in dr. The change of by collision should be taken into accountin the former case.
61. Similar differences are seen in $du/dr$ and $dT/dr$ , which are determined by the second orderterms of and for large .
of the distribution
62. It is also true for a general molecular model that the width in and inof the gain term of the collision integral are of the same order [see M. N. Kogan, “On theequation of motion of a rarefied gas”, J. Appl. Math. Mech. 22, 597 (1958)].
$\xi_{r}$

$T$

$u$

$r$

$\xi_{r}$$\xi_{t}$TABLE I.

Functions $h_{1}(M)$ and $h_{2}(M)$ in Eq. (31).

$\overline{\overline{\frac{Mh_{1}h_{2}Mh_{1}h_{2}Mh_{1}h_{2}}{0.00001.00001.00000.40000.49000.84700.80000.269507088}}}$0.049990.07998
0.12000.16000.20000.24000.28000.32000.3600TABLE II.

0.90830.85820.79660.74040.68910.64210.59910.5596
0.52330.97980.96790.95210.93650.92120.90600.89100.87610.86150.44000.48000.52000.56000.60000.64000.68000.72000.7600The nondimensional density

0.45930.43100.40500.38090.3586
0.33800.31890.30120.2848$\rho/\rho_{w}$0.83260.81840.80430.79040.77650.76280.74920.73560.72220.84000.88000.92000.96000.97000.98009900
1.0000$0$0.25530.24200.22970.21820.21550.21280.21010.20750.69560.68240.66930.65630.65306498
6466
0.6434$0$$0$

versus $r/L$ for various Knudsen numbers$Kn_{w}$.

$P/\rho_{w}$$\ovalbox{\tt\small REJECT} r/_{10.32250^{0}.6^{0}1^{0}4^{5}30^{0.0I0.1110\infty}}LK-..\ovalbox{\tt\small REJECT}_{60320.54270.50680.50070.5000}$

1.00011.0002
1.001
1.002
1.01
1.02
1.05
1.1
1.2
1.4
1.6
1.8
2
5
10
20
50
100
$2^{2}0^{0}0^{0}01000$

0.09734
0.024020.49360.49080.47840.46920.4306
0.4021
0.34800.29200.22370.15000.10950.084100.066830.01006
0.02493

0.49310.49010.47770.46840.42980.40150.34750.29170.22360.15010.10970.08426
0.066990.010100.02506$5.19x10^{-4}$$5.98x10^{-4}$$6.22x10^{-4}$$6.25x10^{-4}$

8. $27x10^{-5}$

9 $54xl0^{-5}$$9.94\cross 10^{-5}$$1.00x10^{-4}$$2.38x10^{-5}$$2.49\cross 10^{-5c}$$2.50x10^{-5}$0.31860.31700.30990.30460.28200.26500.23180.19610.15140.10190.07433
0.05694
0.045130.066400.016310.60020.59080.54730.51540.41110.35810.28570.22900.17000.11140.08047
0.061290.048410.070400.01724$4.05x10^{-4}$$4.28x10^{-4}$$4.40x10^{-4}$$6.46\cross 10^{-5}$$6.81\cross 10^{-5}$$7.01\cross 10^{-5}$$1.61xI0^{-5}$$1.70\cross 10^{-5}$$1.75\cross 10^{-5}$$2.07\cross 10^{-5}$$4.03\cross 10403x10^{-7}1.6Ix10_{-8}^{-6}$$425x10^{-8}4..25x10_{-7}^{-6}170\cross 10$$4.\cdot 37x10_{-7}437xI0175x10_{-8}^{-6}$$2.\cdot 06x10516\cross 10_{-8}^{-6}516\cross 10^{-7}$0.5936
0.5874
0.55650.53200.43970.53680.53390.52070.51000.46170.38620.30750.24420.17920.11630.08365
0.06357
0.050130.07255

0.42520.35720.29080.21520.01775

0.13930.09976
0.07562
0.059530.08572
0.020960.50000.4971
0.48490.4756
0.43640.40720.35150.29370.22370.14890.10820.08279
0.06561$2.38x_{-}10^{-6}596x10_{-7}$$—$$6..25x10_{-7}^{-6}250x10_{-8}625x10$a The isentropic solution (32a) and (32d). With the Knudsen-layer coirection, which is flattened on the
$b$

$c$

sphere, $\rho/Pw=0.6362$ at $r/L=1$ .
The \’uee molecular flow solution (29a).
The data at $r/L=99.94$ .

244
TABLE III. The nondimensional velocity $u/(2RT_{w})^{1/2}$ versus $r/L$ for various Knudsen numbers$Kn_{w}$.

$\ovalbox{\tt\small REJECT}\overline{u/(2RT_{w})^{1/2}}$$\frac{f/-\ovalbox{\tt\small REJECT}_{0.010.1110\infty}}{10.7322040110.41650.50170.55430.56310.5642}$

1.00011.0002

1.001
1.002

0.41050.41690.44940.47630.5876
0.66140.78250.8896
1.00711.1285
1.19641.24101.27281.40041.4292
1.44081.44661.44831.44911.44971.44970.74100.74470.76050.77210.82080.85640.92420.9950
1.2
1.083
1.4
1.183
1.6
1.241
1.8
1.280
21.30851.423
10
1.448
20
1.458
50
1.463
100
1.464
200
1.464
1000
1.464
2000
1.464
1.01
1.02
1.05
1.1

0.42310.42750.45050.47040.56010.62520.74110.8503

0.97371.10181.1731
1.21971.25281.3852
1.4153
1.42771.43401.43591.43681.43751.4376

0.56160.56480.5781
0.5882
0.63100.66300.72490.79030.8789 0.8719
0.9975 0.9625
1.0661 1.0139
1.1114 1.0471
1.1436 1.0702
1.2706 1.1538
1.2988 1.1688
1.3106 1.1742
1.3168 1.1767
1.3187 1.1774
1.3196 1.1778
1.3204 1.1780
1.32050.50720.50980.52180.53170.57810.6156
0.69130.7740$-$0.57110.5742
0.58820.59850.64200.67390.73490.79810.87540.95931.00551.03481.05471.1215
1.13101.13351.1344
1.1346c$–$$-$

0.57200.57530.58940.59980.64340.67540.73620.79920.87610.95901.00461.0333
1.05281.11701.1256

1.12771.1283
1.1284
1.1284
1.1284
1.1284a The isentropic solution (32b) and (32d). With the Knudsen-layer correction, which is flattened on the
$b$

$c$

sphere, $u/(2RT_{w})^{1/2}=0$ 3712 at $r/L=1$ .
The free molecular flow solution (29b).
The data at $r/L=99.94$ .

TABLE IV. The nondimensional temperature$T/T_{w}$versus $r/L$ for various Knudsen numbers$Kn_{w}$.

.

$\ovalbox{\tt\small REJECT} f/n_{w}-10.6434082570.81920.79120.7^{1}8670.78760.7878\ovalbox{\tt\small REJECT} T/T_{w}$1.00011.0002
1.001
1.002
1.01
1.02
1.05
1.1
1.2
1.4
1.6
1.8
2
5.
10
20
50
100
200
1000
2000

0.63820.82150.63600.62650.61930.58840.56450.81890.80420.5162
0.46180.3886
0.29840.24190.20250.17340.04833
0.01895
0.00749
0.00220$8.73x10^{-4}$$3.47\cross 10^{-4}$4.05 $x10^{-5}$$1.61\cross 10^{-5}$0.79080.7286
0.68380.60540.52980.43850.33340.26980.22610.1941
0.058100.026140.01333
0.00704
0.005270.0044700389
0.00382$0$0.81550.81350.8031
0.79350.74480.70570.63070.55370.45830.34850.28230.23710.20420.06491
0.032200.01894
0.012280.01035
0.00946
0.00879

0.00871

0.78630.78410.77490.76780.73590.71020.6556
0.59130.50230.39210.3242
0. $278I$
0.24470.11140.08298
0.07184
0.066220.06455
0.0637506314
0.06307$0$0.78080.77830.76760.75950.72410.69\^o6

0.64050.5771
0.49200.3896
0.32830.28750.25880.15470.13760.13230.13010.12950.12930.1291$-$0.78150.77910.76830.76000.72400.69590.63880.57440.48870.38720.32730.28810.26080.1666
0.15320.14970.14870.78190.77930.76840.76010.72400.69590.63870.57420.48830.38680.32720.28820.26110.16820.15540.15220.15130.1485c 0.1512–0.1512$-$$-$0.15120.1512a The isentropic solution (32c) and (32d). With the Knudsen-layer correction, which is flattened on thesphere, $T/T_{w}=0.8386$ at $r/L=1$ .
$b$

$c$

The free molecular flow solution (29c).
The data at $r/L=99.94$ .

245, terminal Mach number
TABLE V. Nondimensional velocity
at inlinity, frozen temperature.
, and energy flux $W/4\pi p_{w}(2RT_{w})^{1/2}L^{2}$ for various Knudsen numbers
mass flux$u_{\infty}/(2K_{w})^{1/2}$$T_{\infty}/T_{w}$$Q/4\pi\rho_{w}(2\Pi_{w})^{1/2}L^{2}$$M_{\infty}[=u_{\infty}/(5RT_{\infty}/3)^{1/2}]$$Kn_{v}$$\ovalbox{\tt\small REJECT}_{0.236^{w}1^{c}0.5065^{d}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} Kn_{w}u_{\infty}/(2R_{*}T_{w})^{1/2}T/T_{w}M_{\infty}Q/4\pi\rho(2RT_{w})^{1/2}L^{2}W/4\pi p_{w}(2RT_{w})^{1/2}L^{2}01.4640^{\infty_{b}}\infty$

0.005
0.01
0.02
0.05
0.1
0.2
0.5
1
2
5
10

0.00380.00860.01830.0406
0.06300.08670.11410.12900.13880.14580.1484

1.44991.43771.4162
1.36861.3206
1.27001.21081.17811.15641.14071.13470.52220.52900.5374
0.54870.55540.5596
0.56160.56280.56340.24650.25120.25730.26630.27230.27650.27930.2806
0.28130.28170.2819

26
17
11.5
7.44
5.76
4.72
3.927
3.593
3.400
3.273
3.227

0.5637
0. $5uo$0.5642h1.1284c0.1512i3.179$\ovalbox{\tt\small
REJECT}\ovalbox{\tt\small REJECT}\infty$$0.2821^{g}$a From Eqs. (32b) and (32d).
From Eqs. (32c) and (32d).

$d$

$e$

$b$

$c$

Eq. (35a).

$f$

Eq. (35b).
From Eq. (29b).
From Eq. (29C).

$h$

TABLE VI. Typical data of the lattice of$\frac{\frac{Kn_{w}=0.01}{;\hat{r}^{\langle:)}dr\wedge/di}}{016.250x10^{-9}}$

$277$

397
669
$915$

1245

1762
2328

1.001
1.010
1.099
1.499
4.988
99.80
$20I8$$2.677x10^{-5}$$1.241x10^{-4}$$6.833x10^{-4}$$3.179x10^{-3}$$2.496x10^{-2}$0.60257.940$\frac{\frac{0.1}{i\hat{r}^{\langle j)}d_{\Gamma/}^{\wedge}di}}{011.563x10^{-7}}$

523
1000
2091
3073
4394
6462
8720

$(=r_{D}\wedge)$

1.001
1.010
1.100
1.500
5.001
100.1
2010

$1.717x10^{-4}$$7.962x10^{-4}$$6.261x10^{-3}$0.1511
1.981$\frac{Kn_{w}<0.05}{\frac k\theta_{p}^{(k\prime}d\theta/dk,001.750^{(}x10^{-5}}$$1.005x10^{-3}$$1.788x10^{-5}$$1.001x10^{-2}$$3.811x10^{-5}$0.10000.7845$2.310x10^{-4}$$\pi/2$.
$\frac{\frac{10}{i\hat{r}^{\langle:)}d_{\Gamma/di}^{\wedge}}}{016.250x10^{-5}}$$6.858x10^{-6}$$3.103x10^{-5}$$767-$$1.010-$$3.434x10^{-5}-$

$–$

$–$

1476
1989
2654
3682
4432

1.100
1.500
4.997
100.1
1010

$3.148x10^{-4}$

383
629
959
1472

1.100
1.500
4.999
99.94

$1.715x10^{-3}$$5.698x10^{-3}$$1.564x10^{-3}$$1.249x10^{-2}$0.30893.075$-$$(=r_{D}\wedge)$

TABLE V1I. Typical data of the lattice of

2200
2464

$\wedge r$$\frac{\frac{1}{i\hat{r}^{\langle j)}dr\wedge/di}}{013.125x10^{-6}}$$(=r_{D}\wedge)$

57
412
1208

Eq. (30a).
Eq. (30b).

$g$$\theta_{(}$.

$\frac{Kn_{w}\geq 0.05}{\frac{k\theta_{t}^{(k|}d\theta_{(}/dk}{003.734x10^{-6}}}$

253
1172
2837
4995

$1.000x10^{-3}$$4.393x10^{-6}$$1.001x10^{-2}$$1.836x10^{-5}$0.10000.7848$7.S53xlO^{-4}$$556S$$\pi/2$$1.077x10^{-\ell}$$2.663x10^{-3}$$\frac{2560\pi 8.608x10^{-2}}{\ovalbox{\tt\small REJECT}(K,K)=(2560,2464)}\frac{5760\pi 7.966x10^{-2}}{\ovalbox{\tt\small REJECT}(5760,5568)}$$(=r_{D}\wedge)$$6.847x10^{-4}$$3.186x10^{-s}$$2.506x10^{-2}$0.6183,

246$\overline{\sim_{8}\vee\underline{\circ}\overline{\approx}}$$\sim_{\frac{\approx_{\overline{\sim}}}{o}\dot{o}_{||}}$$\frac{\approx\Xi}{\dot{o}_{||}0}$$\sim_{\frac{\approx_{\overline{\sim}}}{o}\dot{o}_{||}}$$(1k^{-}|^{Q}|\underline{\neq_{-.\frac{\overline\circ\circ\approx}{rightarrow v}}^{\omega_{\underline{\#_{\Phi}}}}}\overline{\geq}=_{6=}^{\overline{\sim}}$$\approx_{-\sim}\frac{\triangleright}{\dot\frac{\dot{\omega}}{h}}-\overline{\overline{\sim}}\omega\sim_{\Xi v\infty}\wedge\circ_{\bigcup_{)}}v>_{\Phi}$

8$\frac{\underline{v}}{P}\underline{\underline{\Xi}}\neq_{\geq}^{v}\overline{\circ\infty}\frac{v}{\circ,w}v$$ie\underline{\omega}\omega$

目〉

.

べき

三

9 曇

$\underline{6i_{\frac{v}{-}\underline{\underline{A}}}\backslash }$$t_{1}$

日$H\prec$$\overline{\sim_{\frac{\sim}{\neq_{\geq}^{v}\frac\circ}}}A_{\infty}^{\circ}\overline{\geq}\underline{-}\sim^{-}\Xi_{\sim}^{v}m\cdot-\omega_{\overline{\overline{9}}=}\overline{\circ}$

曽$\overline{\tau_{>}}x^{v}\vee$$\wedge-$$s^{\omega}-\wedge$$H\triangleleft\frac{-}{\in}\cdot r^{\sim}\vee$$H\overline{\circ}$$\wedge-$$\in\cdot\dot{\circ}\succ t\circ\xi\neg\dot{o}$$v\approx$$r\approx vrightarrow$$a$$i\circ$

247

FIG. 1. The profiles of the macroscopic variables $p/p_{w},$ $u/(2RT_{w})^{1/2},$ $T/T_{w}$ , and $M$ in theneighborhood of the sphere for various Knudsen numbers. $(a)Kn_{w}=0-0.05$ and (b) 0.05. The value on the sphere are marked with for $Kn_{w}=0,$ for 0.005, for 0.01,
for0.02, V for 0.05, for 0.1, ’ for 0.2,
for 0.5, A for 1,
for 2, for 5,
for 10, and for$Kn_{w}$$O$$\infty$$O$$E$$\infty$.

$\nabla$$O$$\blacksquare$$\triangle$$\bullet$$\square$

$O$

248

$arrow \mathscr{C}\Xi^{a}$

8

$\ni$

$rightarrow\omega\epsilon$$\sim\aleph$$\overline{\triangleright\omega}\aleph\circ\cdot O\overline{\omega}\overline{v}$$\vee--$

目$\ni\Leftrightarrow$$\geq$$\circ h\approx$$\not\in\backslash \omega\infty$$8\approx$

化$\Rightarrow$$\omega\Leftrightarrow$ジト$\overline{a}ui\triangleleft\backslash$$\approxarrow-\omega$

9$\underline{9}Arightarrow$

8

化へ$\Leftrightarrow\infty$.

$\underline{\Xi}$目$\frac{w}{\omega}\vee’\circ$$\approx\overline{\mathring{\approx}}\approx_{\overline{\epsilon}-\sim^{s_{9}}}arrow.>arrow\alpha \mathscr{C}$下$\Leftarrow$へ$\omega^{-}\wedge\sim\underline{-}h$$\overline{\overline{\mapsto}}h_{\wedge}\infty\sim_{9}\approx$.

”$\approx\underline{v}\backslash \vee$$\vee N\Rightarrow$

8$\frac{C;}{b}\backslash _{8_{\omega}^{\hslash}}^{\sim}i3\underline{s}\circ Q,\vee\wedge$$e/lt^{n}x?fZ)/\infty \mathfrak{n}$$t/$ 鉱249$\underline{\dot{o}}$$\Leftrightarrow\aleph\hat{\omega}’$$\Leftarrow$$arrow\alpha\infty\sim$8$\omega\Leftrightarrow\overline{\infty}-\wedge\ni$

1

$11$

.

$h$

$\infty\Leftrightarrow\succ\approx$逼$k^{\epsilon^{*}}$$\mathfrak{B}.8\omega\vee N$目$\frac{\infty}{\omega}\#^{\Phi}$目て

1\={o}\={o}I$\infty\approx t=$$v\backslash \triangleleft$$\overline{\vee}\circ\cdot$$\approx\Xi-\cdot$

11$\vee\wedge a$

占乙 ;$\frac{\aleph}{\sim N}$$\circ\overline{\frac{\infty}{v}}\neq^{v}arrow$.

$\backslash ^{\epsilon^{*}}$$\approx\underline{g}\approx_{\overline{\sim}}\frac{\infty}{\sim \mathfrak{n}})e^{\ni}\xi_{\neg}^{3}-\wedge$.

$\underline{\overline{o}}_{\aleph^{-}}^{\alpha_{\wedge\sim}}\sim^{h^{\ni}}\alpha$$\vee\approx N$$\underline{o|}$$\underline{v}\wedge_{\S^{\ni}}\wedge\approx_{\sigma\backslash }\infty_{\ni\underline{\Leftrightarrow}}O\vee\Xi v\backslash ^{\ni}\triangleleft\triangleright a$$h_{\vee\triangleleft}^{rh}\Leftrightarrow\S$. .
.

$\ni\backslash v-$

ヌ啗$\omega$$\Leftarrow Q$$\underline{\mathcal{O}}^{\triangleleft t}\backslash \underline{\aleph}\underline{\Rightarrow}I$$\check{\sim\dot{o}}$$\frac{\infty}{\dot{o}}$国$\Phi\not\in\underline{.}$の$\iota?\iota/\iota(JH^{\Phi}l)d\iota p/b\dot{o}$$N^{\wedge}\hat{O}$$\dot{\circ}-$$–\mathscr{O}^{\wedge}$$\subset;\infty^{\wedge}$$O_{arrow}^{\sim}\infty-\wedge$$O\infty$$\hat{N}O$$\circ\infty^{-}$$O\dot{\circ}$$\overline{O}-\wedge$$\dot{\circ}O$$-\infty^{\wedge}$$\mathscr{O}O$$\circ\circ\{\supset$

$O$

$||$

$\aleph^{||}\vee\circ^{\wedge}\epsilon^{s^{\aleph_{\circ}^{\epsilon^{\ni}}}}\frac{\approx 8}{d}$

ぎ冒$\underline{\circ}^{\wedge}$$h^{\ni}-\backslash _{N^{\wedge}}h^{\dashv\hat{\omega}}$$h^{=\underline{o}}-\wedge$.

$\bigwedge_{\mathscr{O}^{\wedge}}\Omega 9$$arrow OO$$\dot{k}O_{\wedge}N^{\wedge}$$\underline{\Leftrightarrow w\infty}\subset 0_{;}\infty-\circ_{\wedge}$$\succ\omega--O_{\wedge}$$h^{\dashv^{O}0}\dot{o}_{\wedge\dot{\circ}}^{\mathscr{O}}$て.

$\circ\circ\infty$$\circ^{\wedge}N$$-\circ\dot{\circ}$$\overline{h}\wedge\circ^{\wedge}-$$\underline{V\infty}$$||\dot{\circ}\circ$$rightarrow-\frac{d}{v}\aleph_{O^{\wedge}}^{O}g^{\ni}\omega$.

$Arightarrow\circ$$\vee v\Xi\underline{o_{\ni^{O^{\wedge}}}}$$1_{\backslash }^{\xi_{\neg}}||$$\underline{o}_{\wedge}\aleph^{\epsilon^{5}}\Xi_{h^{\dashv}}$$’\circ$$h$$r \underline{\circ}\bigvee_{\wedge}\circ$$\sim 8h^{=}$$\overline{\S_{\overline{d}}^{\Phi}}\approx_{\overline{d}_{\wedge^{\wedge}}^{\backslash _{\dashv}}}F\backslash$

へ $O$ つ$\epsilon^{V}\#_{\prec}\mathscr{O}_{-\overline{\sim}}^{\wedge}$$\triangleleftinfty_{\wedge}-8^{\wedge}$$\underline{\infty}\hat{\omega.}\pi_{g^{3}}$$ho$$\prime o$250$\overline{\circ\dot{\circ}}\Phi\infty\circ$$\infty$
$||$

$11$

$\epsilon^{\ni}rightarrow$$\neg\iota\backslash$$arrow d\wedge\circ$$\sim\bigvee_{\wedge}$$\propto$$\triangleleft$$\Phi$$\infty$

$e-$
$||$

$\approx$$\vee\triangleleft\infty$

a

$\}$

$\llcorner$$\bigwedge-\sim\sigma_{L_{\wedge}}$.

$\overline{\underline{\circ}}$$\underline{\infty}$$\sim$$\circ$$\triangleright O$$\not\in$$\underline{V}$$\underline{\approx}$

国$\Leftarrow$$O-$$\Rightarrow$$\sim-$$\approx$$8$$.rightarrow$$\circ$$\sim-$$\Xi$nn$\approx$h畠$\dot{\Phi}$$arrow$$\underline{-\circ}$

雲〉$\underline{\infty\omega}$$||$$-\wedge$$\overline{\underline{\sim}}arrow-\underline{\circ}\backslash \triangleleft$.

$k$$\infty\infty$$\circ$$\overline{\underline{\phi}}$$a\overline{\underline{e}}\vee T\wedge$$\wedge\approx$

$\Xi$

毛

$5$

$\underline{\Xi}$$\overline{\circ-}$$\underline{--}\triangleright^{-}\Phi$$\dot{\circ}$［$\backslash$$arrow$$\circ\Phi$$\cup$$||$$\mapsto$$\dot{\text{卜}}$$\frac{\dot{\circ}}{h}\vee\vee\wedge-\bigwedge_{-\wedge,\circ}$$O$$\Leftrightarrow$何$a$$||\hat{n}$$\Rightarrow$爵

ロ$-\circ$$||$$\approx$$s\sim$“$\sim$$\infty\wedge-$$\approx^{\{}k\vee-$$\infty\infty$$-O$$|\ni O$.

$CL\circ$$N\succ-$ヨ$||$$arrow\underline{\circ}\triangleleft$$\Leftrightarrow$“$\underline{\approx}$A$\approx\vee$$\sim\underline{\circ}-\wedge$$\circ$$||$$.–$$\sim m\triangleleft$$\sim$$b$$\sim\bigwedge_{\overline{\cup}\vee}\wedge\sim$$\underline{O}$$*V$$\underline{\dot{\infty}}$$\frac{\overline\hslash}{\circ}\underline{\bigwedge_{\sim}v}$$\overline{w}$$\approx$$\approx$$\infty\omega$$\Xi\underline{\circ}$$\approx_{\underline{\overline{\circ}}}^{-}\approx^{\underline{-}}\underline{\dot{m}}$$\Leftarrow O$$\ell\S-$$\overline{\mapsto}\infty$

.

$||$

$\Phi\triangleleft$$\Phi R\sim\backslash h$$\frac{\circ}{\triangleright}\underline{\wedge-\approx\wedge}$

251$\dot{\circ}$$\sim\approx$$||$$\Phi$$\ni\Phi$

謹ロ$arrow ds^{II}$$\circ$$k$$n$$\sim$ぐ $-$$\approx-$$\vee\infty\underline{W}$$-|\ni^{\mathscr{O}}$$\triangleright\sigma\iota||$“S)$\llcorner$$\circ$$\sim-$$\cup$—-$d$$\overline{\circ}$$\approx$.–$rightarrow$

$S$

A

$m$

$W$
$V$

–
$\sim$$\Rightarrow$$\circ \mathscr{Q}$$arrow-$

$\in$

$U$
$\circ$$\Phi$$\overline{V\sim}=V$

13

.号
$g$

$\approx$$\underline{\circ}$

$W$

$vv$
$- \frac{\Xi}{\dot{T}}$$\underline{\approx-\circ}$$\underline{\circ}arrow\overline{r}\Phi$$\Leftarrow\sim$$V$科$\sim$$\mapsto\sim$

$||$

$v$
$\dot{\Phi}\#arrow\backslash q$

佳

.

$–$

$\underline{\wedge-\wedge-\cup}$$\approx$$O\overline{\sim}$$||\triangleleft^{-}O$$S$$\backslash ^{\epsilon}\vee-$$11$$\sim dQ$$\sim\circ$

“

–

$n$

$-p$

$f^{3\vee}\neg-$$\approxrightarrow$$-||$$\overline{1}c_{L}^{\ni q}$$\underline{g}\backslash$$\Leftrightarrow\wedge d$$.-$$U\approx$

可$–\underline{\circ}$$rightarrow$$v\infty$$\approx$$-$$arrow\underline{\circ}$$\approx$$-rightarrow$$.=$$\infty$$.-$$\infty$$\sim$$\approx^{-}\sim-$$\wedge\omega$$arrow-A$$O\sim$$\underline{\circ}$$\approx$$\succ v-$$5 arrow-\bigwedge_{-}$$\approx$$\not\in$O. $-$$n\Leftarrow$$\Leftrightarrow$$v$

ロ$\cup$$\Xi$$\dot{"}$$\approx$フ$\omega$$OA\triangleright$$\epsilonarrow\infty$$rightarrow\Phi$$\neq^{v}$$\circ-$$\mapsto\bigwedge_{t6}||$.

$\infty$$\underline{\circ}$$\omega oq$..

$\text{℃^{}\backslash }$$\wedge$$h_{\vee}^{-U}$

252
ロ

$11$

$\underline{\epsilon^{\ni}}$$\sim\sim$$\approx^{\backslash }\backslash$$\wedge$$\frac{}{1}\vee^{\neg}\approx_{q}^{\ni}\mathfrak{t}\dot{\infty}r_{\ni}$$-\sim\cup\underline{\circ}$$\sim_{\dot{\Phi}}^{\Phi}\underline{o}\vee$$\circ\Phi$$arrow||$$\underline{\infty}_{q}$フ$\backslash$$h^{k}$$o^{\vee}\overline{\cup}$$\overline{>\omega}\bigvee_{\sim}$$\tau_{\mathscr{O}^{\wedge}}$$\approx$

ロ$\underline{\circ}\propto$$m_{\dot{N}}$$\Xi^{||}v$$-\aleph$$\approx\backslash$$\approx k$$\triangleright^{v^{\bigvee_{\wedge}}}\neq^{o_{\wedge}}-\circ$

.

$||$

$–\triangleleft\backslash$$\frac{C0}{g_{t}}\wedge\vee-$$-$Il$\aleph^{\not\in}$$\sim\sim$$\sim\aleph^{\backslash }\wedge$$\approx^{\ni}\overline{\aleph}$

–

I

.

$s\infty$$q_{\Phi}$$\underline{k}\underline{O}$$\Leftrightarrow$}$|$$.\underline{\circ}h\triangleleft$$arrow\cup\backslash$$k$$\not\in$$\underline{s}\vee\sim\wedge$$-\circ$フ$\Leftrightarrow$$\sim^{-}$$\alpha$$\approx$$r_{-}\infty^{\wedge}$$b\infty$$\sim\Phi\dot{\circ}$$\dot{\overline{"}}$$\wedge||$$arrow-\aleph$$\underline{\circ}$$\llcorner$$\omega\wedge$$\succ\dot{\vee}$$\overline{\underline{\sim}}\sim^{\wedge}v$.

.

$\underline{C}r$$\approx\infty-$$||$$\Xi v$$\dot{\overline{T}}\backslash \triangleleft$$\underline{\circ}p\wedge$$\omega-$$\mapsto\neq-\wedge$

$11$

.

$\underline{\dot{\circ}}\backslash -)$$k$$\zeta\wedge$

$-$

$h-$
$\sim$

253

FIG. 14. The contour of $2\pi p_{w}^{-1}(2RT_{w})^{3/2}f=0.2$ for various points $r/L$ . (a) $Kn_{w}=0.01$ and (b) $Kn_{w}=0.1$ . The contour of the local$[r/L=1.101(Kn_{w}=0.01), 1.314 (Kn_{w}=0.1)]$ ,$[r/L=1.264(Kn_{w}=0.01)$ ,MaxweUian is shown by:– $[r/L=1.035(Kn_{w}=0.01)]$ ,$[r/L=2.218(Kn_{w}=0.01), 2.067 (Kn_{w}=0.1)]$ . The dashed line —corresponds to Figs. 7(a) and 13(a)1.627 $(Kn_{w}=0.1)$], and$(Kn_{w}=0.01)$ or Figs. 8(c) and 13(b) $(Kn_{w}=0.1)$ . The flow deviates considerably from the local MaxweUian over the whole field when––

$IKn_{w}=0.1$

.

–

254

$(Q_{R}-Q_{\dot{m}})/Q_{\dot{m}n}$lattice dependence ofFIG. 17. The
and $(W_{mu}-W_{\dot{m}n})/W_{\dot{r}}$ . The $ma$]variation ot
over
the whole lattice points onfor different numbers andof thelattice points are marked with hoUow symbols. The] islattice dependence of $\max$ [(Q エー 可 $n)/Q$ 面 $n’(W$$in^{arkedwithb1acksymb\circ1s.LetLSbether_{A_{theresu1tsinSec.VA}}}thefirstparagraphinSec.VD,$$withwh^{\wedge}i^{\theta
1atticesystem,described}m$, and).
are obtained (Note that it depends onis the samelattice system as IS for $Kn$ $=0.01,0.1$ , and 10, andis that wlth
doubleof $IS$ for $Kn_{w}=1$ . M. is a test lattice system with double
andof those of $M_{n-1}$ . The indicates the mlxtmum variation
lattices;ofbetween $M-3$ andbetween $M-2$ andbetweenand $M_{0};O$ between. The. indicates
and$Q_{\dot{m}})/Q$$\max[$(▼ for
可 $n’(W\infty-W_{\dot{r}n})/W_{\dot{n}n}]$ forエー▲ for
for $M0$ ; ◆ for. From separate tests, the results 口 andare not infiuenced if $IS$ is taken as
for $Kn_{w}=1$ .
$(\rho\wedge, u\wedge,)\wedge,$$r\wedge\theta_{(}$$(\rho\wedge,\hat{u},\hat{T})$$\dot{\alpha}mum$$I$$1\leq r\wedge\leq r_{D}\wedge$$(K,\overline{K})$$r\wedge\theta_{(}$$r\wedge\theta_{(}$

$Q$

$B$

$C$

$(K,\overline{K})$$I$.

$Kn_{w}$$O$$M_{0}$$\nabla$$M_{-2}$$M_{-1}$$M_{-1;}$$\Delta$$M_{0}$$Q_{m}$$M-1;\blacksquare$$-W_{\dot{m}})/W_{\dot{m}n}$$M_{0}$$(K,\overline{K})$$(\rho, u\wedge\wedge, \wedge)$一$M_{1}$$M_{-3},\cdot$$M_{1}$$\blacksquare$$M_{0}$$M_{-2},\cdot$