TitleNumerical examination of applicability of the linearizedBoltzmann equation(Mathematical Analysis of Phenomena inFluid and Plasma Dynamics)

Author(s) Sone, Yoshio; Kataoka, Takeshi; Ohwada, Taku; Sugimoto,Hiroshi; Aoki, Kazuo

Citation 数理解析研究所講究録 (1994), 862: 202-218

Issue Date 1994-03

URL http://hdl.handle.net/2433/83852

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

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Numerical examination of applicability of the linearized Boltzmann equation

Yoshio Sone, Takeshi Kataoka, Taku Ohwada, Hiroshi Sugimoto, and Kazuo Aoki

京大・工 曾根 良夫, 片岡 武, 大和田 拓, 杉元 宏, 青木 一生

Department of Aeronautical EngineeringKyoto University, Kyoto 606-01, Japan

Abstract

The validity of the linearized Boltzmann equation in describing the behaviourof rarefied gas flows that deviate only slightly from a uniform equilibrium stateis discussed on the basis of several numerical examples. Various examples of theabove situation are analyzed numerically by the full nonlinear BKW equation andalso by its linearized version, with emphasis on the behaviour for small Knudsennumbers, and the solutions of these equations are compared. When the Knudsennumber is comparable to or smaller than the degree of the deviation from theuniform equilibrium state, the solution of the linearized equation generally differsdecisively from that of the nonlinear equation, however small the degree of thedeviation may be. Situations where the nonhnear effect degenerates are alsonoted.

1. Introduction

The linearized Boltzmann equation, where the second and higher-order terms of theperturbation from a uniform equilibrium solution are neglected, is widely used in analyz-ing rarefied gas flow problems where the state of the gas deviates slightly from a uniformequilibrium state at rest. The situation is usually encountered in gas dynamic problems ofa small system such as in aerosol science and micromachine engineering, where the Machnumber of the flow and the temperature variation, compared with the average temperature,on the boundaries, which are close to each other, are both small. It is, however, known thatin some situations the linearized equation does not give a correct answer, however small thedeviation from a uniform equilibrium state may be ([Cercignani, 1968]; [Sone, 1978]; [On-ishi&Sone, 1983]). They are related to infinite domain problems (e.g., Stokes paradox $[C$ ,1968]). In the analysis of the asymptotic behaviour of steady flows of a rarefied gas for smallKnudsen numbers, Sone ([Sone, 1971, $1984,1987,199la,b]$ ; [Sone &Aoki, 1987]) discussedthe applicability of the linearized Boltzmann equation and pointed out that the nonlineareffect in the Boltzmann equation is, generally, not negligible for any small deviation from auniform equilibrium state if the Knudsen number of the system is comparable to or smallerthan the degree of the deviation. There are, of course, various situations where the effectof nonlinearity degenerates. The above infinite domain examples are also understood bythe general statement if the situation is properly interpreted. Recently Aoki and Masukawa[1994] considered the two surface problem of evaporation and condensation and showed adecisive difference between the numerical solutions of the linear and nonlinear equations forvery small temperature difference of the two boundaries.

The example, however, is a special one, where the flow velocity is uniform in the limitof small difference of the surface temperatures. For more general understanding, in thepresent paper, we consider several different situations whose deviation from a uniform equi-hbrium state is small and investigate the applicability of the linearized Boltzmann equation

数理解析研究所講究録第 862巻 1994年 202-218

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numerically. That is, we analyze the problem numerically by two ways, i.e., by the origi-nal nonlinear equation and by its linearized version, compare the results, and confirm itsapplicability with respect to the parameters: the Knudsen number (Kn) and the parameterthat represents the deviation from a uniform equilibrium state (say, nonuniformity param-eter e). For simplicity of analysis, we adopt the BKW (or BGK) equation ([Bhatnagar, $et$

al., 1954]; [Welander, 1954]; [Kogan, 1958]) as the basic equation. For the present purpose,this is legitimate from comparison of various results of the BKW equation and the standardBoltzmann equation. In the analytical discussion of the applicability of the linearized Boltz-mann equation, which is done in connection with the asymptotic analysis of the Boltzmannsystem for small Knudsen numbers, the BKW equation shows the same behaviour as thestandard Boltzmann equation (e.g., [S&A, 1987]; $[S$ , 1987]; $[S,$ $199la,b]$). The results of thelinearized BKW equation are consistent with recent accurate numerical computations by thehnearized Boltzmann equation for hard-sphere molecules (e.g., [Sone et al., 1990]; [Ohwadaet al., 1989]; $[S, 1991b]$ ; [Takata et al., 1993]). On the boundary the Maxwell type conditionor the conventional boundary condition of evaporation and condensation (e.g., [Cercignani,1987]; $[S, 1987]$ ) is adopted as the kinetic boundary condition.

2. Basic equation and notations

Let $p_{0}$ and $T_{0}$ be the pressure and the temperature of the uniform equilibrium state atrest. When we consider problems with evaporation or condensation on a boundary, $p_{0}$ isthe saturated gas pressure at temperature $T_{0}$ . The density $\rho_{0}$ and the velocity distributionfunction $f_{0}$ of the equilibrium state are given by

(1) $\rho_{0}=p_{0}/RT_{0}$ ,

(2) $f_{0}= \frac{\rho_{0}}{(2\pi RT_{0})^{3/2}}\exp(-\xi_{i}^{2}/2RT_{0})$ ,

where $R$ is the specific gas constant and $\xi_{1}$ is the molecular velocity. We are interested inthe behaviour of the gas for small deviations from this uniform state.

Let $L$ be the characteristic length of the system and let $\ell_{0}$ be the mean free path ofthe equilibrium state, which is the ratio of the mean molecular speed $(8RT_{0}/\pi)^{1/2}$ and themean collision hequency. In the present paper, we use the following nondimensional variablesbased on the above basic variables of the system: Kn $=\ell_{0}/L;x;L$ is the Cartesian coordinatesystem of the physical space; $(r, \theta, x_{3})$ is the cylindrical coordinate system in the $x$ : spacewith the common $x_{3}$ axis; $(2RT_{0})^{1/2}\zeta_{1}$ is the molecular velocity; $\zeta=(\zeta_{i}^{2})^{1/2};f_{0}(1+\phi)$ isthe velocity distribution function; $\rho_{0}(1+\omega)$ is the density of the gas; $(2RT_{0})^{1/2}u$: is theflow velocity; $T_{0}(1+\tau)$ is the temperature; $p_{0}(1+P)$ is the pressure; $n$; is the unit normalvector to the boundary, pointed to the gas; $(2RT_{0})^{1/2}u_{wi}$ is the velocity of the boundarywith $u_{w};n:=0$ (this is required in steady flow problems); $T_{0}(1+\tau_{w})$ is the temperature ofthe boundary; $\rho_{0}(1+\sigma_{ws})$ and $p_{0}(1+P_{ws})$ are, respectively, the saturation gas density andpressure at temperature $T_{0}(1+\tau_{w})$ , determined by the Clausius-Clapeyron relation [Reif,1965]; and $E(\zeta)=\pi^{-3/2}\exp(-\zeta^{2})$ . Thus, $f_{0}=\rho_{0}(2RT_{0})^{-3/2}E(\zeta)$ . The $r$ and $\theta$ componentsof $u_{i}$ and $u_{wi}$ are denoted by the subscripts $r$ and $\theta$ , respectively (e.g., $u_{f},$ $u_{w\theta}$).

In these variables, the nondimensional BKW equation for a steady flow is written as

(3) $\zeta_{i}\frac{\partial\phi}{\partial x_{i}}=\frac{2}{\sqrt{\pi}Iffi}(1+\omega)(\phi_{e}-\phi)$ ,

(4) $\phi_{e}E=\frac{1+\omega}{\pi^{3/2}(1+\tau)^{3/2}}\exp[-\frac{(\zeta.\cdot-u_{i})^{2}}{1+\tau}]-E$,

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where(5a) $\omega=\int\phi Ed\zeta_{1}d\zeta_{2}d\zeta_{3}$ ,

(5b) $(1+ \omega)u_{i}=\int\zeta_{1}\phi Ed(d(d\zeta_{3}$,

(5c) $\frac{3}{2}(1+\omega)\tau=\int(\zeta_{j}^{2}-\frac{3}{2})\phi Ed\zeta_{1}d\zeta_{2}d\zeta_{3}-(1+\omega)u_{j}^{2}$,

(5d) $P=\omega+\tau+\omega\tau$ .

The Maxwell type boundary condition is written as follows:

(6) $\phi(x_{1}, \zeta_{1})=(1-\alpha)\phi(x;, \zeta_{i}-2\zeta_{J}\cdot n_{j}n_{i})+\alpha\phi_{e}(\omega=\sigma_{w}, u;=u_{wi}, \tau=\tau_{w})$ ,

$(\zeta_{1}n:>0)$ ,

(7) $\sigma_{w}=\frac{1}{(1+\tau_{w})^{1/2}}(1-2\sqrt{\pi}\int_{\zeta_{j}n_{j}<0}\zeta_{j}n_{j}\phi Ed\zeta_{1}d\zeta_{2}d\zeta_{3})-1$,

where $\alpha(0\leq\alpha\leq 1)$ is the accommodation coefficient of the boundary. The condition iscalled diffuse reflection when $\alpha=1$ , and specular reflection when $\alpha=0$ . The conventionalcondition of evaporation and condensation on an interface between a gas and its condensedphase, which is also called a complete condensation condition, is as follows:

(8) $\phi(x_{1}, \zeta_{i})=\phi_{e}(\omega=\sigma_{ws}, u_{1}=u_{wi}, \tau=\tau_{w})$ , $(\zeta_{i}n;>0)$ .

In the following analysis, $P_{ws}$ is preferred to $\sigma_{ws}$ as a parameter. It is related to $\sigma_{ws}$ and $\tau_{w}$

as(9) $P_{ws}=\sigma_{ws}+\tau_{w}+\sigma_{ws}\tau_{w}$ .

Let $\phi E$ , the deviation from the uniform state given by Eq. (1), be $O(e)$ . Needless tosay, $u_{w}$: and $\tau_{w}$ should be $O(e)$ for such $\phi E$ to be the solution. Then, the macroscopicvariables $\omega,$ $u_{*}\cdot$ , and $\tau$ are also $O(\epsilon)$ . Neglecting the second and higher-order terms of $O(\epsilon)$

in Eqs. (3)$-(8)$ , we obtain the hnearized equations. The linearized BKW equation is:

(10) $\zeta_{i}\frac{\partial\phi}{\partial x_{i}}=\frac{2}{\sqrt{\pi}Iffi}[\omega+2\zeta_{i}u;+(\zeta^{2}-\frac{3}{2})\tau-\phi]$,

where

(lla) $\omega=\int\phi Ed\zeta_{1}d\zeta_{2}d\zeta_{3}$ ,

(llb) $u;= \int\zeta_{i}\phi Ed\zeta_{1}d\zeta_{2}d\zeta_{3}$ ,

(llc) $\frac{3}{2}\tau=\int(\zeta^{2}-\frac{3}{2})\phi Ed\zeta_{1}d\zeta_{2}d\zeta_{3}$ ,

(lld) $P=\omega+\tau$ .

The linearized Maxwell type condition is:

(12) $\phi(x;, \zeta_{1})=(1-\alpha)\phi(x:, \zeta_{1}-2\zeta_{j}n_{j}n_{i})+\alpha[\sigma_{w}+2\zeta_{j}u_{wj}+(\zeta^{2}-\frac{3}{2})\tau_{w}]$ , $(\zeta_{j}n_{j}>0)$ ,

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(13) $\sigma_{w}=-\frac{1}{2}\tau_{w}-2\pi^{1/2}\int_{(n<0}\zeta_{j}n_{j}\phi Ed\zeta_{1}d\zeta_{2}d\zeta_{3}jj$

The linearized complete condensation condition is:

(14) $\phi(x_{i}, \zeta_{i})=\sigma_{ws}+2\zeta_{j}u_{wj}+(\zeta^{2}-\frac{3}{2})\tau_{w}$ , $(\zeta_{i}n_{i}>0)$ .

The linearized form of Eq. (9) is(15) $P_{ws}=\sigma_{ws}+\tau_{w}$ .

3. Plane Couette flow with evaporation or condensation on the boundaries

In this section we consider the steady behaviour of a gas in the region $0<x_{2}<1$bounded by its two parallel plane condensed phases with different temperatures, one ofwhich is moving in its own plane. Let $u_{wi}=0,$ $\tau_{w}=0$ , and $P_{ws}=0$ at $x_{2}=0$ , and let$u_{wi}=(\epsilon_{1},0,0),$ $\tau_{w}=\epsilon_{2}$ , and $P_{ws}=\epsilon_{3}$ at $x_{2}=1$ . We numerically solve the nonlinear system,Eqs. (3)$-(5d)$ subject to boundary condition (8) at $x_{2}=0$ and $x_{2}=1$ , and the linear system,Eqs. (10)-(11d) with Eq. (14) at $x_{2}=0$ and $x_{2}=1$ and compare the solutions of the twosystems. Our interest is the behaviour of a slightly nonuniform state, i.e., for small valuesof $\epsilon_{1},$ $\epsilon_{2}$ , and $\epsilon_{3}$ . The method of numerical computation is a straightforward application ofthat in [Aoki, et al., 1991]. Thus, it is not repeated here, and only the results of computationare given.

The profiles of $\omega,$ $\tau,$ $u_{1}$ , and $u_{2}$ of the two systems, linear and nonlinear systems, with$\epsilon_{1}=0.02,$ $\epsilon_{2}=0.001$ , and $\epsilon_{3}=0.02$ are shown in Fig. 1 for three Knudsen numbers(Kn $=0.002,0.02$ , and 0.2). Let $\epsilon=\max|\epsilon_{m}|$ , then $\epsilon$ is a measure of deviation from ourreference uniform equilibrium state at rest. This notation will also be used in the followingexamples. When Kn $=0.2$ , where Kn is fairly larger than $\epsilon(=0.02)$ , the profiles of the twosystems in Fig. 1 are very close to each other, and the difference is bounded by $\epsilon^{2}$ . Thus,the linear solution is well qualified as the first order approximation of the nonlinear system.When Kn $=0.02$ , where KJ is comparable to $\epsilon$ , the difference of the two solutions becomesappreciable and it is fairly larger than $\epsilon^{2}$ [Fig. 1 $(c)$ ]. When Kn $=0.002$ , where Kn is fairlysmaller than $\epsilon$ , the two solutions are markedly different, and the difference is obviouslylarger than $\epsilon^{2}$ by far. The linear solution cannot be considered to be an approximation ofthe nonlinear solution. Incidentally, the negative temperature-gradient phenomenon ([Pao,1971]; [Sone &Onishi, 1978]; [Aoki &Cercignani, 1983]; [Hermans &Beenakker, 1986];[Sone et al., 1991]; [A&M, 1994]) is seen in these examples [Fig. 1 $(b)$ ].

When there is neither evaporation nor condensation on the boundaries at $x_{2}=0$ and$x_{2}=1$ , where $u_{2}\equiv 0$ , the situation is different. Figure 2 shows the profiles of $\omega,$ $\tau$ , and$u_{1}$ in the case of diffuse reflection, Eqs. (6) and (7) or Eqs. (12) and (13) with $\alpha=1$ , with$\epsilon_{1}=\epsilon_{2}=0.02$ , i.e., $u_{wi}=0$ and $\tau_{w}=0$ at $x_{2}=0$ and $u_{wi}=(0.02,0,0)$ and $\tau_{w}=0.02$

at $x_{2}=1$ . As in any closed domain problem without evaporation and condensation on theboundary, the solution is not uniquely determined by the diffuse reflection condition; Anothercondition relating to the mass of the gas in the domain is required for the uniqueness. Herewe have chosen the solution with $\sigma_{w}=0$ at $x_{1}=0$ . For three cases of Knudsen numbers,i.e., Kn $=0.002,0.02$ , and 0.2, the deviation of the solution of the hnearized equation fromthat of the nonlinear equation is uniformly very small with respect to the Knudsen number,and the former solution is a good approximation to the latter solution.

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4. Cylindrical Couette flow with evaporation or condensation on theboundaries

Here we consider the behaviour of a gas in the region $1<r<2$ bounded by its twocoaxial circular cylindrical condensed phases with different temperatures, the outer one ofwhich is rotating with a constant angular velocity around its axis. Let $u_{wi}=0,$ $\tau_{w}=0$ , and$P_{ws}=0$ at $r=1$ , and let $u_{w\theta}=\epsilon_{1},$ $u_{wr}=u_{w3}=0,$ $\tau_{w}=e_{2}$ , and $P_{ws}=\epsilon_{3}$ at $r=2$ . Wenumerically solve the nonhnear system, Eqs. (3)$-(5d)$ with Eq. (8) at $r=1$ and $r=2$ , andthe linear system, Eqs. (10)-(11d) with Eq. (14) at $r=1$ and $r=2$ for various sets of theparameters Kn, $\epsilon_{1},$ $\epsilon_{2}$ , and $\epsilon_{3}$ . The method of computation is a straightforward applicationof that in [Sugimoto&Sone, 1992], and only the results of computation are given here.

In Fig. 3, the profiles of $\omega,$ $\tau,$ $u_{f}$ , and $u_{\theta}$ are shown for KJ $=0.005,0.02$ , and 0.2 inthe case $e_{1}=0.02,$ $\epsilon_{2}=0.001$ , and $e_{3}=0.02$ . The difference of the linear and nonlinearsolutions obviously increases as the Knudsen number decreases (Fig. 3). When Kn $=0.2$ ,which is fairly larger than $e(= \max|\epsilon_{m}|)$ , the two solutions are very close; when Kn $=0.02$ ,which is comparable to $\epsilon$ , the difference of $u_{\theta}$ is fairly larger than $\epsilon^{2}$ , and when Kn $=0.005$ ,the difference of $u_{\theta}$ is of the order of $\epsilon$ . The relative difference of $\tau$ increases in a similarway to that of $u_{\theta}$ , but the absolute difference is much smaller since $\epsilon_{2}$ (thus $\tau$ itself) is muchsmaller than $\epsilon_{1}$ (or $u_{\theta}$ ). Incidentally, the negative temperature-gradient phenomenon is seenin these examples [Fig. 3 $(b)$ ].

In Fig. 4, the profiles of $\omega,$ $\tau,$ $u_{r}$ , and $u_{\theta}$ are shown for $\Re=0.005,0.02$ , and 0.2 in thecase $\epsilon_{1}=0.02,$ $e_{2}=0.01$ , and $\epsilon_{3}=0.02$ . The feature of the difference of the two solutionsis similar to that of the previous example, but the difference of $\tau$ is comparable to that of$u_{\theta}$ since $\epsilon_{2}$ is comparable to $\epsilon$ . Incidentally, the negative temperature-gradient phenomenonis not seen in these examples [Fig. 4 $(b)$].

An example without evaporation and condensation on the boundaries is shown in Fig. 5,where the case of $\epsilon_{1}=0.02,$ $\epsilon_{2}=0.01$ and diffusely reflecting boundary are considered. Asin the corresponding problem in Sec. 3, we have chosen the solution with $\sigma_{w}=0$ at $r=1$ .Again, the linear solution is a good approximation to the nonlinear solution for the threeKnudsen numbers Kn $=0.005,0.02$ , and 0.2.

5. Flow past an array of flat plates

Here we consider an example of flows past a body without evaporation and condensation.In order to concentrate our interest on the behaviour in a finite region and to avoid thedifficulty [Sone&Takata, 1992] of numerical computation owing to the discontinuity of thevelocity distribution function around a convex body, we investigate the following somewhatartificial problem in a rectangular domain $(-a<x_{1}<a, 0<x_{2}<b)$ .

(i) Nonhnear problem: The basic equation is given by Eqs. (3)$-(5d)$ . The boundary conditionis as follows: $\phi(\zeta_{2}>0)$ on $(x_{2}=0, -a<x_{1}<a)$ is given by Eqs. (6) and (7) where $u_{wi}=0$ ,$\tau_{w}=e_{2}$ , and

(16a) $\alpha=\frac{1}{2}[1+\cos(2\pi x_{1})]$ , $(-1/2\leq x_{1}\leq 1/2)$ ,

(16b) $\alpha=0$ , (specular reflection), ( $-a<x_{1}<-1/2$ and $1/2<x_{1}<a$);

$\phi(\zeta_{2}<0)$ on $(x_{2}=b, -a<x_{1}<a)$ is given by Eqs. (6) and (7) with $\alpha=0$ (specularreflection); $\phi((1>0)$ on $(x_{1}=-a, 0\leq x_{2}\leq b)$ and $\phi(\zeta_{1}<0)$ on $(x_{1}=a, 0\leq x_{2}\leq b)$ aregiven by the corresponding parts ( $(1\gtrless 0)$ of $\phi_{e}(\omega=0, u_{i}=(\epsilon_{1},0,0), \tau=0)$ .

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(ii) Linear problem: The basic equation is given by Eqs. (10)-(11d). The boundary conditionis as follows: $\phi(\zeta_{2}>0)$ on $(x_{2}=0, -a<x_{1}<a)$ is given by Eqs. (12) and (13) with$u_{wi}=0,$ $\tau_{w}=\epsilon_{2}$ , and Eqs. (16a) and $(16b);\phi(\zeta_{2}<0)$ on $(x_{2}=b, -a<x_{1}<a)$ is givenby Eqs. (12) and (13) with $\alpha=0;\phi(\zeta_{1}>0)$ on $(x_{1}=-a, 0\leq x_{2}\leq b)$ and $\phi(\zeta_{1}<0)$ on$(x_{1}=a, 0\leq x_{2}\leq b)$ are the corresponding parts $(\zeta_{1}\gtrless 0)$ of $2\zeta_{1}\epsilon_{1}$ .

The problem is a model of a uniform flow past an array of flat plates without an angleof attack $(-1/2\leq x_{1}\leq 1/2, x_{2}=2mb, m=0, \pm 1, \cdots)$ . The upstream and downstreamregions are limited at $x_{1}=-a$ and $a$ , since we want to examine nonhnear effects in a finite-domain problem. According to [$S$ &T, 1992], the discontinuity of a velocity distributionfunction on a boundary at the tangential velocities propagates into the gas from convexpoints of the boundary. The present choice, Eq. (16a), of the accommodation coefficientavoids the discontinuity at the leading and trailing edges of the plate, which are the onlyconvex points of the boundary. Thus the velocity distribution function is continuous in thegas.

The flow with the nonuniformity parameters $\epsilon_{1}=0.1$ and $e_{2}=0.1$ in the domain $a=1$ ,$b=1/2$ is computed for three Knudsen numbers $K_{J}=0.02,0.1$ , and 0.5. The profiles of $\omega$ ,$\tau,$ $u_{1}$ , and $u_{2}$ along the sections $x_{1}=0,$ $\pm 0.4$ , and $\pm 0.725$ are shown in Figs. $6a-6d$ . Asin the examples with evaporation and condensation in Secs. 3 and 4, the deviation of thehnear solution from the nonlinear solution increases as the Knudsen number decreases, andthe differences in $\omega$ and $\tau$ of the two solutions obviously exceed $e^{2}(e= \max|\epsilon_{m}|=0.1)$ forKJ $=0.1$ and 0.02, and are $O(\epsilon)$ when Kn $=0.02$ .

6. Discussion

In this paper we considered various rarefied gas flows where the situation is very closeto a uniform equilibrium state at rest, and investigated the flows numerically on the basisof two types of basic equations: the (original nonlinear) BKW equation and its linearizedversion. The results are compared, and the validity of the solution of the linearized equationin describing the flow is examined. The result depends on the Knudsen number of the system.

When the Knudsen number (Kn) is much larger than the nonuniformity parameter $(\epsilon)$ ,the solution of the linearized equation is a good approximation to that of the nonlinearequation. As the Knudsen number decreases, the deviation of the hnear solution from thenonlinear solution generally increases. It is fairly larger than $\epsilon^{2}$ when $Iffi\sim\epsilon$ , and the twosolutions are quite different when $K_{J}\ll\epsilon$ . Thus for Kn $<\sim e$ , the nonlinear solution cannot beobtained by a simple perturbation analysis from the linear solution. In some cases, however,the linear solution is a good approximation to the nonhnear solution irrespective of theKnudsen number (see the second example of Sec. 3 and the last example of Sec. 4). This isdiscussed below.

General theoretical discussion of the importance of the nonlinear term even in the casewhere the system deviates slightly from a uniform state was made in $[S, 1971]$ in connec-tion with asymptotic analysis of the Boltzmann equation for small Knudsen numbers (seealso [$S$ , 1978, 1984, 1991ab]). The present numerical computations give good examples ofthe theoretical discussion. The ratio of the Knudsen number and the nonuniformity pa-rameter determines the validity of the linearized Boltzmann equation. Since the case withsmall nonuniformity parameters is concerned, the ratio takes various values only for smallKnudsen numbers. Therefore, the situation can be clarified by the analysis of the case withsmall Knudsen numbers, which admits a macroscopic description. Thus, the degeneracy ofthe nonlinear effect is easily surveyed by the macroscopic description. That is, the leadingnonlinear term in the macroscopic description is the convection term of the incompress-ible Navier-Stokes system of equations (continuity, momentum, and energy equations), and

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therefore the linearized equation gives a good description in the case where the convectionterm degenerates or is incorporated in the pressure term in the Navier-Stokes system.

The last statement is well exemplified in our numerical computation. In the Couette flowunder diffuse reflection in Sec. 3, the convection term vanishes, and in the cyhndrical Couetteflow under diffuse reflection in Sec. 4, only non vanishing part of the convection term, $u_{\theta}^{2}/r$ ,can be incorporated in the pressure term. In both cases the deviation of the hnear solutionfrom the nonlinear solution is at most of the second order of the nonuniformity parameter(Figs. 2 and 5). In the example in Sec. 5, the flow is nearly in the $x_{1}$ direction, and thereforethe leading convection term of the incompressible Navier Stokes system is $u_{1}\partial u_{1}/\partial x_{1}$ inthe $x_{1}$ -momentum equation and $u_{1}\partial\tau/\partial x_{1}$ in the energy equation. The $u_{1}\partial u_{1}/\partial x_{1}$ can beincorporated in the pressure term, but $u_{1}\partial\tau/\partial x_{1}$ is left as it is. Since the velocity fieldcan be solved independently from the temperature field in the incompressible Navier-Stokessystem, according to the statement the velocity field of the linear equation should be a goodapproximation, but its temperature field may deviate considerably from that of the nonlinearequation. Figures $6a-6d$ support this.

In some infinite-domain problems, the discrepancy of the linearized equation such asStokes paradox $[C, 1968]$ is encountered for arbitrary Knudsen numbers. From the followingreason, this is also the same kind of difficulty of the hnearized equation as that in thepresent examples. In these problems, the solution is supposed to approach a uniform stateat infinity, and the length scale of the variation of the solution increases with the distancefrom a body. Then the effective Knudsen number (the mean free path divided by the locallength scale of variation), which determines the variation of the variables, decreases to vanish,and it becomes much smaller than the small nonuniformity parameter in the far field, andtherefore the criterion on discrepancy of linear solutions applies.

Finally, the computation was carried out by HP 9000730 and MIPS RS 3230 computersat our laboratory and by FACOM VP-2600 computer at the Data Processing Center of KyotoUniversity.

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Sone, Y. and Takata, S., 1992, Discontinuity of the velocity distribution function in ararefied gas arounda convex body and theSlayer at the bottom of the Knudsen layer,Transp. Theor. Stat. Phys. 21, 501-530.

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210

$q$ $s$Q) $0$$–$

$r\infty\Xi+\approx(.\rangle Q)$

$\underline{\overline{o}}\overline{\infty 0}\overline{\omega}$

$\vee\wedge r\circ$

$\overline{\mathring{\circ}^{l}}A_{\ni}^{\Phi}\star t_{\frac{Q)}{Q)}}$

$\dot{o}_{||}\star a^{\partial}\omega\infty\tau^{\dot{\underline{O}}}$

$\backslash n_{r^{\frac{o}{\circ}}}\cdot\overline{\geq}$

$so$

$\urcorner 0^{\frac{\not\supset}{‘ 6}}\circ\dot{o}\cdot\dot{o}^{\infty}\overline{o}^{\wedge\overline{|}_{\overline{\infty^{\underline{\infty}}}}^{E}}v\Phi^{\wedge}$

$||E$ $||$

$\omega_{\circ}^{\circ}r\overline{\S}$

$-\overline{Q)}1-)$

$0\infty\not\in\aleph$

$\dot{o}\alpha$ ae3 a

$\wedge$ $||\overline{\triangleleft}||$

$1\overline{0}^{-}10$

$-\circ-\infty So$

$v610$$.–$a $o^{\wedge}N\Phi$

$\sim_{\frac{-}{o}}\not\in_{arrow r}^{\overline{Q)}\overline{\infty}}\Xi\Phi$

$r\circ 3\in-$

$Q)-$ .$A_{\vee}^{t\lrcorner}$ a

.$\overline{\frac{o}{\underline 0}}\frac{\triangleleft}{\not\in^{Q)}}\approx\wedge+_{\infty_{l1}^{\partial}}^{Q)}\iota_{6}$

$\prec\underline{q)}\dot{d}\infty_{\overline{-\circ}}$.$e-rightarrow$

$\approx^{Q)}\triangleleft A^{q)}\wedge$

$\Leftrightarrowrightarrow\prec$.$oB^{\Phi}$

\\={o}$\underline{\wedge p}$

$\underline{\overline{o}}_{\frac{3}{6}\neq^{ae}}^{}$

$0-\infty$$\overline{\star e^{\partial}}-\omega^{o}\{$

\={o} $\circ!arrow\iota 6$

a $o\underline{(.)}$

$\triangleright Q)l60\infty^{\wedge\sim_{-}}-$

$Ao$$\frac{\star^{l}}{\geq}\infty^{\wedge}0’\prime \mathfrak{l}$

$0\triangleleft$

$\star_{\Phi}\prec\frac{\#_{q\prime}v_{\partial}}{o}\frac{\not\in^{||}\dot{o}}{\infty}^{\wedge}\infty_{\backslash }\succ O\overline{\iota 6}\geq\infty g$

$O_{\frac{Q)}{l\S}}\prime Dg\Rightarrow v\underline{Q\zeta 6}$

$\overline{h}\Leftrightarrow\overline{-}$

$so$$v\approx$

$-$. $\sim_{\approx}r_{\star}=Q$

. $\underline{b}D\Leftrightarrow$

$hR$ \={o}

211

$r_{U}r\circ\infty$$\Leftrightarrow\approx\iota 0$

6 6 $Q\rangle$

$\circ^{\wedge}\underline{T}^{\wedge}\frac{N}{\infty}$

$||\not\in^{Q)}A^{\Phi}$

$\mathring{\aleph}^{1}$ t-$\in-$

$\underline{\overline{\cup}}$

$e\underline{\overline{r\circ}}\dot{g}Qj$

$0$ $\wedge\infty\sim$.

$|| \frac{r_{U}}{v}\infty*$

$\}_{\neg}^{\ni ff}6$

$r \underline{\circ}-\dot{r}3\frac{Q)}{=}$

$\circ^{\ell 6}r\underline{6}.+^{Q}=_{\partial}$

$s^{\tilde{s}^{\dot{O}}\overline{\neq_{\zeta 6}^{O_{l}}}}|.|_{\overline{N}}\backslash$

$\wedge\mathring{\circ}^{1^{\wedge}}+A$

$.-Q)$ $\dot{o}Q$)

$\infty$

$\wedge\infty 6$

$\dot{o}0\circ\triangleleft\dot{o}oo\frac{t.)}{\dot{\sim}}--$

$||$

$||$ $1|$

$t^{\zeta}\circ^{\aleph}g$

$1$

$||-r\circ$$\infty\Leftrightarrow$

$–6$$\frac{\iota 0}{\infty,,q}r_{\Xi}o\Phi\Xi^{\wedge}Q)$

$.=\approxarrow$.$\overline{-\prime\circ}$

$\sim_{-}^{6}\frac{-}{\omega}\infty^{>}\succ\infty$

$\approx\infty\overline{6}$

$0\infty Q)$$P^{\circ}\approx\Leftrightarrow$

$\Leftrightarrow-$

$r_{}=vS$ \={o}$\omega s$

$\Leftrightarrow Q)$

$o\dot{A}r=Q)$

$\Leftrightarrow\prec\cdot\star$

$\underline{o}-\infty$

$\star e^{\underline{O}}o$

$\varpi^{Q)}\omega_{\star}--\overline{\frac{o}{B^{\partial}}}||Q)$

$\frac{\infty}{\mathfrak{B}}\omega\aleph^{\infty\infty_{)}}\overline{qO}t_{\frac{\alpha v_{)}}{Q)}}$

$.-arrow r\Xi-$$\tau 6rightarrow$

$r_{rightarrow}= \overline{\geq}\dot{o}oo\tau\infty_{6}\zeta\omega \mathfrak{c}n\frac{\overline{o}}{\overline{\geq}}$

$\not\in_{Q)}^{o}arrow\partial Q)\geq\prime \mathfrak{o}^{\backslash }\approx \mathfrak{t}||_{\ni}^{\prime\cdot.e}\frac{s}{1}\overline{\infty^{\infty}}\underline{t0\Xi O}$

$\approx ol6$ $\sim\dot{o}0$

$a_{\overline{e}}$ $O\wedge 0\omega v||$

$ff\omega 6N^{\wedge}\overline{to^{S}}o_{\wedge}E$

.$\overline{h}\dot{o}0\frac{r_{U}}{v}-\aleph$

$0$

3$\overline{(\supset 0_{1}}$ $\mathring{o_{1}^{1}\supset}o$

3$\overline{\zeta\supset_{1}\circ}$ $\zeta\supset\circ\circ\infty_{1}$

3$\overline{\{\supset\circ_{1}.}$ $\mathring{(\supset}o_{1}^{t}$

$\dot{p}_{\dashv\approx^{\ni t}}^{\underline{b}\dot{D}^{-\delta}}\circ i_{|.|_{\sim}}\underline{S_{0}}\frac{c\frac{R}{3}}{-t.)}||$

212

$s^{\dot{\underline{O}}}rightarrow=\omega\backslash \dot{\underline{o}}$

$-\infty s$$0\infty\omega\geq$. $rightarrow 60$

$\underline{\wedge\circ}$ $0\underline{.}r\Xi\infty$

$||r\circ$

$\approx\underline{to}$

$r\circ^{\circ}\approx a^{co}\overline{|}_{\dot{\circ}}^{\overline{\infty}}10$

$\overline{o^{\wedge}o}\Phi^{\wedge}||$

$Q’\overline{g}$

$\mathring{o^{i}(\supset}$ $\overline{\zeta\supset\circ}$

$\zeta\supset$

$\tilde{c_{)}\subset.}$ $\overline{o\dot{o}}$

$c$

$\circ\zeta\supset\infty$$\overline{\langle\supset(\supset}$

$c$

$\dot{o}E\underline{\iota 0}$

$i3\approx$ $\approx\approx$

$\approx\approx$$|_{o^{\underline{\dot{U}}}}|r\aleph_{6}$

$\iota ocp\omega$ a$0^{\wedge}\infty$

. $3^{>}q||$

$0-r\circ\underline{10}$

$||-0_{10}$

$-r_{U}$$\iota 0\approx v$

$-(6 \frac{\aleph}{\infty}$

$-\cup\wedge$ . $\underline{\infty\omega}T^{\wedge}\omega$

$\sim_{\approx}^{6}o\overline{e^{\omega}}x:\in-$

$\approx 03^{\triangleright}\dot{\Xi}$

$P^{\circ}-\omega$

$r_{\star U}\Xi_{\partial’}\omega^{\bigvee_{\wedge}}rightarrow\triangleright\zeta n^{>}\ddagger l1$

$0$$\circ C\aleph(\supset_{1}\circ i30_{\}}\triangleleft_{1}\subset 0.$

$(\supset c_{1}o\infty.$

$\langle\supset$

$(\supset_{1}\circ\dot{\circ}N\approx^{g}c_{1}o(\supset\triangleleft$

$c_{1}c^{\rangle}o(\supset$

$0$$(\supset o_{1}c\supset\infty$.

$\approx c_{1}o(\supset\triangleleft$

$\mathring{\circ\circ}(\supset_{1}\zeta$

$.\approx\triangleright\underline{.\underline{G}}\underline{O}-$

$Od\overline{q^{\omega}}\omega\S$

$arrow\partial\prime oQ)$

$(6\infty-A\infty$

$\approx\omega\underline{\circ}_{\overline{O}}^{\wedge}i$

$\sim\omega$

$\approx ocp3r_{arrow}\sim\Xi\dot{d}$

$–$$ol6\omega\infty$

$-\vee\underline{\wedge}$

$\underline{o}.ad^{-}\infty$

$+\cdot-$

6 $\infty$. $r\circ-$

$\dot{o}0\underline{-}$

$\Omega 6\infty^{\wedge}|1$

$\succ 0$. $\prime t|$

Q) $0$

$A_{-}\prec 1\mathring{\circ}^{\wedge\infty_{-}}$

$\langle\subset_{\supset}^{\circ}c_{\supset^{\rangle}}$

(

$\dot{o}c\triangleleft oo$

t-

$\overline{\langle\supset oo}$

$\subset$

$(o\mathring{o}c_{\supset}$

$0\circ o(\supset \mathfrak{t}-\overline{\langle\supset\circ\zeta\supset}$

$0$

$0^{0}\mathring{o(}\supset$

$t\supset\circ N(\supset\triangleright\overline{\supset c(\supset}$

$0$

$*^{\circ}\geq\geq 0_{||}\dot{o}arrow\partial^{\wedge}qg6_{)}$

$\star\omega_{\partial}g\infty^{>}\succ\infty$

$arrow\omega-$

$\approx 0\mathfrak{c}n$ t\S$Or\Omega\omega\underline{d\omega}$

$\overline{6}\Xi\overline{\approx}$. $\approx 0$. $\overline{.}d\approx$

$\underline{\overline{\mathscr{C}}}$

$’\varpi\approx\approx vA^{Q)}$

$\ulcorner\Xi U1arrow$

$\succ.\triangleleft\backslash$

$Oc\dot{o}S_{\omega}^{\approx}\approxarrow^{\frac{o\approx}{\partial}}0q_{)}\dot{s}^{)}t\omega$

$\dot{h}_{arrow\partial}^{\frac{b}{\}\dot{A}}\dot{o}^{\omega}\frac{\approx}{\infty 0}\backslash \frac{q\prime}{Q)}$

213

$\omega s_{}$

Q) $0$

$\dot{E}s$

$arrow 0$

$\overline{\underline{\tau}}$

$\wedge\underline{\overline{o}}\frac{+_{z^{\partial}}}{(o_{o}}\underline{(.)q)}$

$0\infty\dot{o}arrow\simeq\omega t_{\frac{vv}{\omega}}$

$||\circarrow\omega\infty\dot{o}$

$1\mathfrak{d}6\backslash$

$\underline{\circ}r\dot{e}\overline{\geq}$

6 $\underline{-}0$

$(\supset\circ\infty$. $\overline{\circ(\supset}$

$0$

$\circ\dot{\circ}C\aleph$$\overline{\supset\circ.}$

$0$

$\dot{\circ}\circ N$ $\overline{\dot{o}c}$

$\overline{\dot{o}^{\wedge}0}|A_{\underline{\infty}}\infty$

$a\approx$$3\approx$

$a\approx$

$11$

$\overline{\infty}$

$t_{\mathfrak{d}}^{\circ}\omega^{)}q^{\wedge}\circ\dot{o}$

$\circ l^{\sim m}||$

$\dot{o}0\underline{\dot{B}}\overline{\S}$

$||\not\in^{\omega\underline{o}}1$

$\underline{\wedge 0}$

$\underline{t_{\overline{\mathfrak{d}}}}3\infty\Xi S$

$\infty v\overline{\approx}||$

$ae\sim 10$$r\circ\#^{-}$

$\Leftrightarrow 6\circ_{10}$

$0\approx r_{O^{\wedge}}\omega$

$r\circ\overline{v}\underline{\aleph}$

$A_{3}^{\Phi}+B_{=^{\sim}A^{\Phi}}^{\infty}$

$0$$0_{1}\infty\dot{o}oi3^{g}0_{)}^{}\triangleleft_{1}\subset 0$

$\mathring{o_{1}o\dot{o}}$

$0$$0_{1}\infty\dot{o}0\dot{\approx}0_{1}\triangleleft\dot{o}o$

$\mathring{o_{1}\dot{o}0}$

$0$$o_{1}o\infty(\dot{=}a^{s}o_{1}^{t}o\triangleleft(\supset$

$\zeta\zeta\mathring{\circ_{\rangle}\subset_{1}^{\supset}}$

\={o} $\overline{.}E-$

$\#^{-}\dot{\Xi}$

. $\underline{o}r\circ^{\wedge}Q$)

$\star\dot{6}\overline{\omega}arrow\infty$

$\infty\zeta R\infty*$

$\overline{Q)}t_{\neg}$

$’\varpi-6$$\Leftrightarrow P^{\circ}Q)$

$0-\wedge\underline{.\underline{-}}$

$-\varpi$$o\overline{Q)}A^{\Phi}$

$\overline{-\prime\circ}$ $\approx r$

. $\underline{o}3$ \={o}$arrow\partial e-\sim\partial$

$-66$$O-A$$\alpha$ . $*$

$a-\infty$

$\ulcorner\omega\triangleright\frac{+\simeq_{\partial}}{\geq}\dot{o}^{!.\omega}\circ\star\circ\mathring{o}^{1_{r}^{\wedge}}.\underline{\underline{\ominus^{\dot{6}}}}$

$\#_{\Phi}^{\circ}\geq\circ^{\wedge}\mathring{\circ}1\prime l’$

’

$rightarrow 0\triangleleft$

$\approx\omega||\overline{6}$

$r_{R^{\vee}}o^{o}\ovalbox{\tt\small REJECT}\Xi^{\wedge}\omega$

. $\underline{.}\infty\infty$

$\omega\triangleright)$

$a_{\underline{e}}$

$r\circ r\circ(n$

$\underline{-}\approx g\iota 6$

$h\Leftrightarrow\underline{Q)}$

$O\approx--$

$Q)s$$\triangleleft j$

. $\sim_{-}^{tD}\underline{o}$

$\omega\Leftrightarrow v$

$\dot{\overline{h}}R\bigwedge_{arrow\partial}$

ロ $\infty$ $tO$ 寸 $O$ $\infty$ $\not\subset\rangle$ 寸 $O$ $\infty$ $C\rangle$ 寸$o$ $0$ $0$ $arrow$ $0$ $0$ $0$ $arrow$ $0$ $0$ ロ

$o$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $c$ $0$......ロ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ ロ

333

214

$rightarrow 6\prime rv$

$0-\in-$$|| \frac{\sim}{v}\wedge\dot{B}$

$s\not\in v$$t\neg$

$\vee-\circ$$’ \circ s\frac{3}{6}\infty\succ\infty$

\S --6$o_{||}\overline{\infty}r\frac{Q)}{=}$

$s_{\mathring{o}^{1_{rightarrow}}}^{\dot{s}^{o_{\wedge\neq^{v}}}}.-$

. $1_{\wedge\mathring{\circ}^{\wedge}}^{\wedge}$

)$1\circarrow 0_{\partial}6$

$c\triangleleft o(\dot{\supseteq}$ $\overline{o\{\supset}$

$0$

$\zeta\dot{\supseteq}\circ\circ\triangleleft$

$\overline{o\dot{o}}$

$0$

$\tilde{o\langle\supseteq}$ $\overline{\dot{o}0}$ $-or\Xi$$\approx\approx$

$i3\sim$$\approx\approx$

$\dot{o}_{||}0\dot{o}_{||}+_{\Phi}\infty 6$

$t_{0}^{N}\underline{g}_{r\underline{\dot{e}}}$

$\triangleleft\infty-$

$\approx-$

$a$Q)

$\infty 0$.$r_{\underline{\Xi}}\circ$

$|$

$0\approx r\circ$

$||\Leftrightarrow\overline{6}$

$\overline{\omega}H^{tD}Q)\Xi^{\wedge}$

$–\approx v$

$\underline{Q)\infty}\overline{S}arrow\partial\succ\infty^{>}\infty$

$\overline{t6}\omega$

$\sim Q)\dot{a}$

$\approx r\simeq Q)$

$\overline{\vee\circ}$

$\approxarrow\partial\Leftrightarrow$

$r^{\circ}o\overline{o}\dot{\overline{B}}$

$\backslash 0$

$A_{\approx}^{Q)}rightarrow\wedge\circ\triangleleft||arrow v-\simeq$

$08-$ \={o}$\frac{o\approx}{\star_{(.)}l}\underline{arrow\dot{e}}$ \={o}

$\#_{\underline{\Phi}}^{\Phi}\dot{o}_{||}0\frac{\star\overline{\approx^{\partial}}}{o,\infty}\frac{}{.Q)}Q)Q)$

$\infty\omega$ $sv\infty$$\mathfrak{B}^{\approx}’\varpi t\backslash s_{\partial}\star v$

$\dot{\overline{\sim}}6d\star^{\{}\omega_{\partial}^{l1}s^{\dot{\underline{O}}}$

$+_{\frac{A_{\partial}}{\geq\geq}}\circ^{\wedge}|_{\sigma y}|_{\underline{S}}^{r}\overline{U^{6}}A_{\infty}^{O}\overline{\geq}$

$\not\in_{\omega}^{o}\infty s^{s}|||\overline{\infty^{\underline{\infty}}0}$

$+_{\dot{\Phi}}$

$\tilde{s}\Phi^{\wedge}\dot{\circ}$

$0\approx 3\dot{\omega}||$

$\underline{\wedge 6}$

$\frac{o}{\frac{6}{\overline{\sim}}}\infty_{\Phi}Eo_{||^{\frac{\prime\dot{o}}{\not\in^{Q)}}\overline{\frac{()\epsilon}{\aleph\iota 6}}}}^{\wedge}\dot{o}$

$d$ $\ni\infty g$

$A_{b_{r}^{3}}30\circ\wedge||$

$\Leftrightarrow-(0$

6 –

$1 \dot{O}-\wedge^{\wedge}\frac{r\circ}{v}\infty_{1}$

$\frac{b}{\dot{h}}\dot{\mathfrak{v}}k||\not\in_{tarrow}Q)\frac{\aleph}{tD}$

215

$\wedge(\eta g\circ--3$.$||\dot{\vee\Phi}\Phi\underline{\circ 1}\sigma.Q$

$\frac{\aleph By}{o\iota}|$$- \frac{\in}{\Phi}y\Phi q$

$\frac{a}{||}\overline{\frac{\overline‘}{()}}L\overline{[be]}\overline{o}\vdash 3\circ\leq$

$\omega^{P}$ , $\omega^{\aleph}$$\omega^{\aleph}$

$\vee-\Phi\omega oy\neq\Phi\epsilon\sim vy_{1}$

$0$$\dot{\omega}oo$ $\dot{\omega}\circ\circ$ $o\dot{o}\backslash$

$\Phi 9\infty ro\Xi pO\epsilonarrow\infty\sigma\circ_{1}\Phi g=y$

$\leq 0\frac{o}{\Leftrightarrow}a^{\S}$

$s\underline{\epsilon}\succ$. $oo$$\mapsto,$

$0\Leftrightarrowrightarrow$ ,$os\sigma_{\epsilon^{Q}*}\# v$

$\overline{\Phi}\underline{o}\Phi\#\epsilon+$

$\frac{\Phi}{\frac{\Phi}{\Phi^{)}\ulcorner}}\epsilon\succ_{d}\frac{\neq\Phi}{o}E\frac{\Phi\infty\epsilon*\cap}{}\frac{o}{\mathfrak{c}n}\cdot\frac{\iota_{\frac{\circ}{y\Phi\epsilon\sim\infty}}}{\mathfrak{Q}}$

$\overline{\Phi}\underline{\aleph}$ $||$

9$||\vee^{-}$

$\sim\infty\omega_{\vee^{\circ}}\epsilon\simrightarrow$

$\Phi\vdash||$

$\vee B\sim^{\circ}U\triangleright\sim\vee^{-}$

$\epsilon g_{L}xgg\sim$

$|t1$

$\vdash\circ(-\eta$

$\frac{1}{-}\cdot\tilde{\omega}^{J}||$

$\frac{f}{()}L\circ y^{1}\infty^{0)}$

$\epsilon_{\circ}arrow\circ\infty(\dot{\Phi\infty\Xi}\cdot 0||$

$e\sim 0\mapsto$

$r\leq-$9 $s\mu$,$\epsilon+$

$0$

$0.,$$v\epsilon*_{\vee}-\epsilon\star_{4}$

$\dot{P^{\sim}\Phi}\Phi\Xi^{d}\dot{\Phi\Phi}$

$–()oR$$\overline{\Phi}$ : $\Leftrightarrow$

9 $\infty\Phi a\not\in$

$r\infty$$\infty 0\Phi$

$<\prec ss$$\infty\subset L$

$\epsilon*-\cdot\Leftrightarrow$

$\Phi B\overline{\sigma Q}B\not\in$

$’\underline{\circ}\propto\Phi$

$\succ\exists\cap y\infty$

$\Phi\neq r\Phi_{1}-$

$\frac{\infty}{N}\cdot--\cdot F$

$(b\infty\epsilon\tilde{\neq}\Phi||$

$()y^{)}\propto 0\infty\dot{o}0$

\={o}. By $\vee 0$

$0)_{\circ}\overline{-}\vee-$

216

$(\eta\approx oq$$\infty 0^{\cdot}$

$\wedge t\eta B\vee^{-\sigma Q}$

$||-\underline{y}$ .$\circ 1\circ\mathring{\sigma}$

$yBE–$$\aleph\Phi$ .. $hj$

$\overline{\circ)}\vee\Phi$ -i \={o}

$\frac{3}{o||}|$ $\overline{\frac{\Phi}{[be]}}y\circ\epsilon*\infty\leq$ $\omega^{\aleph}$

$\dot{\infty}oo$

$\omega^{\aleph}$

$\dot{c}\mathfrak{n}oo$

$\omega^{\aleph}$

$0\dot{c}\mathfrak{n}$

$—-\cdot\vdash\exists y$

$\frac{y}{r\infty\Phi}\infty\epsilon*\Phi\frac{\circ}{o}y\cap\Phi P\tau y:y-$

$o\epsilon\sim\Leftrightarrow e\prec$

$\underline{\leq}\Phi r\overline{\infty\Phi}\wedge 0$

$\frac{\frac{Or}{}}{.\Phi}*\frac{\frac{o\infty}{\epsilon*\approx}}{o}\cdot\sigma^{o_{Q\underline{\circ}}}s_{\iota^{\epsilon*}}^{y}a^{g}$

$\Phi\Leftrightarrow\epsilon*y$

$\overline{\frac{\Phi}{(\Phi^{\backslash }}}\Phi^{arrow}\frac{o}{r\epsilon}\backslash cv^{\zeta}\alpha_{\}}^{)}\frac{\epsilon_{0}*\Phi}{\mathfrak{Q}}r\Phi$

$\Leftrightarrow\overline{o}arrow||$

$os$$\Leftrightarrow\infty\vee^{-}$

$\underline{y}P_{\underline{\aleph}}\overline{\Phi}||\propto||$

$*\infty\infty\vee\sim 0-$

$\Phi\epsilon*+^{\iota\circ}\vee$

$\vee B\circ y$

$\underline{y}\vee\triangleright\overline{\Phi}$

$\sim g\underline{o}$

’ $\approx$

$-’-”’\neg^{t\circ}\circ\eta+||$

$t\backslash \triangleright C’ 1\infty||$

$\infty\Phi\epsilon\sim y\Phi y-\underline{o}$

$\underline{o\epsilon\succ yPe*}_{*}\underline{\leq}PO\infty-\epsilon\succ\frac{\wedge 0}{P}$

$\neq\Phi\epsilonarrow\dot{P}\Leftarrow\succ_{\vee}yR\Phi\Phi$

$–\Phi\Leftrightarrow$

$\overline{\Phi}()\Leftrightarrow$

$q^{\frac{v}{\infty}}\prec\infty\Phi:0\underline{\Phi\infty}\sim$

$\infty\circ\Leftrightarrow$

$\Phi e+0\approx$

$B–\approx\subset B_{r}$

$\Leftrightarrow\Phi$

$\succ\exists\sigma Q\overline{\infty}$

$\Phi\circ\neq$.$rightarrow-$

$\frac{\infty}{\infty\Phi\aleph}\cdot\infty\Phi gg||$

( $–\cdot$

$\frac{y}{\infty}(b\vee^{\circ}r_{\omega}^{arrow\dot{o}}e$

217

$r\overline{\infty\infty\leq 0}\underline{|}_{\underline{\epsilon}}^{-}--\overline{\sigma_{\circ)0^{Q}}}^{-3}$

$\Leftrightarrow\overline{[be]}\Phi\#$

$\overline{\underline{o}}’\overline{y\circ}\overline{x}$ 可$\epsilon\sim$ $0$

$\overline{\Phi}\Phi\zeta D\vdash\exists\leq$

$\overline{\Phi}\epsilon*P_{tO}\Phi$$\omega^{\aleph}$ $\omega^{\aleph}$

$\omega^{\aleph}$

$\dot{\frac{\Phi}{\Phi()}}-\Phi\Phi\circ\underline{\#\circ}O\mathfrak{c}n\Xi g^{+}e\Re$

$\dot{c}o_{n}o$ $\dot{\circ}\circ_{1}\circ$ $\dot{c}0_{n}$

$\underline{\epsilon}arrow$. $\mathfrak{c}t$) $y$

$\underline{o}$ 巴:$O_{\text{化}}\underline{O}$

$<<y$

$\dot{P}^{\vee}+\sigma_{\epsilon^{Q}}g$

$\Phi\Xi\cdot g$

\={o} $\infty\Phi$ 監$B\circ\Phi\underline{\circ}$

$\overline{\overline{\underline{\Phi y}}}\frac{\epsilon}{\frac{o}{\infty}}\sim\infty\Phi\epsilon*9$

$\infty\aleph$ ロ$\prec-$

$\Phi\inftyarrow||$

$||$

ヨ $\vee 0_{\vee}-$

$y+\circ$$\Leftrightarrow 0||$

ロ $\vee\dot{\triangleright}-$. $y\sim\omega$

$’|$ $s\vee$

’ a $y$

$\overline{\overline{P}}$

憶oes

c) $-J\underline{O}$)

$\epsilonarrow y\triangleright 0$

$\zeta’)\Phi\circ 1||$

$r\epsilonarrow\frac{y}{\Phi}\omega^{0)}$

9 $r!$) $||$

$\epsilonarrow r$

$ooo$$\mapsto\leq-$$\epsilonarrow s-$

$–\Phi\neq$ 窪 $\overline{\underline{o}}$

’

$\overline{\frac{v\Phi}{v)}}(b()or\epsilonarrow\overline{\Phi\Phi}r\epsilonarrow$

嫁 : 禾$\epsilon+\Phi\infty\Phi s$

ヨ $r_{o}L\approx$

$s\infty$

$\vdash\exists-\sim\underline{\Phi}$

$\Phi r_{\overline{\sigma Q}}\overline{\not\in}$

$\circ\Phi_{)_{\aleph}}\frac{\infty}{\aleph}\frac{\approx}{\Phi_{1}g,\circ}\frac{\Phi cB}{\infty}r$

$\overline{\mathfrak{w}||}\Xi\epsilon-\sim_{d}-\overline{g}$

$vB\propto\Phi 0||$

$\aleph 0$ .$\underline{-3^{t\eta}}\underline{gB}$ . $\vee^{\circ}0\omega$

$o||\underline{o}oE^{\overline{\vee}}$

.$\vee\vee\underline{O1}$

218

$P^{l}\overline{\infty^{l}\mathfrak{c}0\leq}\underline{|}_{\omega^{\mathcal{Q}}}\overline{<}_{\overline{\sigma}_{\mathring{k}}^{q_{)}}}Q$

$\wedge 0$

凸 $\overline{\frac{\Phi}{a}}\dagger\tau$

$rightarrow 9$ .$\epsilon+$ \={o}

$\frac{\Phi}{t,.\Phi^{)}}\wedge\overline{\Phi\Phi}\frac{\neq\infty\Phi\circ}{\approx,\epsilon\star}\Xi^{\underline{y}}\omega_{arrow(}^{b}(\epsilon_{\Phi}\tau_{w}^{\Leftrightarrow\leq}o^{b\circ}r_{\iota_{\epsilonarrow}}y$

$\omega^{\aleph}$

$\dot{\circ}\backslash oo$

$\omega^{\aleph}$

$\dot{\circ}\backslash oo$

$\omega^{\aleph}$

$\dot{\omega}0$

\={o} $U1\underline{y}$

$s\underline{v}-$

$oy$$os\approx$$\epsilon r^{\backslash }\sigma Qo$

$F^{d}e*\mu$,

$\frac{b}{o}(\infty\not\in\Phi 9\epsilon\sim\#$

$-E\epsilon\succ(\rangle^{\underline{\circ}}\Phi_{l}$

$\overline{y\Phi}\overline{\frac{o}{y)}}\Phi\infty\epsilon*y$

$\approx\infty\underline{\aleph}\overline{\mathfrak{Q}}$

$\epsilon*\infty\Phi||$$||$

ヨ $\vee 0_{\vee}-$

$y\vdash\circ$

$s\circ||$$oe_{\vee}\dagger\triangleright\mapsto$

$1\prime 11$

$\underline{y}\backslash _{Q}\vee 1$. $a_{y}$

$\overline{\overline{c}}\triangleright_{)}$

憶 $\overline{-r\mathfrak{n}}g$

$\infty\Phi y\epsilon*\circ\triangleright 0_{1}||$

$\dot{P}^{\wedge}\Phi\circ\iota^{(}0^{\eta}$

窪 $p^{1}\sigma||$

$ooo$$P\epsilon\star^{\backslash }-\underline{\leq}--$

$-\Phi y\epsilon\sim O^{\}F\backslash$

$\frac{y}{\infty}-\Phi oo\Phi r\epsilon\succ\Phi\Phi re+$

$\iota<-$$\epsilon\sim\infty\dot{\Phi_{)}q}-\mathbb{R}$

ヨ $\approx oa\approx$

$s$ �

$\succ\exists-a$ 自$P\Leftrightarrow$

$\Phi\sigma Q\overline{\approx}$

$\frac{\infty}{\aleph}\cdot$ : ヨ$o)\Phi\Phi_{()}\infty\infty y\infty\Phi\circ$

$arrow-\cdot-$

$o_{||}\dot{\neq}\epsilon\neg F$

$B\nu\not\subset\Phi 0||$

$\aleph 0$ .一 B $\kappa 0$

$\underline{s^{0)}}\underline{\not\cong}\cdot\vee.\circ$

$o||\Phi\circ R^{\overline{\vee}}$

$\vee\vee\dot{\circ}1$