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 Engineering
Kyoto University, Kyoto 606-01, Japan Abstract
The validity of the linearized Boltzmann equation indescribing the behaviour
of rarefied gas flows that deviate only slightly from a uniform equilibrium state
is discussed on the basis of several numerical examples. Various examples of the
above situation are analyzed numerically by thefull nonlinear BKW equation and
also by its linearized version, with emphasis on the behaviour for small Knudsen
numbers, and the solutions of these equations are compared. When the Knudsen
number is comparable to or smaller than the degree of the deviation from the
uniform equilibrium state, the solution of the linearized equationgenerally differs
decisively from that of the nonlinear equation, however small the degree of the
deviation may be. Situations where the nonhnear effect degenerates are also
noted.
1. Introduction
The linearized Boltzmann equation, where the second and higher-order terms of the
perturbation 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 uniform
equilibrium state at rest. The situation is usually encountered in gas dynamic problems of
a small system such as in aerosol science and micromachine engineering, where the Mach
number 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 that
in some situations the linearized equation does not give a correct answer, however small the
deviation 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 small
Knudsen numbers, Sone ([Sone, 1971,$1984,1987,199la,b]$; [Sone
&Aoki,
1987]) discussedthe applicability of the linearized Boltzmann equation and pointed out that the nonlinear
effect in the Boltzmann equation is, generally, not negligible for any small deviation from a
uniform equilibrium state if the Knudsen number of the system is comparable to or smaller
than the degree of the deviation. There are, of course, various situations where the effect
of nonlinearity degenerates. The above infinite domain examples are also understood by
the general statement if the situation is properly interpreted. Recently Aoki and Masukawa
[1994] considered the two surface problem of evaporation and condensation and showed a
decisive difference between the numerical solutions of the linear and nonlinear equations for
very small temperature difference of the two boundaries.
The example, however, is a special one, where the flow velocity is uniform in the limit
of small difference of the surface temperatures. For more general understanding, in the
present paper, we consider several different situations whose deviation from a uniform
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 its
applicability with respect to the parameters: the Knudsen number (Kn) and the parameter
that 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 ofvarious results ofthe BKW equation and the standard
Boltzmann equation. In the analyticaldiscussion of the applicability ofthe linearized
Boltz-mann equation, which is done in connection with the asymptotic analysis of the Boltzmann
system for small Knudsen numbers, the BKW equation shows the same behaviour as the
standard Boltzmann equation (e.g., [S&A, 1987]; $[S$, 1987]; $[S,$ $199la,b]$). The results of the
linearized BKW equation are consistent with recent accurate numerical computations by the
hnearized Boltzmann equation for hard-sphere molecules (e.g., [Sone et al., 1990]; [Ohwada
et al., 1989]; $[S, 1991b]$; [Takata et al., 1993]). On the boundary the Maxwell type condition
or the conventional boundary condition of evaporation and condensation (e.g., [Cercignani,
1987]; $[S, 1987]$) is adopted as the kinetic boundary condition.
2. Basic
equation
andnotations
Let $p_{0}$ and $T_{0}$ be the pressure and the temperature of the uniform equilibrium state at
rest. When we consider problems with evaporation or condensation on a boundary, $p_{0}$ is
the saturated gas pressure at temperature $T_{0}$. The density $\rho_{0}$ and the velocity distribution
function $f_{0}$ 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 in
the 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 of
the equilibrium state, which is the ratio of the mean molecular speed $(8RT_{0}/\pi)^{1/2}$ and the
meancollision hequency. In the presentpaper, we use thefollowingnondimensional variables
based on the above basic variables of the system: Kn $=\ell_{0}/L;x;L$ is the Cartesian coordinate
system 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; $\zeta=(\zeta_{i}^{2})^{1/2};f_{0}(1+\phi)$ is
the 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 normal
vector to the boundary, pointed to the gas; $(2RT_{0})^{1/2}u_{wi}$ is the velocity of the boundary
with $u_{w};n:=0$ (this is required in steady flow problems); $T_{0}(1+\tau_{w})$ is the temperature of
the boundary; $\rho_{0}(1+\sigma_{ws})$ and $p_{0}(1+P_{ws})$ are, respectively, the saturation gas density and
pressure 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 $\theta$ components
of $u_{i}$ 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) $\zeta_{i}\frac{\partial\phi}{\partial x_{i}}=\frac{2}{\sqrt{\pi}Iffi}(1+\omega)(\phi_{e}-\phi)$,
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 is
called diffuse reflection when $\alpha=1$, and specular reflection when $\alpha=0$. The conventional
condition of evaporation and condensation on an interface between a gas and its condensed
phase, 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 aparameter. 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 to
say, $u_{w}$
:
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:
(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 of
which 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 two
systems. Our interest is the behaviour of a slightly nonuniform state, i.e., for small values
of $\epsilon_{1},$ $\epsilon_{2}$, and $\epsilon_{3}$. The method of numerical computation is a straightforward application of
that in [Aoki, et al., 1991]. Thus, it is not repeated here, and only the results of computation
are 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 ofdeviation from our
reference uniform equilibrium state at rest. This notation will also be used in the following
examples. When Kn $=0.2$, where Kn is fairly larger than $\epsilon(=0.02)$, the profiles of the two
systems 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 becomes
appreciable and it is fairly larger than $\epsilon^{2}$ [Fig. 1
$(c)$]. When Kn $=0.002$, where Kn is fairly
smaller than $\epsilon$, the two solutions are markedly different, and the difference is obviously
larger than $\epsilon^{2}$ by far. The linear solution cannot
be considered to be an approximation of
the 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 the
boundary, thesolutionis not uniquelydetermined bythe diffusereflectioncondition; Another
condition relating to the mass ofthe gas in the domain is required for the uniqueness. Here
we 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 from
that of the nonlinear equation is uniformly very small with respect to the Knudsen number,
4. Cylindrical Couette flow with
evaporation or
condensationon
the boundariesHere 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 of
which 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 $r=2$. We
numerically solve the nonhnear system, Eqs. (3)$-(5d)$ with Eq. (8) at $r=1$ and $r=2$, and
the linear system, Eqs. (10)-(11d) with Eq. (14) at $r=1$ and $r=2$ for various sets of the
parameters Kn, $\epsilon_{1},$ $\epsilon_{2}$, and $\epsilon_{3}$. The method of computation is a straightforward application
of 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 in
the 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 much
smaller than $\epsilon_{1}$ (or $u_{\theta}$). Incidentally, the negative temperature-gradient phenomenon is seen
in 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 the
case $\epsilon_{1}=0.02,$ $e_{2}=0.01$, and $\epsilon_{3}=0.02$. The feature of the difference of the two solutions
is 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 phenomenon
is not seen in these examples [Fig. 4 $(b)$].
An example without evaporation and condensation on the boundariesis shown in Fig. 5,
where the case of$\epsilon_{1}=0.02,$ $\epsilon_{2}=0.01$ and diffusely reflecting boundary are considered. As
in 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 three
Knudsen numbers Kn $=0.005,0.02$, and 0.2.
5. Flow past
an array
of flat platesHere 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 the
difficulty [Sone&Takata, 1992] ofnumerical computation owing to the discontinuity of the
velocity distribution function around a convex body, we investigate the following somewhat
artificial problem in a rectangular domain $(-a<x_{1}<a, 0<x_{2}<b)$.
(i) Nonhnear problem: The basicequation isgiven by Eqs. (3)$-(5d)$. The boundary condition
is 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 $\alpha=0$ (specular
reflection); $\phi((1>0)$ on $(x_{1}=-a, 0\leq x_{2}\leq b)$ and $\phi(\zeta_{1}<0)$ on $(x_{1}=a, 0\leq x_{2}\leq b)$ are
(ii) Linear problem: The basic equationis givenby Eqs. (10)-(11d). The boundary condition
is 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 given
by 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 angle
of attack $(-1/2\leq x_{1}\leq 1/2, x_{2}=2mb, m=0, \pm 1, \cdots)$. The upstream and downstream
regions 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 convex
points of the boundary. The present choice, Eq. (16a), of the accommodation coefficient
avoids the discontinuity at the leading and trailing edges of the plate, which are the only
convex points of the boundary. Thus the velocity distribution function is continuous in the
gas.
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. $6a-6d$. As
in the examples with evaporation and condensation in Secs. 3 and 4, the deviation of the
hnear solution from the nonlinear solution increases as the Knudsen number decreases, and
the differences in $\omega$ and $\tau$ of the two solutions obviously exceed $e^{2}(e= \max|\epsilon_{m}|=0.1)$ for
KJ $=0.1$ 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 close
to a uniform equilibrium state at rest, and investigated the flows numerically on the basis
of two types of basic equations: the (original nonlinear) BKW equation and its linearized
version. The results are compared, and the validity of the solution of the linearized equation
indescribing 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 nonlinear
equation. As the Knudsen number decreases, the deviation of the hnear solution from the
nonlinear solution generally increases. It is fairly larger than $\epsilon^{2}$ when $Iffi\sim\epsilon$, and the two
solutions are quite different when $K_{J}\ll\epsilon$. Thus for Kn$<\sim e$, the nonlinear solution cannot be
obtained bya simple perturbation analysis from the linear solution. In some cases, however,
the linear solution is a good approximation to the nonhnear solution irrespective of the
Knudsen number (see the second example of Sec. 3 and the last example of Sec. 4). This is
discussed below.
General theoretical discussion of the importance of the nonlinear term even in the case
where the system deviates slightly from a uniform state was made in $[S, 1971]$ in
connec-tion with asymptotic analysis of the Boltzmann equaconnec-tion for small Knudsen numbers (see
also [$S$, 1978, 1984, 1991ab]). The present numerical computations give good examples of
the theoretical discussion. The ratio of the Knudsen number and the nonuniformity
pa-rameter determines the validity of the linearized Boltzmann equation. Since the case with
small nonuniformity parameters is concerned, the ratio takes various values only for small
Knudsen
numbers. Therefore, the situation can be clarified by the analysis of the case withsmall Knudsen numbers, which admits a macroscopic description. Thus, the degeneracy of
the nonlinear effect is easily surveyed by the macroscopic description. That is, the leading
nonlinear term in the macroscopic description is the convection term of the
therefore the linearized equation gives a good description in the case where the convection
term degenerates or is incorporated in the pressure term in the Navier-Stokes system.
Thelast statement is well exemplified in our numerical computation. In the Couette flow
under diffuse reflection in Sec. 3, theconvectionterm vanishes, and in the cyhndrical Couette
flow under diffuse reflection in Sec. 4, only non vanishing part ofthe convection term, $u_{\theta}^{2}/r$,
can be incorporated in the pressure term. In both cases the deviation ofthe hnear solution
from 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 therefore
the leading convection term of the incompressible Navier Stokes system is $u_{1}\partial u_{1}/\partial x_{1}$ in
the $x_{1}$-momentum equation and $u_{1}\partial\tau/\partial x_{1}$ in the energy equation. The $u_{1}\partial u_{1}/\partial x_{1}$ can be
incorporated in the pressure term, but $u_{1}\partial\tau/\partial x_{1}$ is left as it is. Since the velocity field
can be solved independently from the temperature field in the incompressible Navier-Stokes
system, according tothe statement the velocity field of the linear equation should be a good
approximation, but its temperature field may deviate considerably from that of thenonlinear
equation. Figures $6a-6d$ support this.
In some infinite-domain problems, the discrepancy of the linearized equation such as
Stokes paradox $[C, 1968]$ is encountered for arbitrary Knudsen numbers. From the following
reason, this is also the same kind of difficulty of the hnearized equation as that in the
present examples. In these problems, the solution is supposed to approach a uniform state
at infinity, and the length scale of the variation of the solution increases with the distance
from a body. Then the effective Knudsen number (the mean free path divided by the local
length scale ofvariation), which determines the variation of the variables, decreases to vanish,
and it becomes much smaller than the small nonuniformity parameter in the far field, and
therefore the criterion on discrepancy of linear solutions applies.
Finally, the computation was carried out by HP 9000730 and MIPS RS 3230 computers
atour laboratory and by FACOM VP-2600 computer at theData Processing Center of Kyoto
University.
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Sone, Y. and Takata, S., 1992, Discontinuity of the velocity distribution function in a
rarefied gas arounda convexbody and theSlayer at the bottom of the Knudsen layer,
Transp. Theor. Stat. Phys. 21, 501-530.
Sone, Y., Takata, S., and Ohwada, T., 1990, Numerical analysis of the plane Couette flow
of a rarefied gas on the basis of the linearized Boltzmann equation for hard-sphere
molecules, $Eur$. J. Mech., B/Fluids 9, 273-288.
Sugimoto, H. and Sone, Y., 1992, Numerical analysis ofsteady flows of a gas evaporating
from its cyhndrical condensed phase on the basis of kinetic theory, Phys. Fluids A 4,
419-440.
Takata, S., Sone, Y., and Aoki, K., 1993, Numerical analysis ofa uniform flow of a rarefied
gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules,
Phys. Fluids A 5, 716-737.
$q$ $s$ 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$$\urcorner 0^{\frac{\not\supset}{‘ 6}}\circ\dot{o}\cdot\dot{o}^{\infty}\overline{o}^{\wedge\overline{|}_{\overline{\infty^{\underline{\infty}}}}^{E}}v\Phi^{\wedge}$
$||E$ $||$ $\omega_{\circ}^{\circ}r\overline{\S}$ $-\overline{Q)}1-)$ $0\infty\not\in\aleph$ $\dot{o}\alpha$ ae 3
a
$\wedge$ $||\overline{\triangleleft}||$ $1\overline{0}^{-}10$ $-\circ-\infty So$ $v610$ $.–$ a $o^{\wedge}N\Phi$ $\sim_{\frac{-}{o}}\not\in_{arrow r}^{\overline{Q)}\overline{\infty}}\Xi\Phi$ $r\circ 3\in-$ $Q)-$ . $A_{\vee}^{t\lrcorner}$a
.
$\overline{\frac{o}{\underline 0}}\frac{\triangleleft}{\not\in^{Q)}}\approx\wedge+_{\infty_{l1}^{\partial}}^{Q)}\iota_{6}$ $\prec\underline{q)}\dot{d}\infty_{\overline{-\circ}}$ . $e-rightarrow$ $\approx^{Q)}\triangleleft A^{q)}\wedge$ $\Leftrightarrowrightarrow\prec$.
$oB^{\Phi}$ \\={o} $\underline{\wedge p}$ $\underline{\overline{o}}_{\frac{3}{6}\neq^{ae}}^{}$ $0-\infty$ $\overline{\star e^{\partial}}-\omega^{o}\{$ \={o} $\circ!arrow\iota 6$a
$o\underline{(.)}$ $\triangleright Q)l60\infty^{\wedge\sim_{-}}-$ $Ao$ $\frac{\star^{l}}{\geq}\infty^{\wedge}0’\prime \mathfrak{l}$ $0\triangleleft$$\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}
$\Leftrightarrow\approx\iota 0$ 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 $\overline{\{\supset\circ_{1}.}$ $\mathring{(\supset}o_{1}^{t}$
$s^{\dot{\underline{O}}}rightarrow=\omega\backslash \dot{\underline{o}}$ $-\infty s$ $0\infty\omega\geq$ . $rightarrow 60$ $\underline{\wedge\circ}$ $0\underline{.}r\Xi\infty$ $||r\circ$ $\approx\underline{to}$ $r\circ^{\circ}\approx a^{co}\overline{|}_{\dot{\circ}}^{\overline{\infty}}10$ $\overline{o^{\wedge}o}\Phi^{\wedge}||$ $Q’\overline{g}$ $\mathring{o^{i}(\supset}$ $\overline{\zeta\supset\circ}$ $\zeta\supset$ $\tilde{c_{)}\subset.}$ $\overline{o\dot{o}}$ $c$ $\circ\zeta\supset\infty$ $\overline{\langle\supset(\supset}$ $c$ $\dot{o}E\underline{\iota 0}$ $i3\approx$ $\approx\approx$ $\approx\approx$ $|_{o^{\underline{\dot{U}}}}|r\aleph_{6}$ $\iota ocp\omega$
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$$\omega s_{}$ Q) $0$ $\dot{E}s$ $arrow 0$ $\overline{\underline{\tau}}$ $\wedge\underline{\overline{o}}\frac{+_{z^{\partial}}}{(o_{o}}\underline{(.)q)}$ $0\infty\dot{o}arrow\simeq\omega t_{\frac{vv}{\omega}}$ $||\circarrow\omega\infty\dot{o}$ $1\mathfrak{d}6\backslash$ $\underline{\circ}r\dot{e}\overline{\geq}$ 6 $\underline{-}0$ $(\supset\circ\infty$. $\overline{\circ(\supset}$ $0$ $\circ\dot{\circ}C\aleph$ $\overline{\supset\circ.}$ $0$ $\dot{\circ}\circ N$ $\overline{\dot{o}c}$ $\overline{\dot{o}^{\wedge}0}|A_{\underline{\infty}}\infty$ $a\approx$ $3\approx$ $a\approx$
$11$ $\overline{\infty}$ $t_{\mathfrak{d}}^{\circ}\omega^{)}q^{\wedge}\circ\dot{o}$ $\circ l^{\sim m}||$ $\dot{o}0\underline{\dot{B}}\overline{\S}$ $||\not\in^{\omega\underline{o}}1$ $\underline{\wedge 0}$ $\underline{t_{\overline{\mathfrak{d}}}}3\infty\Xi S$ $\infty v\overline{\approx}||$ $ae\sim 10$ $r\circ\#^{-}$ $\Leftrightarrow 6\circ_{10}$ $0\approx r_{O^{\wedge}}\omega$ $r\circ\overline{v}\underline{\aleph}$ $A_{3}^{\Phi}+B_{=^{\sim}A^{\Phi}}^{\infty}$ $0$ $0_{1}\infty\dot{o}oi3^{g}0_{)}^{}\triangleleft_{1}\subset 0$ $\mathring{o_{1}o\dot{o}}$ $0$ $0_{1}\infty\dot{o}0\dot{\approx}0_{1}\triangleleft\dot{o}o$ $\mathring{o_{1}\dot{o}0}$ $0$ $o_{1}o\infty(\dot{=}a^{s}o_{1}^{t}o\triangleleft(\supset$ $\zeta\zeta\mathring{\circ_{\rangle}\subset_{1}^{\supset}}$
\={o}
$\overline{.}E-$ $\#^{-}\dot{\Xi}$ .$\underline{o}r\circ^{\wedge}Q$) $\star\dot{6}\overline{\omega}arrow\infty$ $\infty\zeta R\infty*$ $\overline{Q)}t_{\neg}$ $’\varpi-6$ $\Leftrightarrow P^{\circ}Q)$ $0-\wedge\underline{.\underline{-}}$ $-\varpi$ $o\overline{Q)}A^{\Phi}$ $\overline{-\prime\circ}$ $\approx r$ .$\underline{o}3$ \={o} $arrow\partial e-\sim\partial$$-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)$$a_{\underline{e}}$ $r\circ r\circ(n$
$\underline{-}\approx g\iota 6$ $h\Leftrightarrow\underline{Q)}$ $O\approx--$ $Q)s$ $\triangleleft j$
.
$\sim_{-}^{tD}\underline{o}$ $\omega\Leftrightarrow v$ $\dot{\overline{h}}R\bigwedge_{arrow\partial}$ロ $\infty$ $tO$ 寸 $O$ $\infty$ $\not\subset\rangle$ 寸 $O$ $\infty$ $C\rangle$ 寸 $o$ $0$ $0$ $arrow$ $0$ $0$ $0$ $arrow$ $0$ $0$ ロ
$o$. $0$ $0$. $0$ $0$. $0$ $0$. $0$ $0$. $0$ $c$. $0$
ロ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ ロ
$0-\in-$
$|| \frac{\sim}{v}\wedge\dot{B}$
$s\not\in v$
$t\neg$
$\vee-\circ$ $’ \circ s\frac{3}{6}\infty\succ\infty$
\S --6
$o_{||}\overline{\infty}r\frac{Q)}{=}$ $s_{\mathring{o}^{1_{rightarrow}}}^{\dot{s}^{o_{\wedge\neq^{v}}}}.-$ .$1_{\wedge\mathring{\circ}^{\wedge}}^{\wedge}$ ) $1\circarrow 0_{\partial}6$$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}$
.
$||\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-$$\infty 0^{\cdot}$ $\wedge t\eta B\vee^{-\sigma Q}$
$||-\underline{y}$. $\circ 1\circ\mathring{\sigma}$ $yBE–$ $\aleph\Phi$ .. $hj$ $\overline{\circ)}\vee\Phi$ -i \={o}
$\frac{3}{o||}|$ $\overline{\frac{\Phi}{[be]}}y\circ\epsilon*\infty\leq$ $\omega^{\aleph}$
$\dot{\infty}oo$ $\omega^{\aleph}$ $\dot{c}\mathfrak{n}oo$ $\omega^{\aleph}$ $0\dot{c}\mathfrak{n}$ $—-\cdot\vdash\exists y$
$\frac{y}{r\infty\Phi}\infty\epsilon*\Phi\frac{\circ}{o}y\cap\Phi P\tau y:y-$
$o\epsilon\sim\Leftrightarrow e\prec$
$\underline{\leq}\Phi r\overline{\infty\Phi}\wedge 0$
$\frac{\frac{Or}{}}{.\Phi}*\frac{\frac{o\infty}{\epsilon*\approx}}{o}\cdot\sigma^{o_{Q\underline{\circ}}}s_{\iota^{\epsilon*}}^{y}a^{g}$
$\Phi\Leftrightarrow\epsilon*y$
$\overline{\frac{\Phi}{(\Phi^{\backslash }}}\Phi^{arrow}\frac{o}{r\epsilon}\backslash cv^{\zeta}\alpha_{\}}^{)}\frac{\epsilon_{0}*\Phi}{\mathfrak{Q}}r\Phi$
$\Leftrightarrow\overline{o}arrow||$ $os$ $\Leftrightarrow\infty\vee^{-}$ $\underline{y}P_{\underline{\aleph}}\overline{\Phi}||\propto||$ $*\infty\infty\vee\sim 0-$ $\Phi\epsilon*+^{\iota\circ}\vee$ $\vee B\circ y$ $\underline{y}\vee\triangleright\overline{\Phi}$ $\sim g\underline{o}$ ’ $\approx$ $-’-”’\neg^{t\circ}\circ\eta+||$
$t\backslash \triangleright C’ 1\infty||$
$\infty\Phi\epsilon\sim y\Phi y-\underline{o}$
$\underline{o\epsilon\succ yPe*}_{*}\underline{\leq}PO\infty-\epsilon\succ\frac{\wedge 0}{P}$ $\neq\Phi\epsilonarrow\dot{P}\Leftarrow\succ_{\vee}yR\Phi\Phi$ $–\Phi\Leftrightarrow$ $\overline{\Phi}()\Leftrightarrow$ $q^{\frac{v}{\infty}}\prec\infty\Phi:0\underline{\Phi\infty}\sim$ $\infty\circ\Leftrightarrow$ $\Phi e+0\approx$ $B–\approx\subset B_{r}$ $\Leftrightarrow\Phi$ $\succ\exists\sigma Q\overline{\infty}$ $\Phi\circ\neq$
.
$rightarrow-$ $\frac{\infty}{\infty\Phi\aleph}\cdot\infty\Phi gg||$ ( $–\cdot$ $\frac{y}{\infty}(b\vee^{\circ}r_{\omega}^{arrow\dot{o}}e$$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}$
$\circ\Phi_{)_{\aleph}}\frac{\infty}{\aleph}\frac{\approx}{\Phi_{1}g,\circ}\frac{\Phi cB}{\infty}r$ $\overline{\mathfrak{w}||}\Xi\epsilon-\sim_{d}-\overline{g}$ $vB\propto\Phi 0||$ $\aleph 0$ . $\underline{-3^{t\eta}}\underline{gB}$. $\vee^{\circ}0\omega$ $o||\underline{o}oE^{\overline{\vee}}$ . $\vee\vee\underline{O1}$
$P^{l}\overline{\infty^{l}\mathfrak{c}0\leq}\underline{|}_{\omega^{\mathcal{Q}}}\overline{<}_{\overline{\sigma}_{\mathring{k}}^{q_{)}}}Q$ $\wedge 0$ 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 $P\Leftrightarrow$ $\Phi\sigma Q\overline{\approx}$ $\frac{\infty}{\aleph}\cdot$
