熱対流問題の解に対する計算機援用証明
Some
Computer
Assisted Proofs for Solutions of the Heat
Convection
Problems
渡部
善隆
\dagger
中尾
充宏
\ddagger
山本
野人
*
西田
孝明
*
Yoshitaka
Watanabe
Mitsuhiro T.Nakao
Nobito Yamamoto
Takaaki Nishida
\dagger
九州大学情報基盤センター
\ddagger
九州大学大学院数理学研究院
*
電気通信大学情報工学科
*
京都大学大学院理学研究科
概要
This is
acontinuation
of
our
previous
results [7].
In
[7], the authors
considered
the
tw0-dimensional
Rayleigh-B\’enard convection and proposed
an
approach
to
prove
the
ex-sistence
of
the
steady-state solutions based
on
the
infinite dimensional
fixed-point
theorem
using
Newton-like
operator
with
the
spectral
approximation
and
the
constructive
error
es-timates.
We
numerically
verified several exact non-trivial
solutions which correspond
to
the
bifurcated
solutions from the
trivial
solution. This
paper
shows
more
detailed
results
of verification for the given Prandtl and Rayleigh
numbers,
which
enables
us
to study the
global bifurcation structure. All numerical examples discussed
are
taken
into
account of
the
effects of rounding
errors
in
the floating point computations.
1
The
Rayleigh-B\’enard
Problems
Consider
aplane
horizontal
layer
$(0\leq z\leq h)$
of
an
incompressible viscous fluid heated from
below. At the lower boundary:
$z=0$
the layer of fluid is maintained at temperature
$T+\delta T$
and the temperature
of
the
upper
boundary
$(z=h)$
is
$T$
(see Fig.1).
Fig.1. Geometry of the
convection
problem.
All variations
with respect to
$y$
-direction
are
assumed to
vanish,
then according to the
Oberbeck-Boussinesq
approximations
$[1, 3]$
,
the equations
governing convection
in alayer in the
tw0-dimensional
(x-z)
are
described
as
follows:
$\{$
$u_{t}+uu_{x}+wu_{z}$
$=$
$w_{t}+uw_{x}+ww_{z}$
$=$
$u_{x}+w_{z}$
$=$
$\theta_{t}+u\theta_{x}+w\theta_{z}$
$=$
$-p_{x}/\rho_{0}+\nu\Delta u$
,
$-(p_{z}+g\rho)/\rho_{0}+\nu\Delta w$
,
(1)
0,
$\kappa\Delta\theta$.
数理解析研究所講究録 1286 巻 2002 年 17-26
17
In the above system (1),
$(u, 0, w)$
is the velocity vector
field
in the respective
direction
$(\#, y, z);p$
is the
pressure
field;
0is
the temperature;
$\rho$is the
fluid
density;
$\rho 0$
is the density at temperature
$T+\delta T;\nu$
is the kinematic
viscosity;
$g$
is the gravitational acceleration;
$\kappa$is the
coefficient
of
thermal diffusivity;
$*_{\xi}:=\partial/\partial\xi(\xi=x, z, t)$
;and
$\Delta:=\partial^{2}/\partial x^{2}+\partial^{2}/\partial z^{2}$
.
The
Oberbeck-Boussinesq
approximation also requires that the
fluid
density is
to
be independent of pressure and depends
linearly
on
the temperature
$\theta$,
therefore
$\rho$
can
be
represented
by
$\rho-\rho_{0}=-\rho_{0}\alpha(\theta-T-\delta T)$
,
where ais the
coefficient of
thermal expansion.
The
Oberbeck-Boussinesq
equations (1)
have astationary solution:
$u^{*}=0$
,
$w^{*}=0$
,
$\theta^{*}=T\mathit{4}$
$\delta T$
$- \frac{\delta T}{h}z$
,
$p^{*}=p_{0}-g \rho_{0}(z+\frac{\alpha\delta T}{2h}z^{2})$
representing the purely heat conducting
state,
where
$p_{0}$
is aconstant. By setting
\^u
$:=u$
,
$\hat{w}:=w$
,
$\hat{\theta}:=\theta^{*}-\theta$
,
$\hat{p}:=p^{*}-p$
,
the perturbed equations:
$\{$
$\hat{u}_{t}+\hat{u}\hat{u}_{x}+\hat{w}\hat{u}_{z}$
$=$
$\hat{p}_{x}/\rho_{0}+\nu\Delta\hat{u}$
,
$\hat{w}_{t}+\hat{u}\hat{w}_{x}+\hat{w}\hat{w}_{z}$
$=$
$\hat{p}_{z}/\rho_{0}-g\alpha\hat{\theta}+\nu\Delta\hat{w}$
,
$\hat{u}_{x}+\hat{w}_{z}$
$=$
0,
$\hat{\theta}_{t}+\delta T\hat{w}/h+\hat{u}\hat{\theta}_{x}+\hat{w}\hat{\theta}_{z}$
$=$
$\kappa\Delta\hat{\theta}$,
(2)
are
obtained.
Moreover,
transforming
to
dimensionless
variables:
$tarrow\kappa t$
,
$uarrow\hat{u}/\kappa$
,
$warrow\hat{w}/\kappa$
,
$\thetaarrow\hat{\theta}h/\delta T$
,
$parrow\hat{p}/(\rho_{0}\kappa^{2})$
of
(2),
the
dimensionless
equations:
$\{$
$u_{t}+uu_{x}+wu_{z}$
$=$
$p_{x}+P\Delta u$
,
$w_{t}+uw_{x}+ww_{z}$
$=$
$p_{z}-PR$
$\theta+P\Delta w$
,
$u_{x}+w_{z}$
$=$
0,
$\theta_{t}+w+u\theta_{x}+w\theta_{z}$
$=$
$\Delta\theta$(3)
are
led,
where
$R$
$:= \frac{\delta T\alpha g}{\kappa\nu h}$is the Rayleigh number and
$\mathcal{P}$
$:= \frac{\nu}{\kappa}$
is the Prandtl number.
xThe Rayleigh
number
is sometimes defined
by
$72=(\delta T\alpha gh^{3})/(\kappa\nu)$
when
the
dimensionless
equations
are
reduced
to the
domain
of
$0\leq z\leq 1$
.
2Afixed-point
formulation
We shall find the steady-state
solutions,
$u_{t}$
,
$w_{t}$
and
$\theta_{t}$are
equated
to
0in
(3),
and
assume
that all fluid motion is
confined
to the rectangular
region
$\Omega:=\{0<x<2\pi/a, 0<z<\pi\}$
for
agiven
wave
number
$a>0$
.
Let
us
impose periodic boundary
condition
(period
$2\pi/a$
)
in
the
horizontal
direction,
stress-ffee boundary conditions
$(u_{z}=w=0)$
for
the
velocity
field
and Dirichlet boundary conditions
$(\theta=0)$
for the temperature field
on
the
surfaces
$z=0$
,
$\pi$
,
respectively.
Furthermore,
we assume
the following
evenness
and
oddness conditions:
$u(x, z)=-u(-x, z)$ ,
$w(x, z)=w(-x, z)$
,
$\theta(x, z)=\theta(-x, z)$
.
We introduce the stream function
$\Psi$
, through the definition
$u=-\Psi_{z}$
,
$w=\Psi_{x}$
so
that
$u_{x}+w_{z}=0$
.
Cross-differentiating the equation of motion in
(3)
in order to eliminate
the
pressure
$p$
and
setting
$\Theta:=\sqrt{\mathcal{P}\mathcal{R}}\theta$
,
we
obtain
$\{$
$P\Delta^{2}\Psi$
$=$
$\sqrt{\mathcal{P}\mathcal{R}}\ominus_{x}-\Psi_{z}\Delta\Psi_{x}+\Psi_{x}\triangle\Psi_{z}$
in
$\Omega$,
$-\Delta$
$=$
$-\sqrt{P\mathcal{R}}\Psi_{x}+\Psi_{z}_{x}-\Psi_{x}_{z}$
in
$\Omega$.
(4)
From the boundary conditions imposed
above,
the
stream
function Iand departure of
tem-perature from linear profile
$$
can
be represented by the following double Fourier series:
$\Psi=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}A_{mn}\sin(amx)\sin(nz)$
,
$\ominus=\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}B_{mn}\cos(amx)\sin(nz)$
.
(5)
By
(5),
we
introduce following function
spaces
for
$k\geq 0$
:
$X^{k}:= \{\Psi=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}A_{mn}\sin(amx)\sin(nz)|A_{mn}\in R$
,
$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}((am)^{2k}+n^{2k})A_{mn}^{2}<\infty\}$
,
$\mathrm{Y}^{k}:=\{\ominus=\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}B_{mn}\cos(amx)\sin(nz)|B_{mn}\in R$
,
$\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}((am)^{2k}+n^{2k})B_{mn}^{2}<\infty\}$
which
are
considered
as
closed
subspaces
of usual
$k$
-th order
Sobolev space
$H^{k}(\Omega)$
.
For
$M_{1}$
,
$N_{1}$
,
$M_{2}\geq 1$
and
$N_{2}\geq 0$
,
we
indicate arelation
$N:=(M_{1}, N_{1},M_{2}, N_{2})$
and
define
the
finite dimensional approximate
subspaces
by
$s_{N}^{(1)}$
$=$
$\{\Psi_{N}=\sum_{m=1}^{M_{1}}\sum_{n=1}^{N_{1}}\hat{A}_{mn}\sin(amx)\sin(nz)|\hat{A}_{mn}\in R\}$
,
$S_{N}^{(2)}$
$=$
$\{_{N}=\sum_{m=0}^{M_{2}}\sum_{n=1}^{N_{2}}\hat{B}_{mn}\cos(amx)\sin(nz)|\hat{B}_{mn}\in R\}$
,
$S_{N}$
$=$
$S_{N}^{(1)}\mathrm{x}S_{N}^{(2)}$
,
and denote
an
approximate
solution
of
(4)
by
$\text{\^{u}}_{N}$$:=(\hat{\Psi}_{N},\hat{}_{N})\in S_{N}$
which
is
obtained by
an
appropriate
method. Then
setting
$\{$
$f_{1}(\Psi, \ominus)$
$:=$
$\sqrt{\mathcal{P}R}_{x}-\Psi_{z}\Delta\Psi_{x}+\Psi_{x}\Delta\Psi_{z}$
,
$f_{2}(\Psi, )$
$:=$
$-\sqrt{PR}\Psi_{x}+\Psi_{z}\ominus_{x}-\Psi_{x}_{z}$
,
$\Psi=\hat{\Psi}_{N}+w^{(1)}$
,
$=\hat{}_{N}+w^{(2)}$
,
(4)
is
rewritten
as
the problem
to
find
$(w^{(1)}, w^{(2)})\in X^{4}\cross \mathrm{Y}^{2}$
satisfying
(6)
$\{_{-\Delta w^{(2)}}^{\mathcal{P}\Delta^{2}w^{(1)}}==f_{2}(\hat{\Psi}_{N}+w(1)^{\wedge},\ominus_{N}+w^{(2)})+\Delta\hat{}_{N}f_{1}(\hat{\Psi}_{N}+w^{(1)^{\wedge}},\ominus_{N}+w^{(2)})-\mathcal{P}\Delta^{2}\hat{\Psi}_{N}$
inin
$\Omega\Omega.$’
Note
that
$(w^{(1)}, w^{(2)})$
is
expected to
be small
if
$\text{\^{u}}_{N}$is
an
accurate
approximation.
Defining
$w$
$=$
$(w^{(1)},w^{(2)})$
,
$h_{1}(w)$
$=$
A
$(\hat{\Psi}_{N}+w^{(1)},\hat{\Theta}_{N}+w^{(2)})-\mathcal{P}\Delta^{2}\hat{\Psi}_{N}$
,
$h_{2}(w)$
$=$
$f_{2}(\hat{\Psi}_{N}+w^{(1)},\hat{\Theta}_{N}+w^{(2)})+\Delta\hat{}_{N}$
,
$h(w)$
$=$
$(h_{1}(w), h_{2}(w))$
,
by
virtue of
Sobolev
embbeding theorem and
the
definition
of
$f1$
and
$f_{2}$
,
$h$
is abounded
continuous map fro
$\mathrm{m}$$X^{3}\cross \mathrm{Y}^{1}$
to
$X^{0}\cross \mathrm{Y}^{0}$
.
Moreover,
it is
easily
shown that for
all
$(g_{1}, g_{2})\in$
$X^{0}\cross \mathrm{Y}^{0}$
,
the
linear
problem:
$\{$
$\Delta^{2}\overline{\Psi}$$=$
$g_{1}$
in
$\Omega$,
$-\Delta\overline{\Theta}$$=$
$g_{2}$
in
$\Omega$(7)
has aunique solution
$(\overline{\Psi},\overline{\Theta})\in X^{4}\cross \mathrm{Y}^{2}$
.
When this
mapping is
denoted by
$\overline{\Psi}=(\Delta^{2})^{-1}g_{1}$
and
$\overline{\Theta}=(-\Delta)^{-1}g_{2}$
,
an
operator:
$\mathcal{K}:=(\mathcal{P}^{-1}(\Delta^{2})^{-1}, (-\Delta)^{-1})$
:
$X^{0}\cross \mathrm{Y}^{0}arrow X^{3}\cross \mathrm{Y}^{1}$
is acompact
map
because
of
the compactness
of
the
imbedding
$H^{4}(\Omega)arrow H^{3}(\Omega)$
,
$H^{2}(\Omega)rightarrow$
$H^{1}(\Omega)$
and the boundedness
of
$(\Delta^{2})^{-1}$
:
$X^{0}arrow X^{4}$
,
$(-\Delta)^{-1}$
:
$\mathrm{Y}^{0}arrow \mathrm{Y}^{2}$
.
Therefore,
(6)
is
rewritten
by afixed-point equation:
$w=Fw$
(8)
for
the compact operator
$F:=\mathcal{K}\circ h$
on
$X^{3}\cross \mathrm{Y}^{1}$
,
and
Schauder’s
fixed-point theorem asserts
that,
for
anonempty, closed,
bounded and
convex
set
$W\subset X^{3}\cross \mathrm{Y}^{1}$
,
if
$FW\subset W$
(9)
holds,
then
there
exists
afixed-point
of
(8)
in
$W$
.
Aconcrete
computer algorithm to construct
acandidate set
$W$
which
satisfies
(9)
is proposed in [7].
3Numerical
Examples
In the
verification
step,
interval arithmetic is used
to
take account
of the
effects
of rounding
errors
in the
floating
point computations.
We
use
Fortran
90
library
$\mathrm{I}\mathrm{N}\mathrm{T}\mathrm{L}\mathrm{I}\mathrm{B}_{-}90$coded by
Kearfott
[5]
with
DIGITAL
Fortran
V5.4-1283
on
Compaq Alpha
Server
GS320
(Alpha
21264
731MHz; Tru64
UNIX
V5.1)
3.1
The
trivial
solution
It is clear that the problem
(4)
has
atrivial
solution
$\Psi=\ominus=0$
for
all
$\mathcal{P}$and
71.
Fig.2
shows
the isotherm of the temperature
$T+ \delta T-\frac{\delta T}{h}z$
when
$T=0$
and
$\delta T=5$
.
Fig.2 The isotherm of the temperature: stationary solution.
It
is
known
that for small
7?
the fluid conducts
heat
diffusively, and at acritial point
$Rc$
,
heat
is transposed through the fluid by
convection.
It
has
been shown by Joseph [4] that
(3)
has
aunique
trivial solution for
$\mathcal{R}<\prime \mathcal{R}c$
.
However,
the global
structure
of
bifurcated solutions
after
the
critical Rayleigh point
$Rc$
has
not been known theoretically.
3.2
First
bifurcated
solutions from the trivial solution
In 1916, Rayleigh [6]
considered
the linearized stability and found the critical Rayleigh number
as
follows
$\mathcal{R}_{C}=\inf_{m,n}\frac{(a^{2}m^{2}+n^{2})^{3}}{a^{2}m^{2}}=6.75$
$(m=1, n=1, a=1/\sqrt{2})$
.
The
usual bifurcation
theory
implies
that the stationary
bifurcation
occurs
from the above
critical point. We select
$a=1/\sqrt{2}$
and
$\mathcal{P}$$=10$
in the following numerical experiments. After
the
critical Rayleigh number
$\mathcal{R}c=6.75$
,
we
obtain two non-trivial approximate solutions for
various Rayleigh numbers
72
of the form:
$M_{1}$
$N_{1}$
$M_{2}$
$N_{2}$
$\hat{\Psi}_{N}=\sum\sum\hat{A}_{mn}\sin(amx)$
$\sin(nz)$
,
$\hat{}_{N}=\sum\sum\hat{B}_{mn}$
COS
$\{Amn\}$ $\sin(nz)$
$m=1n=1$
$m=0n=1$
for
some
Mi,
$M_{2}$
,
$N_{1}$
and
$N_{2}$
by
Fourier-Galerkin
method combined with Newton-Raphson
iteration.
Fig.3 shows the velocity field
$(-(\hat{\Psi}N)_{z}, (\hat{\Psi}_{N})_{x})$
at
$R$
$=50$
,
$\mathcal{P}=10$
,
$M_{1}=N_{1}=M_{2}=$
$N_{2}=10$
, respectively.
We
illustrate the particular value
of
coefficients,
under the figures, which
has
the maximum absolute value in
$\{\hat{A}_{mn}\}$
and
$\{\hat{B}_{mn}\}$
,
respectively.
$arrow\simarrow---arrow\sim--\simarrowarrow-arrowarrow-$
$\dagger I||/’\prime_{lr-\backslash }J’\simrightarrow\sim\backslash \backslash ’arrowarrowarrowarrow\sim\backslash$
$\backslash \{:,’\wedge-arrowarrow-\sim\backslash \backslash \backslash x_{4\mathrm{t}}^{arrow\sim\sim\backslash }|\dagger I$ $|/\prime\prime\prime\sim-arrowarrowarrow-\backslash \wedge\simarrow-\backslash \backslash \prime\prime-\backslash ^{\iota 1}\iota\{|\dagger d’/d,,\sim\backslash l’\sim\sim\sim\sim\backslash \backslash |’arrowarrowarrow-arrow\sim\backslash$
’
$|\dagger‘\backslash \sim\simarrowarrowarrow\vee’\backslash \backslash \backslash -\sim\sim\{\iota_{\backslash \backslash -\prime}^{\backslash }\prime\prime\prime \iota^{\mathrm{t}}\prime\prime\downarrow \mathrm{I}\}\}\backslash \mathrm{I}\prime \mathrm{I}’\backslash \backslash -\sim\sim\backslash arrowarrowarrowarrow\sim\backslash -\prime pl,$
$t-\backslash \backslash \prime\prime\prime/\dagger \mathrm{f}$
‘
$|\dagger$
},
$’ \dagger’\backslash \wedge-\prime d/,\mathrm{t};\backslash \simarrowarrowarrowarrowarrow\vee d\backslash \backslash \backslash -\prime\prime\sim\sim\backslash \backslash \backslash arrowarrow-\cdotarrow-’|\backslash \backslash -\sim\sim\dagger\iota_{\iota_{\backslash \neg\prime}}^{\tau\}}\bigwedge_{J/}’\dagger\dagger|\dagger$,
$-\simarrow---arrow\sim$
$\hat{A}_{11}\approx 15.37$
$\hat{A}_{11}$
z-15.37
Fig.3 The velocity field of the first bifurcated solution.
Fig.4 shows the isotherm of the temperature
$\theta’=\delta T(1-z/\pi-\ominus/\sqrt{R\mathcal{P}}\pi)+T$
when
$T=0$
,
$\delta T=5$
.
Fig.4
The isotherm
of the temp
3.3
Second bifurcated solutions from
the
trivial solution
After
the Rayleigh number
$R$
$= \frac{(a^{2}m^{2}+n^{2})^{3}}{a^{2}m^{2}}=13.5$
$(m=2,n=1, a=1/\sqrt{2})$
,
we
obtain two non-trivial approximate
solutions
which
are
expected
to be second bifu
solutions ffom
the trivial solution. Fig. and
Fig.6
show the velocity
field
at
$72=5\mathrm{t}$
$10$
,
$M_{1}=N_{1}=M_{2}=N_{2}=10$
and the isotherm of the
temperature,
respectively
$\sim\sim\simarrow\wedge$
$-arrow\vee\wedge-$
$\sim\sim\veearrow-$
$-arrow\vee\wedge-$
$’||,\prime\prime\prime \mathrm{t}$
$’-,\wedge\cdot,,I\prime\prime\sim-l,\backslash ,\prime\prime-arrowarrow,I‘’\vee\sim,\sim\backslash ’\prime\prime-\backslash ^{1}\iota-\mathrm{I}|_{\iota,\dagger}^{l^{-\prime\prime}}l’,4\downarrow\iota_{-\dagger|\mathrm{I}_{\backslash -\prime d\backslash \backslash }\backslash }-\iota \mathfrak{l}|_{4-\prime}\dagger\downarrow-\mathrm{t}^{\backslash -r\downarrow\dagger\backslash }-dl\mathrm{t}\backslash -’-\backslash \sim\sim\vee\sim--\sim\vee\wedge--’\sim\sim\sim--arrowarrowrightarrow-\sim\prime I\backslash \backslash arrow\wedge\backslash \wedge\vee\vee I\backslash \backslash \sim-\backslash \backslash \prime\prime\dagger \mathrm{t}\backslash -\backslash ^{\grave{\mathrm{t}\mathrm{t}}\iota\grave{\iota}ll_{\vee-\backslash \prime}}\iota_{4-\prime}^{d-\backslash }\backslash \backslash ’-\backslash |||’t’\mathrm{t}\mathrm{t}_{\backslash -\prime l\iota\downarrow-\prime_{l}}^{\prime-^{4}1|1\iota_{-\dagger|}}-\backslash \downarrow|\downarrow l-\backslash ||-\prime\prime\prime|\backslash$
,
$\hat{A}_{21}$
z-7.026
$\hat{A}_{21}\approx 7.026$
Fig.5 The velocity field
of
the
second bifurcated solution.
$7?=14$
$72=20$
$72=30$
$72=40$
$72=50$
$72=60$
Fig.6 The isotherm of the temperature for the second bifurcated solution.
3.4
Third
bifurcated solutions from
the trivial
solution
After the Rayleigh number
$R$ $= \frac{(a^{2}m^{2}+n^{2})^{3}}{a^{2}m^{2}}=1331/36$
$(m=3, n=1, a=1/\sqrt{2})$
,
we
obtain two non-trivial approximate solutions which
are
expected
to be
third
bif
solutions from the trivial solution. Fig. and Fig.8 show the velocity field at
$\mathcal{R}=!$
$10$
,
$M_{1}=N_{1}=M_{2}=N_{2}=10$
and the isotherm of the temperature, respectively
$\dagger\downarrow,\mathrm{t}_{\backslash }|\mathrm{t}|\prime\prime\backslash \prime\prime\prime\backslash ’-\sim\prime\prime^{4l}4’\backslash \sim\prime\prime’\iota\backslash -",’\prime\prime\iota_{l1}^{\downarrow\dagger^{4_{r\backslash }}}\vee\backslash 4-\iota\iota_{i^{\backslash \prime}\dagger}’\sim\backslash ’,\iota\iota_{l_{d\prime\prime}}^{d,\backslash }\downarrow|l\backslash \prime\prime\iota_{1}^{||}’\prime\prime\dagger$$\mathrm{t}^{-4},\prime\prime|,,|,\cdot,,,,|\dagger\prime^{\backslash 4}\iota\iota_{\iota-}\dagger \mathrm{t}_{\backslash \prime}^{\prime r_{d}\iota\prime}\downarrow \mathrm{t}\backslash \prime\prime-\backslash 1d-14\backslash \prime\prime\wedge-\grave{\iota}l1l\vee\sim\backslash ’\backslash -\prime d\iota\backslash -’\backslash \sim-’\prime\prime\backslash -\prime 4\iota_{\backslash -\backslash \vee}I\iota_{\backslash \prime}\prime^{1}-\backslash -1\iota\iota^{\prime-\prime\prime\prime}4|1^{d\sim,}\dagger’\backslash ^{\iota\downarrow,\backslash }\iota\dagger^{l}\downarrow t_{1\prime}^{4\backslash }’\dagger\dagger^{\iota_{4}},t\downarrow^{4\prime}\mathrm{t}|4\backslash \backslash -,\prime\prime\prime\backslash \downarrow\prime\prime\prime|\mathrm{t}’\backslash ’$
’
$I\backslash \neg-\prime\prime\backslash -\prime\prime 1\backslash -\prime\prime\backslash \backslash \cdot\prime 4\backslash -\prime\prime\backslash \prime\prime I---\sim---arrow---\mathrm{I}_{-\sim\vee}\ddagger--$
$\hat{A}_{31}\approx$
-2.029
$A\wedge 31\approx 2.029$
Fig.7 The velocity
field
of the third
bifurcated
solution.
$R$
$=37$
$72=40$
$72=50$
$7?=60$
Fig.8 The isotherm
of
the temperature for
the third
bifurcated
solution.
3.5
Another non-trivial solutions
We also
obtain
four different non-trivial
approximate solutions
after
$R$
$=32.5$
.
According to
this
observation,
we
expected
the
existence of another
bifurcated
solutions
from
the
non-trivial
solutions which
seem
to
be
on
the second
bifurcation
branch
(cf.Fig.11).
For example,
we
observed
the phenomena
as
shown in Fig.9 and Fig.10
for
the
case
that
$R$
$=50$
,
$\mathcal{P}$$=10$
,
$M_{1}=$
$N_{1}=M_{2}=N_{2}=10$
.
$-\vee\vee\vee\vee----\cdot$
-
\sim \sim \sim \sim \sim
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