Bifurcation of a Flow of a Gas between Rotating
Coaxial Circular Cylinders with Evaporation and
Condensation
Yoshio Sone† and Toshiyuki Doi‡
†
230–133 Iwakura-Nagatani-cho Sakyo-ku, Kyoto 606-0026, Japan; e-mail: [email protected]
‡
Department of Applied Mathematics and Physics, Tottori University, Tottori 680-8552, Japan
Abstract. Bifurcation of a flow of a gas between two rotating coaxial circular cylinders made of the
condensed phase of the gas is studied analytically on the basis of the asymptotic equations and boundary
conditions derived from the Boltzmann system for small Knudsen numbers. The bifurcation relation and
behavior of solution near the bifurcation point are obtained, and the effect of evaporation and condensation
on the cylinders on the bifurcation is clarified.
1
INTRODUCTION
The behavior of a gas between two rotating coaxial circular cylinders is a classical problem of fluid
dynamics in relation to bifurcation of flow.[1, 2] Recently, we extended the problem, on the basis of kinetic
theory, to the case when the cylinders are made of the condensed phase of the gas, on which evaporation
or condensation takes place,[3, 4] and presented the bifurcation of flow in an axially symmetric and uniform
system. In Ref. 5, the restriction of axial uniformity being eliminated, the problem is studied numerically
by the direct-simulation Monte-Carlo method and a Taylor-roll type of flow is shown to exist stably in
addition to the two above-mentioned axially uniform solutions. In the numerical computation, the range of
the parameters studied and the analysis in the neighborhood of a bifurcation point are inevitably limited.
Thus, in the present paper, we will study the problem without axial uniformity by combination of analysis
and numerical computation with interest in the behavior in the neighborhood of a bifurcation point.
Consider a gas between two rotating coaxial circular cylinders made of the condensed phase of the gas,
where evaporation or condensation takes place. Let the radius, temperature, and circumferential velocity of
the inner cylinder be, respectively, LA , TA , and VθA (≥ 0), and let the corresponding quantities of the outer
cylinder be LB , TB , and VθB ; the saturated gas pressure at temperature TA is√denoted by pA√and that at TB
by pB . The problem is characterized by the parameters LB /LA (= r̂B ), VθA / 2RTA , VθB / 2RTA , TB /TA ,
pB /pA , Kn (= `A /LA ) , and the kind of the gas (or the molecular model), where R is the specific gas constant
and `A is the mean free path of the gas in the equilibrium state at rest with temperature TA and pressure
pA . The ratio `A /LA is called the Knudsen number. In a series of our works, we have been studying the flow
of this system, with special interest in bifurcation of flow, on the basis of the Boltzmann equation limiting
the class of solution to the axially symmetric solution. In the present paper, taking a small quantity ε, we
study the asymptotic behavior as ε → 0 of the axially symmetric solution of the problem described above
when the above six parameters are limited to the following case:

√
√
r̂B − 1 = O(1),
VθA / 2RTA = εuθA , VθB / 2RTA = εuθB , 
(1)
√

TB /TA − 1 = ετB , pB /pA − 1 = εPB ,
Kn = 2ε/ π,
where uθA (≥ 0), uθB , τB , and PB are of the order of unity. We will see, in the present analysis, that an
axially nonuniform solution bifurcates in the above range of parameters from the axially uniform solution,
which is unique in this range as noted in Ref. 4. From the result in the present work, the axially nonuniform
solution bifurcated may be considered to continue to exist in the range of the parameters studied in Ref. 4.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
646
2
BASIC EQUATION AND BOUNDARY CONDITION
Asymptotic theory that describes the time-independent behavior of a gas around the condensed phase of
the gas for small Knudsen numbers is developed, and the fluid-dynamic-type equations and its associated
boundary conditions that describe the asymptotic behavior are derived.[6] Let the cylindrical coordinate
system with the axis of the cylinders as the axial direction, the flow velocity (in the cylindrical coordinates),
pressure, and temperature of the gas be, respectively, LA (r̂, θ, ẑ), ε(2RTA )1/2 (ur , uθ , uz ), pA (1 + εP ), and
TA (1 + ετ ). The equations and the boundary conditions governing the limiting values of the variables ur ,
uθ , uz , P, and τ as ε → 0 in an axially symmetric state (∂/∂θ = 0) for the parameter range (1) are obtained
by the following system with the same notations for the limiting values: The governing equations are
∂P
∂P
=
= 0,
∂ r̂
∂ ẑ
⇒ P : uniform
1 ∂ ur r̂ ∂uz
+
= 0,
r̂ ∂ r̂
∂ ẑ
ur
(2)
(3a)
·
µ
¶
¸
∂ ur
∂ ur
u2
1 ∂ P1
γ1 1 ∂
∂ ur
ur
∂ 2 ur
+ uz
− θ =−
+
r̂
− 2 +
,
∂ r̂
∂ ẑ
r̂
2 ∂ r̂
2 r̂ ∂ r̂
∂ r̂
r̂
∂ ẑ 2
·
µ
¶
¸
∂uθ
γ1 1 ∂
∂ uθ
uθ
∂ 2 uθ
ur ∂ uθ r̂
+ uz
=
r̂
− 2 +
,
r̂ ∂ r̂
∂ ẑ
2 r̂ ∂ r̂
∂ r̂
r̂
∂ ẑ 2
·
µ
¶
¸
∂ uz
∂ uz
1 ∂P1
γ1 1 ∂
∂ uz
∂ 2 uz
ur
+ uz
=−
+
r̂
+
,
∂ r̂
∂ ẑ
2 ∂ ẑ
2 r̂ ∂ r̂
∂ r̂
∂ ẑ 2
·
µ
¶
¸
∂τ
∂τ
γ2 1 ∂
∂τ
∂2τ
ur
+ uz
=
r̂
+ 2 ,
∂ r̂
∂ ẑ
2 r̂ ∂ r̂
∂ r̂
∂ ẑ
(3b)
(3c)
(3d)
(3e)
where P1 is the next-order term in ε of P, divided by ε; γ1 and γ2 are constants depending on molecular
models, for example, γ1 = 1.2700424, γ2 = 1.922284 (a hard-sphere gas) or γ1 = γ2 = 1 (the BKW model).
When the property of the cylinder surface is locally isotropic and the distribution of the reflected molecules
has a finite complete condensation part, the boundary conditions on the cylinders for Eqs. (2)–(3e) are given
in the following form:
P = C4∗ ur ,
τ = d∗4 ur ,
uθ = uθA , uz = 0 at r̂ = 1,
∗
∗
P = PB − C4 ur , τ = τB − d4 ur , uθ = uθB , uz = 0 at r̂ = r̂B ,
(4a)
(4b)
where C4∗ and d∗4 are constants depending on molecular models and kinetic boundary conditions. For the
complete condensation condition, for example, C4∗ = −2.1412 and d∗4 = −0.4557 (a hard-sphere gas) or
C4∗ = −2.13204 and d∗4 = −0.44675 (the BKW model).
Equations (2)–(3e) are of the same form as the leading set of equations derived from the Navier-Stokes
equations for a compressible fluid when the Mach number and the (relative) temperature variation are of the
same order of smallness and the Reynolds number is of the order of unity. Equations (3a)–(3d) are of the
same form as the Navier-Stokes equations for an incompressible fluid, but Eq. (3e) is a little different from the
energy equation of incompressible Navier-Stokes equations. The difference is due to the fact that the work
done by pressure is of higher order in an incompressible fluid. The energy equation for an incompressible
fluid with the internal energy multiplied by 5/3 (or the thermal conductivity multiplied by 3/5) is of the
same form as Eq. (3e). The boundary conditions (4a) and (4b) are derived by the Knudsen-layer analysis.
From Eqs. (3a)–(3d) and Eqs. (4a) and (4b) excluding their second relations, the velocity field is determined independently of the temperature field. Thus, we will mainly discuss the velocity field. Further, if
this system is rewritten with the variables (ur , uθ , uz )/γ1 , P/γ1 C4∗ , and P1 /γ12 and the parameters uθA /γ1 ,
uθB /γ1 , and PB /γ1 C4∗ , the system does not contain the quantities that depend on molecular models and/or
kinetic boundary conditions. Thus the results in these quantities obtained in the following analysis are
independent of them.
647
3
3.1
BIFURCATION ANALYSIS
Axially symmetric and uniform solution
First consider the case where the behavior of the gas is axially uniform (or ∂/∂ ẑ = 0). Then, the axially
uniform solution, indicated by the subscript U, of Eqs. (2)–(3e) under the boundary conditions (4a) and (4b)
is given in the following form:
³
γ1 b
c0 ´
r̂B
urU =
,
uθU = γ1 c1 r̂1+b +
,
uzU = 0,
PU =
PB ,
τU = d1 r̂β + d0 ,
(5)
2 r̂
r̂
r̂B + 1
Ã
!
µ
¶
1+b
r̂B
uθA
1
r̂B
2r̂B PB
r̂B uθB
uθA
uθB
,
c
=
, c0 = 2+b
b=
−
−
,
1
∗
2+b
r̂B + 1 γ1 C4
γ1
γ1
γ1
γ1
r̂B − 1
r̂B − 1
Ã
!
µ
¶
β+1
d∗4 (r̂B
+ 1)
γ1
1
d∗4
−1
d1 = β
τB − ∗ PB ,
d0 = β
τB − ∗
PB ,
β = b,
γ2
C4
C4 (r̂B + 1)
r̂B − 1
r̂B − 1
where the solutions at the apparent singularities b = 0 and −2 are obtained as the limit of the above solution.
The mass flux M from the inner cylinder to the outer per unit length is given by
µ
¶
2πLA (2RTA )1/2 pB
2πLA (2RTA )1/2 PB
ε= ∗
M=
−1 .
(6)
1 + LA /LB C4∗
C4 (1 + LA /LB ) pA
Evaporation takes place on the inner (outer) cylinder and condensation does on the outer (inner) cylinder
when PB /C4∗ > 0 (< 0), where C4∗ < 0 according to accurate results for a hard-sphere gas and the BKW
model and approximate results for other molecular models.
3.2
Bifurcation point
Here, we consider a time-independent solution periodic with period 2πLA /α in the axial direction and
investigate whether this solution bifurcates from the axially uniform solution and the behavior of the solution
near the bifurcation point, if any. Let the bifurcation point be at uθA = uθAb , uθB = uθBb , τB = τBb , and
PB = PBb and the axially uniform solution fU at this point be fU b . The solution in the neighborhood of the
bifurcation point (or when uθA − uθAb , uθB − uθBb , PB − PBb , and τB − τBb are of the order of δ 2 , say) is
found to be expressed in the following form:
f (r̂, ẑ) = fU b (r̂) + δf11 (r̂) cos αẑ + δ 2 [f20 (r̂) + f21 (r̂) cos αẑ + f22 (r̂) cos 2αẑ] + · · · ,
2
uz (r̂, ẑ) = δW11 (r̂) sin αẑ + δ [W21 (r̂) sin αẑ + W22 (r̂) sin 2αẑ] + · · · ,
(7)
(8)
where f = ur , uθ , P1 , or τ. Equations (7) and (8) with Eq. (5) being substituted into Eqs. (2)–(4b), it is
found that the coefficient functions f11 (r̂), f20 (r̂), etc. of r̂ are formally determined from the lowest order
and that each Fourier component function is determined independently at each order of δ. Let Umn (r̂) and
Vmn (r̂) be fmn (r̂), respectively, corresponding to uθ and ur . From Umn (r̂) and Vmn (r̂), the other variables
are determined. That is, nαWmn = −(r̂Vmn )0 /r̂, where (∗)0 =d(∗)/dr̂. The temperature τ can be shown to
be determined uniquely from the velocity.1 Thus, the bifurcation can be discussed only with the velocity
field, and τB does not influence the bifurcation. The mass flux M from the inner cylinder to the outer
remains the same as the axially uniform solution up to the order of δ.
The component functions U11 (r̂) and V11 (r̂) are the solution of the following homogeneous boundary-value
problem of ordinary differential equations:
Lθ (U11 ) + qθ V11 = 0,
Lr (V11 ) + qr U11 = 0;
0
U11 (r̂) = V11 (r̂) = V11
(r̂) = 0 at r̂ = 1 and r̂ = r̂B ,
1 The
(9)
(10)
uniqueness is easily seen to be equivalent to the proposition that the following system has only the trivial solution:
d2 g(r̂)/dr̂2 + (κ/r̂)dg(r̂)/dr̂ − α2 g(r̂) = 0, g(1) = g(r̂B ) = 0; κ = 1 − β.
R
From the system it is easily derived that 1r̂B [α2 g 2 + (dg/dr̂)2 ]r̂κ dr̂ = 0, from which the proposition, i.e., g = 0, follows.
648
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600
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400
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!#/$
.
!
200
40
0
−5
0 5
0 ) *+,- 5
−5
(a)
(b)
PBb /(−γ1 C4∗ )
for various α when the outer cylinder
FIGURE 1. Bifurcation curve I: The relation uθAb /γ1 versus
is at rest uθBb = 0 (r̂B = 2). (a) Wider range of uθAb /γ1 and (b) the lowest (or first) branch for α = π/2, π, and 2π.
Table 1. (uθAb /γ1 )m , αm , and aA /aO , aB /aO , and aP /aO at αm for uθBb = 0, r̂B = 2, and various PBb /(−γ1 C4∗ ).
PBb
(−γ1 C4∗ )
−5
−2
−1
0
1
2
5
(uθAb /γ1 )m
αm /π
aA /aO
aB /aO
aP /aO
48.870
35.890
34.201
34.093
35.846
39.858
70.599
1.1408
1.0356
1.0157
1.0067
1.0110
1.0314
1.2181
−7.359
7.270
5.531
5.301
6.176
9.491
−6.142
6.038
−4.831
−3.062
−2.061
−0.916
1.810
−11.26
−44.95
17.52
5.177
−4.061
−17.33
−50.35
97.27
where
Lθ (U ) = U 00 + (1 − b)r̂−1 U 0 − [(1 + b)r̂−2 + α2 ]U,
¡
¢
Lr (V ) = V 0000 + (2 − b)r̂−1 V 000 − 3r̂−2 + 2α2 V 00 + [3(1 + b)r̂−3 − 2α2 (1 − b/2)r̂−1 ]V 0
(11a)
+ [−3(1 + b)r̂−4 + 2α2 (1 − b/2)r̂−2 + α4 ]V,
¡
¢
qθ = −4c1 (1 + b/2)r̂b , qr = −4α2 c1 r̂b + c0 r̂−2 .
(11b)
(11c)
The homogeneous boundary-value problem has nontrivial solution only when the parameters uθAb /γ1 ,
uθBb /γ1 , PBb /γ1 C4∗ , r̂B , and α satisfy a relation, say
Fb (uθAb /γ1 , uθBb /γ1 , PBb /γ1 C4∗ , r̂B , α) = 0.
(12)
The bifurcation relation when the outer cylinder is at rest is shown in Figs. 1 and 2 and Table 1. In Fig. 1,
the relation uθAb /γ1 versus PBb /γ1 C4∗ is shown for various α when uθBb = 0 and r̂B = 2. There are infinitely
many uθAb /γ1 for a PBb /γ1 C4∗ , the first three of which are shown in Fig. 1 (a).2 The lowest branches of the
curves for three α are shown in a larger scale in panel (b) of Fig. 1. The minimum value of uθAb /γ1 with
respect to α at a given PBb /γ1 C4∗ is denoted by (uθAb /γ1 )m and the minimum point by αm . They are shown
in the second and third columns in Table 1. In Fig. 2, the bifurcation curves for the cases r̂B = 1.5 and
3, corresponding to Fig. 1 (b), are shown (there are infinitely many other curves above the curves in the
figures). The bifurcation curves when the outer cylinder is rotating are shown in Figs. 3 and 4. In Fig. 3,
the relation uθAb /γ1 versus PBb /γ1 C4∗ for three sets of (r̂B , α) is shown for various uθBb (there are infinitely
many curves above the curves in the figures). The minimum (uθAb /γ1 )m of uθAb /γ1 with respect to α with
the other parameters fixed is shown as the curves (uθAb /γ1 )m versus uθBb /γ1 for various PBb /(−γ1 C4∗ ) in
2 “Infinitely
many” is the plausible result suggested by numerical study.
649
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−5
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−5
0 5
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(a)
(b)
FIGURE 2. Bifurcation curve II: The first branch of the relation uθAb /γ1 versus PBb /(−γ1 C4∗ ) for various α when
the outer cylinder is at rest uθBb = 0. (a) r̂B = 1.5 and (b) r̂B = 3.
#($ *)+, "
# $ )/, - # $ )0, . #$&% ' - - - . . . - . . !
FIGURE 3. Bifurcation curve III: The first branch of the relation uθAb /γ1 versus PBb /(−γ1 C4∗ ) for various uθBb /γ1 .
(a) r̂B = 1.5 and α = 2π, (b) r̂B = 2 and α = π, and (c) r̂B = 3 and α = π/2.
panel (a) of Fig. 4; the curve αm (the minimum point) versus uθBb /γ1 is shown for various PBb /(−γ1 C4∗ ) in
panel (b) of Fig. 4. The effect of PBb /(−γ1 C4∗ ) or that of evaporation and condensation appears strongly
when uθBb < 0 or the two cylinders are rotating in opposite directions.
When the relation (12) is satisfied, the boundary-value problem given by Eqs. (9) and (10) has a nontrivial
solution. Some examples on the first branch for the case (α = π, r̂B = 2) of the profiles (U11 , V11 )/||f11 ||
R r̂
2
2
versus r̂, where ||f11 || = [ 1 B (U11
+ V11
)r̂dr̂]1/2 , are shown in Fig. 5. The flow field (V11 , W11 ) on the
r̂ẑ plane is a circulating flow of a single or multiple rolls depending on whether there is a node of V11
in 1 < r̂ < r̂B . When PBb = 0 and uθBb = 0 (or there is neither evaporation nor condensation on the
cylinders in the unperturbed flow and the outer cylinder is at rest), a single roll lies with its center in the
R r̂
2
central part of (1, r̂B ), and the E⊥ (= 1 B U11
r̂dr̂) of the perturbed circumferential motion is comparable
R r̂B 2
2
to Eq [= 1 (V11 + W11 )r̂dr̂] of the motion in the r̂ẑ plane. When the outer cylinder is rotating, its effect
appears differently for uθBb > 0 and for uθBb < 0. When uθBb > 0, the shape of the roll does not change
much from that for uθBb = 0, but the ratio E⊥ /Eq decreases to vanish with the increase of |uθBb |. On the
other hand, when uθBb < 0, the center of the roll moves towards the inner cylinder with increase of |uθBb |
and multiple rolls (two, three, · · · ) appear with its further increase; the ratio E⊥ /Eq increase with |uθBb |.
When PBb 6= 0, the flow field is affected by the radial convection of the unperturbed flow, and the roll or rolls
move downstream. The roll on the side of the outer cylinder disappears with further increase of convection
or |PBb |; this example is shown in Fig. 6. The variation of E⊥ /Eq with uθBb is qualitatively similar to that
for PBb = 0.
650
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! + -,
0 20
−20
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−20
(a)
(b)
FIGURE 4. Bifurcation curve IV: the outer cylinder is rotating (r̂B = 2). (a) (uθAb /γ1 )m versus uθBb /γ1 r̂B and
(b) αm versus uθBb /γ1 r̂B The symbol (∗)m indicates the minimum with respect to α; the line (uθAb )m = uθBb r̂B is
shown by the chain line – - –.
!
!
!
" #
R r̂
2
2
+V11
)r̂dr̂]1/2 (α = π, r̂B = 2, the
FIGURE 5. The various profiles (U11 , V11 )/||f11 || versus r̂, where ||f11 || = [ 1 B (U11
first branch of the bifurcation curves). (a) PBb /(−γ1 C4∗ ) = −10, (b) PBb /(−γ1 C4∗ ) = 0, and (c) PBb /(−γ1 C4∗ ) = 10.
The dashed lines - - - are U11 /||f11 || and the solid lines —– are V11 /||f11 ||.
3.3
Solution in the neighborhood of a bifurcation point
The bifurcation point has been determined. However, the boundary-value problem for U11 (r̂) and V11 (r̂)
[Eqs. (9) and (10)] being homogeneous, U11 (r̂) and V11 (r̂), the leading term of the bifurcated solution, are
undetermined by a constant factor at this stage. It requires higher-order analysis to determine this factor,
which is carried out in this subsection.
The equations for U21 (r̂) and V21 (r̂) are of the same form as those for U11 (r̂) and V11 (r̂); the boundaryvalue problem for Um1 (r̂) and Vm1 (r̂) (m ≥ 3) is inhomogeneous, and its homogeneous part is of the same
form as that for U11 (r̂) and V11 (r̂). Thus, the inhomogeneous part of the boundary-value problem for Um1 (r̂)
and Vm1 (r̂) (m ≥ 3) must satisfy some relation (solvability condition)3 for the solution Um1 (r̂) and Vm1 (r̂)
to exist. The homogeneous part of the boundary-value problem for Umn (r̂) and Vmn (r̂) (n 6= 1) has no
nontrivial solution, unless an additional relation among uθAb /γ1 , uθBb /γ1 , PBb /γ1 C4∗ , r̂B , and α is satisfied.
From the solvability condition of the boundary-value problem for U31 (r̂) and V31 (r̂), the undetermined
norm (for example, δkf11 k defined in the last paragraph of Section 3.2) of the solution δ(U11 (r̂), V11 (r̂)) is
3 The
solution of the adjoint problem of Eqs. (9) and (10) should be orthogonal to the inhomogeneous part.
651
1
0.5
0
1
1.5
2 1
1.5
(a)
2 1
1.5
(b)
2
(c)
FIGURE 6. Collapse of a roll for large |PBb | by the radial convection of the unperturbed flow (α = π, r̂B = 2,
uθBb /γ1 = −300, the first branch curve). (a) PBb /(−γ1 C4∗ ) = −5 , (b) PBb /(−γ1 C4∗ ) = 0, and (c) PBb /(−γ1 C4∗ ) = 20.
50
50
0
0
−50
−5
0 5
(a)
−50
−5 0
2
(b)
PBb /(−γ1 C4∗ )
FIGURE 7. The coefficients aA /aO , aB /aO , and aP /aO versus
(r̂B = 2, and α = π) on the first
branch. (a) uθBb = 0 and (b) uθBb /γ1 = −40. The parameter uθAb /γ1 is given in Figs. 1 and 3. The vertical chain
lines – - – - at PBb /(−γ1 C4∗ ) = −3.53365 and 3.12345 in (a) and at PBb /(−γ1 C4∗ ) = −5.63002 and 0.41251 in (b) are
the common asymptotes of the curves.
determined by the following equation:
(δkf11 k)[γ1 (aA ∆uθA + aB ∆uθB − aP ∆PB /C4∗ ) − aO (δkf11 k)2 ] = 0,
µ
¶2
δkf11 k
aB ∆uθB
aP ∆PB
aA ∆uθA
that is,
+
+
=
,
or δkf11 k = 0.
γ1
aO γ1
aO γ1
aO (−γ1 C4∗ )
(13)
(14)
where ∆uθA = uθA − uθAb , ∆uθB = uθB − uθBb , and ∆PB = PB − PBb , and the coefficients aA /aO , aB /aO ,
and aP /aO , which depend on the definition of the norm, are constants determined by the parameters uθAb /γ1 ,
uθBb /γ1 , PBb /(−γ1 C4∗ ), α, and r̂B . Their examples are given in Fig. 7 and in the fourth to sixth columns
of Table 1. The case aO = 0 should be analyzed separately and requires the analysis to higher order, and
then (δkf11 k/γ1 )4 , instead of (δkf11 k/γ1 )2 , is found to be the linear combination of ∆uθA /γ1 , ∆uθB /γ1 , and
∆PB /(−γ1 C4∗ ).
The second relation of Eq. (14) corresponds to the axially uniform solution. The first relation gives the
norm of bifurcated solution in the neighborhood of the bifurcation point. The δkf11 k/γ1 can be real or
time-independent bifurcated solutions are possible only when
aB ∆uθB
aP ∆PB
aA ∆uθA
+
+
> 0.
a O γ1
aO γ1
aO (−γ1 C4∗ )
652
(15)
1
1
0.5
0
1
0.5
1.5
2
0
1
(a)
1.5
2
(b)
FIGURE 8. Flow field (projection of streamlines on the r̂ẑ plane) (r̂B = 2, α = π) near a bifurcation point. (a) At
(∆uθA /γ1 = 0.01, ∆uθB = ∆PB = 0) from the bifurcation point [uθAb /γ1 = 34.207, uθBb = 0, PBb /(−γ1 C4∗ ) = −1] on
the first branch in Fig. 1. (b) At (∆uθA /γ1 = 0.01, ∆uθB = ∆PB = 0) from the bifurcation point [uθAb /γ1 = 134.973,
uθBb = 0, PBb /(−γ1 C4∗ ) = −1] on the second branch. The arrow indicates the direction of flow.
That is, the bifurcated solution is possible only when the variation of the parameters uθA , uθB , and PB
from uθAb , uθBb , and PBb satisfies the relation (15). For example, consider the case on the first branch for
r̂B = 2, uθBb = 0, and α = π. The value aA /aO is positive in −3.53365 < PBb /(−γ1 C4∗ ) < 3.12345 [Fig. 7
(a)], and therefore the bifurcated solution exists for positive ∆uθA /γ1 , that is, it extends to the upper side of
the bifurcation curve uθAb /γ1 versus PBb /(−γ1 C4∗ ) in Fig. 1 and in the other range of PBb /(−γ1 C4∗ ) [within
Fig. 7 (a)], the solution extends to the lower side. The bifurcation curve corresponds to the neutral curve (or
the boundary of linearly stable and unstable regions in the parameter space) of the axially uniform solution
for the class of solutions with time-scale of variation of the order of LA /(2RTA )1/2 ε or L2A /ν, where ν is the
kinematic viscosity. The side above the first branch in Fig. 1 is unstable one. The above-mentioned direction
should not be confused with the linearly stable or the unstable side. The complete time-dependent analysis
is not given in the present paper, but it is better to explain a little more.
The basic equations describing a time-dependent problem where the time-scale variation is of the order
of LA /(2RTA )1/2 ε or L2A /ν are given by the set of equations (2)–(3e) with ∂ur /∂ t̂, ∂uθ /∂ t̂, ∂uz /∂ t̂, and
∂(τ −2P/5)/∂ t̂ being added, respectively, to the left-hand sides on Eqs. (3b)–(3e), where t̂ = t(2RTA )1/2 ε/LA
(or = 2tν/L2A γ1 ) with t being the time.[6] From the linear-stability analysis of the axially uniform solution
on the basis of this set of equations, each of the Fourier components with respect to ẑ of the perturbation
develops independently with t̂ and consists of infinitely many components, each of which also develops
independently and exponentially with t̂ [say, exp(κn t̂) (n = 1, 2, · · · )]. The amplifying factor κn changes its
sign from negative to positive when the parameter uθAb /γ1 passes the n-th branch of the bifurcation curves
in Fig. 1 from below. For the complete time-dependent study, solutions with shorter time scale of variation
should be considered on the basis of the Boltzmann equation.
Finally, examples of the flow field, expressed with the terms up to the order of δ, in the neighborhood of
bifurcation points are shown in Fig. 8. The panel (a) shows some of the streamlines at a point near the first
bifurcation curve in Fig. 1, and the panel (b) shows those near the second one.
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Sone, Y., Handa, M., and Sugimoto, H., Transp. Theory Stat. Phys. 31, (2002) (to be published).
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