Transp Porous Med (2012) 92:633–647
DOI 10.1007/s11242-011-9925-4
Weakly Nonlinear Stability Analysis
of Temperature/Gravity-Modulated Stationary
Rayleigh–Bénard Convection in a Rotating
Porous Medium
B. S. Bhadauria · P. G. Siddheshwar ·
Jogendra Kumar · Om P. Suthar
Received: 2 September 2011 / Accepted: 10 December 2011 / Published online: 28 December 2011
© Springer Science+Business Media B.V. 2011
Abstract The effect of time-periodic temperature/gravity modulation on thermal instability in a fluid-saturated rotating porous layer has been investigated by performing a weakly
nonlinear stability analysis. The disturbances are expanded in terms of power series of amplitude of convection. The Ginzburg–Landau equation for the stationary mode of convection is
obtained and consequently the individual effect of temperature/gravity modulation on heat
transport has been investigated. Further, the effect of various parameters on heat transport
has been analyzed and depicted graphically.
Keywords Ginzburg–Landau equation · Porous medium · Temperature modulation ·
Gravity modulation · Rotation
List of Symbols
Latin Symbols
A Amplitude of streamline perturbation
d Height of the fluid layer
g Acceleration due to gravity
B. S. Bhadauria (B)
Department of Applied Mathematics and Statistics, School for Physical Sciences,
Babasaheb Bhimrao Ambedkar University, Lucknow 226025, UP, India
e-mail: [email protected]
P. G. Siddheshwar · O. P. Suthar
Department of Mathematics, Bangalore University, Bangalore, Karnataka, India
e-mail: [email protected]
O. P. Suthar
e-mail: [email protected]
B. S. Bhadauria · J. Kumar
Department of Mathematics, Faculty of Science, DST—Centre for Interdisciplinary
Mathematical Sciences, Banaras Hindu University, Varanasi 221005, UP, India
e-mail: [email protected]
123
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B. S. Bhadauria et al.
kc
η2
Nu
p
Wavenumber
kc2 + π 2
Nusselt number
Pressure
Prandtl number, κδνT
Darcy number, dK2
Pr Vadasz number, Da
Pr
Da
Va
Ra
Ta
T
T
t
(x, y, z)
d Kz
Rayleigh–Darcy number, βT gT
2 2 δνκT
Taylor–Darcy number, δνd
Temperature
Temperature difference across the fluid layer
Time
Space co-ordinates
Greek Symbols
δ Porosity
α Coefficient of thermal expansion
κ Thermal diffusivity
μ Dynamic viscosity ν
ρ
ω
δ1
δ2
φ
τ
Kinematic viscosity, ρμ0
Fluid density
Modulation frequency
Amplitude of temperature modulation
Amplitude of gravity modulation
Phase angle
Perturbation parameter
τ = 2 t (small time scale)
Other Symbol
∇1 2
∂2
∂x2
+
∂2
∂z 2
Subscripts
b Basic state
c Critical value
0 Value of the un-modulated case
Superscripts
Perturbed quantity
∗ Dimensionless quantity
123
Weakly Nonlinear Stability Analysis
635
1 Introduction
The classical Rayleigh–Bénard convection due to bottom heating is widely known and is a
highly explored phenomenon. It has been extensively studied in a porous domain also (see
Nield and Bejan 2006). It has numerous applications in many practical problems. A comprehensive exposition on its applications in various fields is given in Nield and Bejan (2006),
Ingham and Pop (2005) and Vafai (2005). The study of fluid convection in a rotating porous
medium is also of great practical importance in many branches of modern science, such
as centrifugal filtration processes, petroleum industry, food engineering, chemical engineering, geophysics, and biomechanics. Several studies on the onset of convection in a rotating
porous medium have been reported (Friedrich 1983; Patil and Vaidyanathan 1983; Palm and
Tyvand 1984; Jou and Liaw 1987a,b; Qin and Kaloni 1995; Vadasz 1996a,b, 1998; Vadasz
and Govender 2001; Straughan 2001; Desaive et al. 2002; Govender 2003).
In most studies related to thermal instability in a rotating porous medium saturated with
a Newtonian fluid, steady temperature gradient is considered. However it is not so in many
practical problems. There are many interesting situations of practical importance in which
the temperature gradient is a function of both space and time. This non-uniform temperature
gradient (temperature modulation) can be determined by solving the energy equation with
suitable time-dependent thermal boundary conditions and can be used as an effective mechanism to control the convective flow. Innumerable studies are available which explain how
a time-periodic boundary temperature affects the onset of Rayleigh-Bénard convection. An
excellent review related to this problem is given by Davis (1976).
The study of Venezian (1969) or the Floquet theory has been extensively followed in the
thermal convection problem in porous media when the boundary temperatures are timeperiodic (Caltagirone 1976; Chhuon and Caltagirone 1979; Antohe and Lage 1996;
Malashetty and Wadi 1999; Malashetty and Basavaraja 2002, 2003; Malashetty et al. 2006;
Malashetty and Swamy 2007; Bhadauria 2007a,b; Bhadauria and Sherani 2008; Bhadauria
and Srivastava 2010). However, fewer studies are available on convection in a rotating porous
media with temperature modulation of the boundaries [see Bhadauria 2007c,d; Bhadauria
and Suthar 2009 and references therein].
Another problem that leads to variable coefficients in the governing equations of thermal instability in porous media is the one involving vertical time-periodic vibration of the
system. This leads to the appearance of a modified gravity, collinear with actual gravity,
in the form of a time-periodic gravity field perturbation and is known as gravity modulation or g-jitter in the literature. Malashetty and Padmavathi (1997), Rees and Pop (2000,
2001, 2003), Govender (2005a,b), Kuznetsov (2006a, b), Siddhavaram and Homsy (2006);
Strong (2008a,b); Razi et al. (2009); Saravanan and Purusothaman (2009), Saravanan and
Arunkumar (2010), Malashetty and Swamy (2011), and Saravanan and Sivakumar (2010,
2011) document some aspects of the problem.
The studies so far reviewed concern linear stability of the thermal system in a nonrotating/rotating porous media in the absence/presence of temperature/gravity modulation,
and hence address only questions on onset of convection. If one were to consider heat and
mass transports in porous media in the presence of temperature/gravity modulation, then the
linear stability analysis is inadequate, and the nonlinear stability analysis becomes inevitable. In the light of the above, we make a weakly nonlinear analysis of the problem using the
Ginzburg–Landau equation and in the process quantify the heat and mass transports in terms
of the amplitude governed by the Ginzburg–Landau equation. The Ginzburg–Landau equation has been solved numerically, and consequently the effect of modulation on the Nusselt
number is studied.
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B. S. Bhadauria et al.
2 Mathematical Formulation
We consider a horizontal porous layer saturated with an incompressible viscous Boussinesq
fluid confined between two parallel horizontal planes z = 0 and z = d. The porous layer is
heated from below and cooled from above. A Cartesian frame of reference is chosen in such
a way that the origin lies on the lower plane and the z axis as vertically upward. The system
is rotating about z axis with a uniform angular velocity . The governing equations for the
system are given by
∇ · q = 0,
1 ∂ q
ρ
2
1
ν
× q − q ,
= − ∇ p + g − δ ∂t
ρ0
ρ0
δ
K
∂T
1
γ
+ (
q · ∇)T = κT ∇ 2 T,
∂t
δ
ρ = ρ0 [1 − βT (T − T0 )] ,
(1)
(2)
(3)
(4)
=
where q is the velocity, ρ is the density of the fluid, p is hydrodynamic pressure, (0, 0, ) is the angular velocity, T is the temperature, and κT is the thermal conductivity of
the fluid.
We assume the externally imposed boundary temperature to oscillate with time, according
to the relations (Venezian 1969):
T = T0 +
T 1 + 2 δ1 cos(ωt)
2
T 1 − 2 δ1 cos(ωt + φ)
= T0 −
2
at z = 0,
(5)
at z = d,
where T0 is some reference temperature, and ω is the modulation frequency. The quantity
2 δ1 is the amplitude of modulation. We consider small amplitude modulation such that both
and δ1 are small.
We use the following scalings to non-dimensionalize the governing equations 1–3
(x, y, z) = d(x ∗ , y ∗ , z ∗ ), (U, V, W ) =
T = (T ) · T ∗ .
δκT ∗
d2
(U , V ∗ , W ∗ ), t = t ∗ ,
d
κT
p=
δμκT ∗
p ,
K
(6)
The dimensionless form of the governing equations is given by
1 ∂q √
1
+ T a k̂ × q + q = −∇ p − Ra Da
− (T − T0 ) k̂,
V a ∂t
βT T
∂T
γ
+ (q · ∇) T = ∇ 2 T.
∂t
(7)
(8)
At the basic state, the fluid is assumed to be quiescent, and thus the basic state quantities are
given by
ρ = ρb (z, t);
123
p = pb (z, t); T = Tb (z, t); qb = (0, 0, 0).
(9)
Weakly Nonlinear Stability Analysis
637
The basic state equations from (2)–(4) are
d pb
(10)
= −ρb g,
dz
∂ Tb
∂ 2 Tb
γ
= 2 .
(11)
∂t
∂ z
Solution of the above Eq. 11 subject to the non-dimensionalized thermal boundary conditions
(5) is given by
Tb (z, t) = (1 − z) + 2 δ1 F(z, t),
where
(12)
F(z, t) = Re A(λ)eλz + A(−λ)e−λz e−iωt ,
ω
1 e−iφ − e−λ
; λ = (1 − i)
A(λ) =
.
λ
−λ
2 e −e
2
(13)
We now impose infinitesimal perturbations to the basic state in the form:
q = qb + Q , T = Tb + T , ρ = ρb + ρ,
(14)
where primes denote the perturbation quantities. Now from Eq. 7, we have
∂p
1 ∂U √
− T aV + U = − ,
V a ∂t
∂x
∂p
1 ∂W
+W =−
+ RaT.
V a ∂t
∂z
From Eqs. 15 and 16, we eliminate pressure term to obtain an equation in the form:
1
∂ ∂U
∂W
∂T
−
= −Ra
,
V a ∂t ∂z
∂x
∂x
(15)
(16)
(17)
Also we have:
√
1 ∂V
(18)
+ T aU + V = 0,
V a ∂t
Introducing stream function as U = ∂/∂z and W = −∂/∂ x in the Eqs. 8, 17, and 18
and using Tb in the reduced system, we get the equations as
√ ∂V
1 ∂∇ 2
∂T
+ ∇2 − T a
+ Ra
= 0,
(19)
V a ∂t
∂z
∂x
√ ∂
1 ∂
+ 1 V + Ta
= 0,
(20)
V a ∂t
∂z
∂F
∂ (, T ) ∂
∂
− ∇1 2 T =
+
−1 + 2 δ1
.
(21)
γ
∂t
∂ (x, z)
∂x
∂z
3 Stability Analysis
3.1 Linear Stability Analysis for Marginal Stationary Convection in Unmodulated System
To make this article self-contained, we give a brief account of important linear stability analysis results. The linear stability results can be obtained by neglecting the nonlinear terms in
123
638
B. S. Bhadauria et al.
the above equations. The perturbed quantities are expanded by using normal mode technique
as
⎫
(x, z) = A sin(kc x) sin(π z) ⎬
T (x, z) = B cos(kc x) sin(π z) .
(22)
⎭
V (x, z) = C sin(kc x) cos π z)
In the above expressions, kc is the wave number. The Rayleigh number for stationary is given
by
η2 η2 + π 2 T a
st
Ra =
,
(23)
kc 2
where η2 = π 2 + kc2 , and the solution of , T and V is obtained as
⎫
(x, z, t) = A sin(kc x) sin(π z)
⎪
⎪
⎪
⎪
kc
⎬
T (x, z, t) = − 2 A cos(kc x) sin(π z)
.
η√
⎪
⎪
π Ta
⎪
⎭
A sin(kc x) cos(π z) ⎪
V (x, z, t) = −
η2
(24)
We note at this point that the oscillatory convection is possible (see Vadasz 1998), but we
focus our attention on only stationary convection in this article.
3.2 Nonlinear Stability Analysis
We use the time variations only at the slow time scale by considering τ = 2 t.
⎡
⎤
√
2 ∂
∂
∂
2
2
∇
−
−
∇
Ra
T
a
1
⎢ V a ∂τ 1
∂x
∂z ⎥
⎢
⎥
∂
∂
F
∂
⎢
⎥
0
γ 2
− ∇1 2
⎢ − −1 + 2 δ1
⎥
⎢
⎥
∂z ∂ x
∂τ
⎣
⎦
2
√
∂
∂
Ta
0
+1
∂z
V a ∂τ
⎡
⎤
⎡ ⎤
0
⎢ ∂(, T ) ⎥
⎢
⎥.
⎦
⎣
× T =⎣
⎦
∂(x,
z)
V
0
Now we use the following asymptotic expansion in Eqs. 19–21:
⎫
Ra = Ra0 + 2 Ra2 + · · · ⎪
⎪
⎬
= 1 + 2 2 + · · ·
,
2
T = T1 + T2 + · · · ⎪
⎪
⎭
2
V = V1 + V2 + · · ·
(25)
(26)
where Ra0 is the critical Rayleigh number in unmodulated case. Substituting Eq. 26 in Eq. 25
and comparing like powers of on both side, we get the solutions at different orders.
At first order:
⎡
√
∂
∂ ⎤
−∇1 2 Ra0
− Ta
⎡
⎤ ⎡ ⎤
⎢
∂x
∂z ⎥ 1
0
⎢
⎥
∂
⎢
⎥ ⎣ T1 ⎦ = ⎣ 0 ⎦ .
(27)
−∇1 2
0
⎢ ∂x
⎥
⎣√
⎦ V1
0
∂
0
1
Ta
∂z
123
Weakly Nonlinear Stability Analysis
639
The above equations correspond to linear stability equations for stationary mode of convection. The solutions of the above equations can be given by [using Eq. 24]
⎫
1 = A(τ ) sin(kc x) sin(π z)
⎪
⎪
⎬
kc
T1 = − 2 A(τ ) cos(kc x) sin(π z)
,
(28)
⎪
η√
⎪
⎭
V1 = −π T a A(τ ) sin(kc x) cos(π z)
The system (27) gives the critical value of Rayleigh number and corresponding wave number
as obtained in the previous section.
At the second order, we have
⎡
√
∂
∂ ⎤
−∇1 2 Ra0
− Ta
⎡
⎤ ⎡
⎤
⎢
∂x
∂z ⎥ 2
R21
⎢
⎥
∂
⎢
⎥ ⎣ T2 ⎦ = ⎣ R22 ⎦ ,
(29)
−∇1 2
0
⎢ ∂x
⎥
⎣√
⎦
V
R
2
23
∂
0
1
Ta
∂z
where
R21 = 0,
R22 =
(30)
∂(1 , T1 )
πk 2
= − 2c [A(τ )]2 sin(2π z),
∂(x, z)
2η
(31)
and
R23 = 0.
One can obtain second-order solutions as
2 = 0
T2 = −
V2 = 0
(32)
⎫
⎪
⎪
⎬
kc2
[A(τ )]2 sin(2π z) .
⎪
8πη2
⎪
⎭
(33)
The horizontally averaged Nusselt number N u(τ ) for the stationary mode of convection (the
preferred mode in this problem) is given by
kc 2π/kc
(1
−
z
+
T
)
dx
2
z
2π x=0
z=0 .
N u(τ ) =
(34)
kc 2π/kc
2π x=0 (1 − z)z dx
z=0
Substituting Eq. 33 in Eq. 34 and simplifying, we get
N u(τ ) = 1 +
kc2 [A(τ )]2
.
4η2
At the third order, we have
⎡
√
∂
∂
−∇1 2 Ra0
− Ta
⎢
∂x
∂z
⎢
∂
⎢
−∇1 2
0
⎢ ∂x
⎣√
∂
0
1
Ta
∂z
(35)
⎤
⎡
⎤ ⎡
⎤
⎥ 3
R31
⎥
⎥ ⎣ T3 ⎦ = ⎣ R32 ⎦ ,
⎥
⎦ V3
R33
(36)
123
640
B. S. Bhadauria et al.
where
∂ T1
1 ∂(∇ 2 1 )
−
,
∂x
Va
∂τ
∂1 ∂ F
∂ T1
∂1 ∂ T2
+ δ1
−γ
,
=
∂ x ∂z
∂ x ∂z
∂τ
1 ∂ V1
=−
.
V a ∂τ
R31 = −Ra2
(37)
R32
(38)
R33
Substituting 1 , T1 , T2 , V1 , and V2 from Eqs. 28 and 33 in Eqs. 37–39, we get
2
k2
η dA
− R2 c2 A sin(kc x) sin(π z),
R31 =
V a dτ
η
kc dA
k 3 A3
∂F
+ δ1
kc A − c 2 cos(2π z) cos(kc x) sin(π z),
R32 = γ 2
η dτ
∂z
4η
√
π T a dA
R33 =
sin(kc x) cos(π z).
V aη2 dτ
(39)
(40)
(41)
(42)
The solvability condition for the existence of the third-order solution is given by
1 2π/k
c
ˆ 1 R31 + Ra0 T̂1 R32 − V̂1 R33 ]dxdz = 0,
[
(43)
z=0 x=0
ˆ 1 , T̂1 , and V̂1 are the solutions of the adjoint system of the first order, given by
where ⎫
ˆ 1 = −A(τ ) sin(kc x) sin(π z)
⎪
⎪
⎪
⎪
kc
⎬
T̂1 = − 2 A(τ ) cos(kc x) sin(π z)
.
(44)
η
√
⎪
⎪
π Ta
⎪
⎪
A(τ ) sin(kc x) cos(π z) ⎭
V̂1 =
η2
Using Eqs. 40–42 and 44 in the Eq. 43 and simplifying, we get the Ginzburg–Landau equation
for stationary instability with a time-periodic coefficient in the form:
1
kc2
π 4 kc2 T a
γ dA(τ )
A3 (τ ) = 0, (45)
+
+ 2
− F (τ )A(τ ) +
Va
η
dτ
8η2
2V a 2 η2 + π 2 T a
where
F (τ ) =
Ra2
− δ2 I (τ ),
Ra0
(46)
and
1 I (τ ) =
0
∂ F(z, τ ) 2
sin (π z) dz.
∂z
(47)
The solution of Eq. 45, subject to the initial condition A(0) = a0 where a0 is a chosen initial
amplitude of convection, can be obtained by means of a numerical method. In calculations,
we may assume Ra2 = Ra0 to keep the parameters to the minimum.
123
Weakly Nonlinear Stability Analysis
641
4 Time-Periodic Gravity field
Now we study the effect of a periodically varying gravitational field. For this, we assume
g = g0 (1 + 2 δ2 cos(ωt)),
(48)
where g0 is some reference value of the gravitational force, ω is the frequency of modulation,
τ is the slow time scale defined in the previous section, and the gravitational force is given
by g = (0, 0, −g). It should be noted here that the basic temperature in the present case is
considered to be steady, and is given by
Tb = 1 − z.
(49)
Using above expression in the Eqs. 19–21, we get the equation corresponding to gravity modulation. The linear stability analysis can be performed exactly in the same way as described
in the above section, and the same results (Eqs. 23 and 35) can be readily drawn.
Further the system (25) reduces to the form:
⎡ 2
⎤
√
∂
2
2
2 δ cos(ωt)] ∂ − T a ∂
−
∇
Ra[1
+
∇
1
2
⎢ V a ∂τ 1
∂x
∂z ⎥
⎢
⎥
∂
∂
⎢
⎥
⎢
⎥
γ 2
− ∇1 2
0
⎢
⎥
∂x
∂τ
⎣
⎦
2
√
∂
∂
Ta
0
+1
∂z
V a ∂τ
⎡
⎤
⎡ ⎤
0
⎢ ∂(, ) ⎥
⎥
× ⎣T ⎦=⎢
(50)
⎣ ∂(x, z) ⎦ .
V
0
Also, we obtained here the same result at the first and second orders as reported in Eqs. 29
and 34. In particular, the Nusselt number for stationary mode of convection can be obtained
as
N u(τ ) = 1 +
kc2 [A(τ )]2
.
4η2
(51)
At the third order, the expression in Eq. 30, in this case, are obtained as
R31 = −[Ra2 + Ra0 δ2 cos(ωτ )]
1 ∂(∇1 2 1 )
∂ T1
−
,
∂x
Va
∂τ
(52)
∂ T1
∂1 ∂ T2
−γ
,
(53)
∂ x ∂z
∂τ
1 ∂ V1
R33 = −
.
(54)
V a ∂τ
For the above given values of R31 , R32 , and R33 , the solvability condition (Eq. 44) produces
the Ginzburg–Landau amplitude equation given by
1
γ dA(τ )
kc2
π 4 kc2 T a
A3 (τ ) = 0, (55)
+ 2
− F (τ )A(τ ) +
+
Va
η
dτ
8η2
2V a 2 η2 + π 2 T a
R32 =
where
F (τ ) =
Ra2
+ δ2 cos(ωτ ).
Ra0
(56)
123
642
B. S. Bhadauria et al.
5 Results and Discussions
In this article, we have studied the effect of temperature and gravity modulations on thermal
instability in a rotating fluid saturated porous layer. A weakly nonlinear stability analysis has
been performed to investigate the effect of temperature/gravity modulation on heat transport.
The temperature modulation has been considered in the following three cases:
1. In-phase modulation (IPM) (φ = 0),
2. Out-phase modulation (OPM) (φ = π) and
3. Lower-boundary modulated only (LBMO) (φ = −i∞),
The effect of temperature modulation has been depicted in Figs. 1, 2, 3 while that of gravity
modulation in Fig. 4. From the Figs. 1, 2, 3, we observe the following general result for
Nusselt number N u:
N u IPM < N u LBMO < N u OPM .
The parameters that arise in the problem are T a, V a, φ, δ1 , and δ2 and these parameters
influence the convective heat transports. The first two parameters relate to the fluid and the
structure of the porous medium, and the last three concern the two external mechanisms of
controlling convection.
Due to the assumption of low porosity medium, the values considered for V a are 100, 150
and 200. Because small amplitude modulations are considered, the values of δ1 and δ2 lie
between 0 and 0.5. Further, the modulation of the boundary temperature and the vertical
time-periodic fluctuation of gravity are assumed to be of low frequency. At low range of
frequencies, the effect of frequencies on onset of convection as well as on heat transport is
minimal. This assumption is required to ensure that the system does not pick up oscillatory
(a)
(b)
2.5
2.0
2.5
2.0
Nu
Nu
0.02, 0.05, 0.08
1.5
Va
100, Ta
60,
2
1.0
0.00
1, 2, 5
1.5
0.05, Ta
1.0
0.05
0.10
0.15
0.20
0.00
0.05
0.10
60, Va
100
0.15
0.20
(d) 3.0
(c)
2.5
60
Ta 30
Nu
Nu
2.5
2.0
100
1.5
200
150
Va 100
1.5
0.05, Va
100,
2
1.0
0.00
2.0
0.05, Ta
1.0
0.05
0.10
0.15
0.20
0.00
0.05
0.10
60,
2
0.15
Fig. 1 Variation of Nusselt number N u with time τ for different values of (a) δ1 , (b) ω, (c) T a, (d) V a
123
0.20
Weakly Nonlinear Stability Analysis
(a)
643
2.7
(b) 2.6
2.6
2.5
2.4
2.4
Nu
Nu
2.5
2.3
0.02
0.05
2.2
2.1
Va
100, Ta
2
5
2.2
60,
2.1
2
2.0
0.05, Ta
60, Va
100
2.0
0
5
10
15
20
0
(d)
(c)
5
10
15
20
3.0
2.5
2.5
2.0
Ta 30
60
100
200
150
Nu
Nu
1
2.3
0.08
2.0
1.5
Va 100
1.5
0.05, Va
100,
0.05, Ta
2
1.0
60,
2
1.0
0
5
10
15
0
20
5
10
15
20
Fig. 2 Variation of Nusselt number N u with time τ for different values of (a) δ2 , (b) ω, (c) T a, (d) V a
(a) 2.6
(b) 2.60
2.55
2.50
2.4
Nu
Nu
2.5
0.02
0.05
Va
100, Ta
0.08
1
2.40
2
5
2.35
2.3
60,
2.30
2
2.2
0.05, Ta
60, Va
100
2.25
0
5
10
15
20
0
5
10
15
20
15
20
(d) 3.0
(c)
2.5
2.5
2.0
60
Ta 30
100
Nu
Nu
2.45
1.5
2.0
Va 100
150
200
1.5
0.05, Va
100,
2
0.05, Ta
1.0
60,
2
1.0
0
5
10
15
20
0
5
10
Fig. 3 Variation of Nusselt number N u with time τ for different values of (a) δ1 , (b) ω, (c) T a, (d) V a
convective mode at the onset due to modulation in a situation that is conducive otherwise to
stationary mode.
In Fig. 1a–d, we have plotted the Nusselt number N u with respect to time τ for the case
of IPM. From the figures, we find that for small time τ , the value of N u does not alter and
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644
B. S. Bhadauria et al.
(a) 2.7
(b) 2.6
2.6
2.5
2.4
2.4
Nu
Nu
2.5
2.3
0.02
0.05
0.08
2.2
2.1
Va
100, Ta
25,
5
2
2.1
2
0.05, Ta
25, Va
100
2.0
0
5
10
15
20
0
3.0
(d) 3.0
2.5
2.5
2.0
Ta 30
60
Nu
Nu
1
2.2
2.0
(c)
2.3
100
1.5
5
10
15
20
15
20
200
2.0
Va 100
150
1.5
0.05, Va
100,
2
0.05, Ta
1.0
60,
2
1.0
0
5
10
15
20
0
5
10
Fig. 4 Variation of Nusselt number N u with time τ for different values of (a) δ1 , (b) ω, (c) T a, (d) V a
remains almost constant, then it increases on increasing τ , and finally becomes steady on
further increasing the time τ .
From the Fig. 1a, b, we observed that on increasing amplitude of modulation δ1 and frequency of modulation ω, the value of Nusselt number N u does not alter. Thus, the effects of
increasing δ1 and ω have negligible effect on rate of heat transfer in case of IPM. From Fig. 1c,
it has been examined that on increasing Darcy–Taylor number T a, the value of Nusselt number decreases, thus decreasing the rate of heat transport and hence stabilizing the system.
The effect of Vadasz number V a on N u is observed in Fig. 1d. From the figure, we find that
N u increases with increasing V a, thus advancing the convection. Further, we observed that
the results obtained in this case are qualitatively similar to those of unmodulated case.
Figure 2a–d shows the plots of N u with time τ for the case of OPM. From Fig. 2a, we find
that the effect of increasing modulation amplitude δ1 on N u is to increase the magnitude of
N u, i.e., rate of heat transport increases. Figure 2b shows that, on increasing the frequency of
modulation ω, the magnitude of N u does not change, but wavelength of oscillations become
shorter with increasing ω. From the Fig. 2c, we observe that on increasing T a, the value of
N u decreases, thus the effect of increasing T a is to decrease the rate of heat transfer, and
hence to suppress the convection. Figure 2d displays the effect of V a on N u. From the figure,
we examine that on increasing V a, N u increases, therefore, the effect of increasing V a is
to increase the rate of heat transfer across the porous layer.
It is obvious from the figure that
N u/δ1 =0.02 < N u/δ1 =0.05 < N u/δ1 =0.08 .
In Fig. 3a–d, we depict the variation of Nusselt number N u with time τ for lowerboundary temperature modulation only. From the figure, we find qualitatively similar results
to that in Fig. 2a–d.
123
Weakly Nonlinear Stability Analysis
645
In Fig. 4a–d, we depict the variation of N u with respect to slow time τ under gravity modulation. From Fig. 4a, we find that an increment in amplitude of modulation δ2 , increases
N u; however, the wavelength of oscillations remains unaltered. From Fig. 4b, which shows
the effect of frequency of modulation ω on N u, we observe that on increasing ω, the value
of N u remains unchanged wherein the wavelength of oscillations decreases. The effect of
rotation has been shown in Fig. 4c, and is found to have stabilizing effect. However, from
Fig. 4d, we observe that on increasing Vadasz number V a, the value of N u increases, therefore advancing the convection. Further, the results in Fig. 4a–d are found qualitatively similar
to that of Figs. 2a–d and 3a–d.
6 Conclusion
In the present study, the effects of temperature/gravity modulation in a rotating fluid saturated porous layer have been analyzed by performing nonlinear stability analysis using
Ginzburg–Landau equation. On the basis of the above discussion following conclusions are
drawn:
(1)
(2)
(3)
(4)
(5)
N u IPM < N u LBMO < N u OPM for temperature modulation.
The effect of increasing Taylor number T a on N u is to decrease the heat transport.
The effect of increasing Vadasz number V a is to advance the convection for in-phase
and out-phase as well as for the case when only lower plate is modulated.
On increasing the magnitude of modulation and frequency of modulation for the case
of in-phase temperature modulation, we find negligible effect on N u, while there is
significant effect in the cases of OPM and LBMO.
The results of gravity modulation are found qualitatively similar to the cases of OPM
and LBMO.
Acknowledgements Part of this study was done during the lien period sanctioned by Banaras Hindu University, Varanasi, India to the author BSB to work as Professor of Mathematics at Babasaheb Bhimrao Ambedkar
University, Lucknow. The authors are grateful to the referees for their most useful comments.
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