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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11164-11171
© Research India Publications. http://www.ripublication.com
11164
Effect of internal heating on weakly non-linear stability analysis of
Rayleigh-Bénard Convection in a vertically oscillating Micropolar fluid
Vasudha Yekasi
Department of Mathematics, Christ University, Bangalore, India.
S Pranesh
Department of Mathematics, Christ University, Bangalore, India.
Shahnaz Bathul
Department of Mathematics, Jawaharlal Nehru Technological University, Hyderabad, India.
Abstract
In this paper, we study the combined effect of internal heating
and time-periodic gravity modulation on thermal instability in
a micropolar fluid layer, heated from below. The time-
periodic gravity modulation, considered in this problem can
be realized by vertically oscillating the fluid layer. A weak
non-linear stability analysis has been performed to get
expression for Nusselt number using Ginzburg–Landau
equation derived from Lorenz equations. Effects of various
parameters such as internal Rayleigh number, Prandtl number,
amplitude and frequency of gravity modulation have been
analysed on heat transport. It is found that the response of the
convective system to the internal Rayleigh number is
destabilizing. Further, it is found that the heat transport can be
controlled by suitably adjusting the amplitude and frequency
of gravity modulation.
INTRODUCTION
In recent years, the dynamics of micropolar fluids has been a
popular area of research for engineers and scientists who
works with hydrodynamic problems (Lukaszewicz[1],
Eringen[2], Sheremet[3], Miroshnichenko[4], Dupuy[5],
Gibanov[6], Boukrouche[7], Benes[8]) . Several interesting
aspects are revealed during the study of these problems which
we cannot found in Newtonian fluids. These micropolar fluid,
which is non-symmetric to stress tensor, consists of randomly
oriented particles and each element has translation as well as
rotational motions. Eringen[2] was the first who formulated
general theory of the micropolar fluids. Siddheshwar and
Pranesh [9,10], Pranesh et al [11-14] have carried out
investigations in micropolar fluid under different situations.
In the research of fluid dynamics, much attention was drawn
to controlling the convection by considering the effects of
time periodic vertical oscillations or gravity modulation or g-
jitter. The complex body forces which make these oscillating
forces to arise occurs in different ways. This time periodic
vertical oscillations have applications in experiments of space
laboratory, crystal growth field, large-scale convection of
atmosphere and so on. Nelson[15], Wadih[16,17] carried out
experiments under microgravity conditions with materials
processing or physics of fluids in an arbitrary space craft.
Effect of g-jitter on the stability of a heated fluid layer was
first studied by Gershuni[18] and Gresho[19]. Some studies
related to time periodic vertical oscillations are
Siddheshwar[19,20], Pranesh [12,13], Bhadauria [21],
Lyubimova[22] and Maria et al[23].
In earth’s mantle, radioactive decay of elements produces heat
which induces convection [24], in atmosphere, motion takes
place due to heat absorption from sunlight [25] and the
convection resulted from volumetric heat produced by nuclear
fusion reaction is important in studying many astronomical
events [26]. These are the geophysical, astrophysical and
industrial processes which we come across convection
induced by internal heat generation. Literature that reports
internal heat generation is Bhattacharya [27], Takashima[28],
Pranesh et al [13,14,29,30] and recently Maria et al[23].
Several works have been done to study the onset of
convection and heat transfer in micropolar fluid with gravity
modulation. But, for non-linear analysis of micropolar fluid
with internal heat generation case, no study is being reported
under gravity modulation, where heat transport, in terms of
amplitude and frequency of modulation, is regulated in the
system because of vibrations produced in the system.
Objectives of this work are
(i) Deriving Ginzburg Landau equation from Lorenz
equations (Siddheswar[31]).
(ii) Quantifying the heat transport in the system by
considering different values to the parameters.
MATHEMATICAL FORMULATION
Consider an infinite horizontal layer of Boussinesquian ,
micropolar fluid of depth d, where the fluid is heated from
below with the internal heat generation existing within the
fluid system. Let ΔT be the temperature difference between
the lower and upper surfaces with the lower boundary at a
higher temperature than the upper boundary. These
boundaries maintained at constant temperature. A Cartesian
system is taken with origin in the lower boundary and z-axis
vertically upwards (see figure 1).
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11164-11171
© Research India Publications. http://www.ripublication.com
11165
Figure 1. Schematic diagram of the flow configuration
The governing equations are
Continuity Equation :
0. q , (1)
Conservation of Linear Momentum :
,)2(ˆ. 20
qktgpqqtq
(2)
,1)( 0 tCosgtg (3)
Conservation of Angular Momentum :
2.. 2
0
qqt
I , (4)
Conservation of Energy :
02
0
.. TTQTTC
TqtT
, (5)
Equation of State :
00 1 TT , (6)
where, is the velocity, 𝜌0 is density of the fluid at
temperature T = 𝑇0, p is the pressure, 𝜌 is the density, is
acceleration due to gravity, 𝑔0 is the mean gravity, ε is the
small amplitude of gravity modulation, γ is the frequency, ζ is
coupling viscosity coefficient or vortex viscosity, is the
angular velocity, I is moment of inertia, λ' and η' are bulk
and shear spin viscosity coefficients, T is the temperature, χ is
the thermal conductivity, β is micropolar heat conduction
coefficient, α is coefficient of thermal expansion, Q is the
internal heat source and t is time.
Basic State:
The basic state of the fluid is quiescent and is described by
).(),(),(,0,0,0),0,0,0( zTTzzppqq bbbbb
(7)
Substituting equation (7) into basic governing equations (1)-
(6), we obtain the following quiescent state solutions :
0ˆ)(10 ktCosgdz
dPb
b , (8)
02
2
TTQdz
Td b , (9)
)1 00 TTbb . (10)
Solving equation (9)subject to the boundary conditions
TTTb 0 at z = 0 and 0TTb at z = d,
we obtain
2
2
0
1
QdSin
dzQdSin
TTTb
. (11)
Stability Analysis:
The stability of the basic state is analyzed by introducing the
following perturbations
'''' ,,,, TTTpppqqq bbbbb
(12)
where, the prime indicates that the quantities are infinitesimal
perturbations.
Substituting equation (12) into the equations (1)-(6) and using
the basic state solutions, we get linearized equations
governing the infinitesimal perturbations in the form:
0. ' q , (13)
,)2(
ˆ1.
2
00
q
ktCosgpqqtq
(14)
2
.. 20
q
qt
I , (15)
TQT
CW
QdSin
dzQdCosQd
dT
zT
tT
2
02
22
1
,
(16)
.'0' T (17)
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11164-11171
© Research India Publications. http://www.ripublication.com
11166
Substituting equation (17) in equation (14) and eliminating
pressure by cross differentiation, we get
.
21
2
2
2
2
22
200
0
2
2
2
2
22
0
yxxzzy
xw
zu
xTtCosg
xw
zu
z
zxw
zuw
xw
zxuu
xw
zu
t
yyyy
(18)
Writing y-component of the equation (15) we get
.2
200
y
yyyy
xw
zu
zw
xu
ztI
(19)
We consider only two dimensional disturbances and thus
restrict ourselves to the xz-plane.
We now introduce the stream functions in the form
,,x
wz
u
(20)
which satisfies the continuity equation (13).
On using equations (20) in equations (16), (18) and (19), we
get
,,
)2(1
20
2
400
20
JxTtCosg
ty
(21)
,,
2
0
220
y
yyy
JIt
I
(22)
,,,0
2
0
TJTJC
QTTxCxz
TtT
y
yb
(23)
where, J stands for Jacobian. Let
.,,
,,,,,,,
2
*
2
*'
*
*
2
'*
'''***
dd
TTT
dd
ttdz
dy
dxzyx
yy
(24)
where, * denotes the non-dimensionalised quantity.
Substituting equation (24) into equations (21)-(23) we get the
following dimensionless (on dropping the asterisks for
simplicity);
,1
))(1(,Pr
1
21
41
22
yNNxTtCosRJ
t
(25)
,,Pr
2Pr
2
12
12
32
y
yyy
JNNNN
tN
(26)
,,,5
25
0
TJTJNRiTTNW
zT
tT
y
y
(27)
where,
0
Pr
(Prandtl Number),
300 Tdg
R (Rayleigh Number),
1N (Coupling Parameter),
22dIN (Inertia Parameter),
23d
N
(Couple Stress Parameter),
20
5dC
N
(Micropolar Heat Conduction Parameter),
2QdRi (Internal Rayleigh Number)
and
.
10
RiSinzRiCosRi
zT
Equations (25) to (27) are solved subject to free-free,
isothermal and no spin boundary conditions, given by
04
4
2
2
T
zzy
𝑎𝑡 𝑧 = 0 𝑎𝑛𝑑 𝑧 = 1.
Linear Stability Theory:
In this section, we discuss the linear stability analysis, which
is of great utility in the local nonlinear stability analysis to be
discussed further on. The linearized version of equations (25)-
(27), after neglecting the Jacobian , is
,
))(1(1Pr
1
21
221
yNxTtCosRN
t
(28)
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11164-11171
© Research India Publications. http://www.ripublication.com
11167
,2Pr
211
23
2
NNNt
Ny (29)
,502
yNWz
TTRi
t
(30)
Eliminating T and 𝜔𝑦 from equations (28) to (30), we get an
equation for 𝜓 in the form
,12Pr
1Pr
1
2Pr
02511
23
22
6221
221
21
23
2
zTfNNNNfN
xTR
Rit
NNt
Rit
NNt
N
(31)
where, 𝑓 = Real part of (𝑒−𝑖𝛺𝑡) and 𝑓 ′ = (−𝑖𝛺) Real part of
(𝑒−𝑖𝛺𝑡). In the dimensionless form, the velocity boundary
conditions for solving equation (31) are obtainable from
equations (28)-(30) in the form
08
8
6
6
4
4
2
2
zzzz
at z = 0,1. (32)
Finite Amplitude convection:
The finite amplitude analysis is carried out here via Fourier
series representation of stream function 𝜓, the spin 𝜔𝑦 and the
temperature distribution T. Linear analysis gives only the
condition for onset of convection, it is insufficient to explain
the rate of heat transport, therefore non-linear analysis is
carried out.
The first effect of non-linearity is to distort the temperature
field through the interaction of 𝜓 and T. The distortion of
temperature field will correspond to a change in the horizontal
mean, i.e., a component of the form 𝑆𝑖𝑛(2𝜋𝑧) will be
generated. Thus, a minimal double Fourier series which
describes the finite amplitude convection is given by
𝜓(𝑥, 𝑦, 𝑡) = 𝐴(𝑡)𝑆𝑖𝑛(𝜋𝛼𝑥)𝑆𝑖𝑛(𝜋𝑧)
𝜔𝑦(𝑥, 𝑦, 𝑡) = 𝐵(𝑡) 𝑆𝑖𝑛(𝜋𝛼𝑥)𝑆𝑖𝑛(𝜋𝑧) (33)
𝑇(𝑥, 𝑦, 𝑡) = 𝐸(𝑡)𝐶𝑜𝑠(𝜋𝛼𝑥)𝑆𝑖𝑛(𝜋𝑧) + 𝐹(𝑡)𝑆𝑖𝑛(𝜋𝑧)
where, A, B, E and F are the amplitudes to be determined
from the dynamics of the system. Since the spontaneous
generation of large scale flow has been discounted, the
functions 𝜓 and 𝜔𝑦 do not contain an x-independent term.
Substituting equation (33) into equations (25)-(27) and
following standard procedure, we obtain the following non-
linear non- autonomous system (generalized Lorenz model) of
differential equations:
)(Pr
)(Pr1)(Pr))(1(
)(
1
212
tBN
tAkNtEk
tCosRtA
, (34)
),(Pr2
)(Pr
)(Pr
)(2
1
2
21
2
23 tB
NN
tAN
kNtB
NkN
tB
(35)
),()()()(
)()()()(
25
2
205
0
tFtAtFtBN
tERiktAz
TtBNz
TtE
(36)
),()(2
)()(2
)()(4)(
2
52
2
tEtA
tEtBNtRiFtFtF
(37)
where, over dot denotes time derivative with respect to t. It is
important to observe that the non-linearities in equations (34)-
(37) stem from the convective terms in the energy equation
(4) as in the classical Lorenz system.
The Ginzburg-Landau equation from Lorenz model
In this section, we derive the Ginzburg- Landau equation from
Lorenz equations (34)-(37), from which we have
,21 BYBYA (38)
,
][1
][1
1
Pr][12
1
41
2
BtCosR
kN
AtCosR
kNAtCosR
kE
(39)
,
1
2
05
0
25
2
ERik
Az
TBNz
TCABN
F
(40)
Using equation (38) in equation (39) and then using the
resulting equation along with equation (38) in (40), we get
,
][1
543
tCosBYBYBY
E
(41)
8 9 10 11 12 13
6 7 14 15 16,
Y B Y Y Sin t B Y Y Sin t Y BF
Y B Y B Y Sin t Y Y B
(42)
where ,
,Pr 2
1
21
kNNY
,22
1
32
kNN
Y
,Pr
21
3R
kYY
,
Pr
Pr1 12
124
RYkNY
Y
,
Pr
PrPr1 212
21
5R
kNYkNY
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11164-11171
© Research India Publications. http://www.ripublication.com
11168
,12
6 YY
,22
52
7 YNY
,
][1
38 tCos
YY
,][1
32
49 tCos
YRikYY
,
][12
310
tCosY
Y
,][1
42
511 tCos
YRikYY
,
][12
412
tCosY
Y
,10
50
13 Yz
TN
zT
Y
,
][12
514
tCosY
Y
][1
52
15 tCosYRik
Y
,
20
50
16 Yz
TN
zT
Y
Substituting equations (38), (41) and (42) in equation (37), we
get a third order equation in B after neglecting the terms of the
type .,,,,,,,2
2
3
32
22
2
2
3
3
4
4
tBB
tBB
tBB
tB
tBB
tB
tB
tB
,3321 BPBP
tBP
(43)
where , 𝑃1 = 𝑀1 + 𝑀4 + 𝑀5 + 𝑀8 + 𝑀9 ,
𝑀1 = [1 + 𝜀 Cos[Ω𝑡] [M2 𝐶𝑜𝑠[Ω𝑡]+M3 Sin[Ωt]],
𝑀2 = 2𝑌14𝑌6𝜀 + 2Y12Y7Ω ,
𝑀3 = 2Y11Y7Ω𝜀+ 2(4𝜋2 − 𝑅𝑖)𝑌14𝑌6 + 2(4𝜋2 − 𝑅𝑖)Y12Y7,
𝑀4 = 2(4𝜋2 − 𝑅𝑖)[1 + 𝜀 Cos(Ω𝑡)]2 [Y15Y6 + Y16Y6] ,
𝑀5 = [1 + 𝜀 Cos(Ω𝑡)]3 [𝑀6Sin[Ωt] + 2(4𝜋2 − 𝑅𝑖)M7] ,
M6 = 2Y15Y6𝜀Ω ,
𝑀7 = Y16Y6 + Y13Y7,
M8 = 2(4𝜋2 − 𝑅𝑖)[Y15Y6 + Y11Y7][1 + 𝜀 Cos(Ω𝑡)]2 ,
M9 = [4Y14Y6𝜀Ω + 4Y12Y7𝜀Ω]Sin2[Ωt] ,
𝑃2 = −2(4𝜋2 − 𝑅𝑖)[Y15Y7[1 + 𝜀 Cos(Ω𝑡)]2 + Y16Y7[1 +𝜀 Cos(Ω𝑡)]3] − 4Y14Y7𝜀ΩSin2[Ωt] + 𝑀10 ,
𝑀10 = [1 + 𝜀 Cos(Ω𝑡)][−2(4𝜋2 − 𝑅𝑖)𝑌14𝑌7𝑆𝑖𝑛[Ω𝑡] −2𝑌14𝑌7Ω𝐶𝑜𝑠[Ω𝑡] − 2Y15Y7𝜀ΩSin[Ωt],
𝑃3 = [𝑌2 − 𝑁5]𝜋2𝛼𝑌5𝑌72[1 + 𝜀 Cos(Ω𝑡)]2] .
Equation (43) is obviously the Ginzburg-Landau equation for
non-linear convection in a micropolar fluid with vertical
oscillation and heat source. By using equation (43) in equation
(42), we get F.
Heat Transport
The influence of gravity modulation with internal heat source
on heat transport which is quantified in terms of Nusselt
number (Nu) is defined as follows:
𝑁𝑢 =𝐻𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡 𝑏𝑦 (𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛+𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛)
𝐻𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡 𝑏𝑦 (𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛)
0
2
0
0
2
0
)1(2
)1(2
z
k
z
z
k
z
dxzk
dxTzk
, (44)
where, subscript in the integrand denotes the derivative with
respect to z.
Substituting equation (33) in equation (44) and completing the
integration, we get
𝑁𝑢 = 1 − 2𝜋𝐹(𝑡).
RESULTS AND DISCUSSIONS
In the study of thermal instability in a fluid layer, external
regulation of convection is important. In this paper, time
depended vertical oscillation (gravity modulation) is
considered as an external force for convection. The non-linear
analysis is carried out in order to study the effects of vertical
oscillations and internal heating on heat transport in a
micropolar fluid. The fourth ordered non-autonomous Lorenz
model is obtained by using truncated Fourier series. Ginzburg-
Landau equation is then derived from Lorenz model. The
effect of Coupling Parameter 𝑁1, Inertia Parameter
𝑁2, Couple Stress Parameter 𝑁3, Micropolar Heat Conduction
Parameter 𝑁5, Prandtl number Pr , internal Rayleigh number Ri, amplitude 𝜀 and frequency Ω of the gravity modulation
are analyzed and the results are depicted in figures (2)-(9).
From the figures, we oberve that for small time, Nusselt
number, Nu is less than one, which indicates that the heat
transfer is due to conduction, as time increases, Nu also
increases and becomes greater than one, which shows that
convective region is in place. Further increase in time, the
graphs remains oscillatory which means that early chaos is
precipated.
From figures (2)- (9), we observe that
(i) Increase in internal Rayleigh number
increases the amount of heat transfer.
(ii) increase in 𝑁1 , decrease the rate of heat
transfer and thus stabilizes the system. This
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11164-11171
© Research India Publications. http://www.ripublication.com
11169
is on account of the presence of suspended
particles, the critical Rayleigh number
increases and hence the heat transport
decreases.
(iii) increase in 𝑁2, increases inertia of the fluid
due to the suspended particles and then
increases the amount of heat transfer.
(iv) increase in 𝑁3 increases couple stress of
the fluid and decreases gyrational velocities.
So, increases the measure of heat transfer.
(v) When 𝑁5 increases, the heat induced into
the fluid, due to this microelements, also
increase, thus reduces the heat transfer.
(vi) Pr reduces the measure of heat transfer.
(vii) amplitude 𝜀 of the gravity modulation
increase the measure of heat transfer
(viii) frequency Ω of the modulation decrease the
measure of heat transfer
CONCLUSION
(i) By adding suspended particles into
Newtonian fluids, heat transport decreases
and thus stabilizes the system.
(ii) 𝑁1, Ω and Pr decreases the rate of heat
transfer.
(iii) 𝑁2, 𝑁3 , 𝑁5, 𝜀 and Ri increases the rate of
heat transfer.
(iv) Heat transport can be controlled effectively
by gravity modulation.
ACKNOWLEDGEMENTS
Authors, S Pranesh and Vasudha Yekasi, would like to
acknowledge the management of Christ University for their
support in completing the work. Author, Shahnaz Bathul,
would like to acknowledge the management of Jawaharlal
Technological University for their support in completing the
work.
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11164-11171
© Research India Publications. http://www.ripublication.com
11170
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Figures:
Figure 2: Plot of Nusselt number 𝑁𝑢 versus time t for
various values of internal Rayleigh number Ri for
.Ω0.2,ε10,Pr1,N 2,N0.5,N0.1,N 5321 2
Figure 3: Plot of Nusselt number 𝑁𝑢 versus time t for
various values of Coupling Parameter 𝑁1 for
2.Ri2,Ω0.2,ε10,Pr1,N 2,N0.5,N 532
Figure 4: Plot of Nusselt number 𝑁𝑢 versus time t for
various values of Inertia Parameter 𝑁2 for 𝑁1 = 0.1, 𝑁3 =2, 𝑁5 = 1, 𝑃𝑟 = 10, 𝜀 = 0.2, Ω = 2, 𝑅𝑖 = 2.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11164-11171
© Research India Publications. http://www.ripublication.com
11171
Figure 5: Plot of Nusselt number 𝑁𝑢 versus time t for
various values of Couple Stress Parameter 𝑁3 for 𝑁1 =0.1, 𝑁2 = 0.5, 𝑁5 = 1, 𝑃𝑟 = 10, 𝜀 = 0.2, Ω = 2, 𝑅𝑖 = 2.
Figure 6: Plot of Nusselt number 𝑁𝑢 versus time t for
various values of Micropolar Heat Conduction Parameter
𝑁5 for 𝑁1 = 0.1, 𝑁3 = 2, 𝑁2 = 0.5, 𝑃𝑟 = 10, 𝜀 = 0.2, Ω =2, 𝑅𝑖 = 2.
Figure 7: Plot of Nusselt number 𝑁𝑢 versus time t for
various values of Prandtl number 𝑃𝑟 and internal Rayleigh
number Ri for 𝑁1 = 0.1, 𝑁2 = 0.5, 𝑁3 = 2, 𝑁5 = 1, 𝜀 =0.2, Ω = 2, 𝑅𝑖 = 2.
Figure 8: Plot of Nusselt number 𝑁𝑢 versus time t for
various values of amplitude of modulation 𝜀 for 𝑁1 =0.1, 𝑁2 = 0.5, 𝑁3 = 2, 𝑁5 = 1, 𝑃𝑟 = 10, Ω = 2, 𝑅𝑖 = 2.
Figure 9: Plot of Nusselt number 𝑁𝑢 versus time t for
various values of frequency of modulation Ω for 𝑁1 =
0.1, 𝑁2 = 0.5, 𝑁3 = 2, 𝑁5 = 1, 𝜀 = 0.2, 𝑅𝑖 = 2, , 𝑃𝑟 = 10.
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