Effect of internal heating on weakly non-linear stability

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11164-11171

© Research India Publications. http://www.ripublication.com

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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).

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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)



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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)



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,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



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,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



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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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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


various values of Coupling Parameter 𝑁1 for

2.Ri2,Ω0.2,ε10,Pr1,N 2,N0.5,N 532


various values of Inertia Parameter 𝑁2 for 𝑁1 = 0.1, 𝑁3 =2, 𝑁5 = 1, 𝑃𝑟 = 10, 𝜀 = 0.2, Ω = 2, 𝑅𝑖 = 2.



11171


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.


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.


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.


various values of amplitude of modulation 𝜀 for 𝑁1 =0.1, 𝑁2 = 0.5, 𝑁3 = 2, 𝑁5 = 1, 𝑃𝑟 = 10, Ω = 2, 𝑅𝑖 = 2.


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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