Thermal instability in a rotating nanofluid layer: A revised model

Ain Shams Engineering Journal (2015) xxx, xxx–xxx

Ain Shams University

Ain Shams Engineering Journal

www.elsevier.com/locate/asejwww.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

Thermal instability in a rotating nanofluid layer: A

revised model

* Corresponding author. Tel.: +82 2 2123 2816.

E-mail addresses: [email protected] (D. Yadav),

[email protected] (G.S. Agrawal), jinholee@yonsei.

ac.kr (J. Lee).

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

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Please cite this article in press as: Yadav D et al., Thermal instability in a rotating nanofluid layer: A revised model, Ain Shams Eng J (2015), http://dx10.1016/j.asej.2015.05.005

Dhananjay Yadav a, G.S. Agrawal b, Jinho Lee a,*

a School of Mechanical Engineering, Yonsei University, Seoul 120 749, South Koreab Institute of Computer Application, Mangalayatan University, Aligarh, India

Received 20 January 2015; revised 22 April 2015; accepted 13 May 2015

KEYWORDS

Nanofluid;

Brownian motion;

Thermophoresis;

Thermal instability;

Rotation;

Critical Rayleigh number

Abstract In this paper, the analysis of thermal instability of rotating nanofluid layer is revised with

a physically more realistic boundary condition on the nanoparticle volumetric fraction i.e. the

nanoparticle flux is assumed to be zero rather than prescribing the nanoparticle volumetric fraction

on the rigid impermeable boundaries. This shows that the nanoparticle fraction value at the bound-

ary adjusts accordingly. In this respect, the present model is more realistic physically than those of

the previous investigations. The numerical computations are presented for water-based nanofluids

with alumina and copper nanoparticles. For Alumina-water nanofluid, zero flux nanoparticle

boundary condition has more destabilizing effect than the constant nanoparticle boundary condi-

tions, while reverse for copper–water nanofluid. The effect of rotation is found to have a stabilizing

effect. Further, volumetric fraction of nanoparticles /�0, the Lewis number Le, the density ratio Rq

and modified diffusivity ratio NA are found to destabilize the system.� 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction

In the current decade, nanofluids have started replacing ordi-nary fluids as a heat transfer medium due to their enhanced

abilities as heat exchangers and transporters. The term nano-fluid refers to the suspensions of nanoscale particles in the base

fluid. In occurrence of a very few percents of nanoparticles, asignificant increase of the heat transfer characteristics ofnanofluids has been found [1,2]. The convective heat transfer

characteristics of nanofluids depend on the thermo-physicalproperties of the base fluid and the ultra fine particles, the flowpattern and flow structure, the volume fraction of the sus-pended particles, the dimensions and the shape of these parti-

cles. The properties of nanofluids have made them potentiallyuseful in many practical applications in industrial, commercial,residential and transportation sectors [3–5].

Buongiorno [6] was the first researcher who proposed amodel for convective transport in nanofluids incorporatingthe effects of Brownian diffusion and thermophoresis. This

model was applied to study the thermal instability problemby Tzou [7,8] and observed that nanofluid is less stable thanregular fluid. Later, this problem was revisited by Nield and

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Nomenclature

a dimensionless wave number

ac critical wave numberc specific heatd diameter of nanoparticlesDB Brownian diffusion coefficient

DT thermophoretic diffusion coefficient~g acceleration due to gravityk thermal conductivity

Le Lewis numberNA modified diffusivity ratioNB modified specific heat increment

p pressurePr Prandtl number~v velocity of nanofluidRa thermal Rayleigh number

Rc critical Rayleigh numberRq density ratiot time

T temperatureTa Taylor numberx; y; zð Þ space co-ordinates

Greek symbols

a thermal diffusivityb coefficient of thermal expansion

l viscosity

q density of the nanofluidq0 reference density of nanofluidqp density of nanoparticlesqcð Þ heat capacity

/ volume fraction of the nanoparticles/�0 reference scale for the nanoparticle fractionnz z-component of vorticity

X angular velocityr2

P horizontal Laplacian operatorr2 Laplacian operator

Superscripts

* non-dimensional variables0 perturbed quantities

Subscripts

p particleb basic state0 lower boundary1 upper boundary

2 D. Yadav et al.

Kuznetsov [9] by taking different types of non-dimensionalparameters. An extension to the porous medium was made

by Nield and Kuznetsov [10], Kuznetsov and Nield [11] andChand and Rana [12,13]. The effect of internal heat sourceon the onset of nanofluid convection was examined by

Yadav et al. [14]. They found that the basic temperature distri-bution and the basic volumetric fraction of nanoparticle distri-bution deviate from linear to non-linear in the presence of

internal heating, and the critical Rayleigh number decreaseswith an increase in the internal heat source strength. The effectof magnetic field on the onset of nanofluid convection wasstudied by Yadav et al. [15,16] and Chand and Rana [17].

Nield and Kuznetsov [18] investigated the onset of double-diffusive nanofluid convection in a nanofluid layer. An exten-sion to the porous medium was made by Kuznetsov and Nield

[19], Yadav et al. [20], Rana et al. [21] and Rana and Agarwal[22]. Thermal conductivity and viscosity variation on the onsetof convection in nanofluid were studied by Nield and

Kuznetsov [23], Yadav et al. [24,25] and Umavathi et al.[26]. They obtained that the consequence of these factors wasto increase the critical value of the Rayleigh number. Non-linear two-dimensional convection in a nanofluid saturated

porous medium was studied by Bhadauria and Agarwal [27].During the last few decades, thermal instability in a rotat-

ing fluid layer has considerable interest due to its wide range

of applications in physics and engineering, including rotatingmachineries such as nuclear reactors, petroleum industry, bio-chemical and geophysical problems [28]. To attain the

improved performance of such applications the use of nanoflu-ids with higher thermal conductivities can be considered as a

Please cite this article in press as: Yadav D et al., Thermal instability in a rotating10.1016/j.asej.2015.05.005

working medium. Thermal instability problem for rotatingnanofluid layer was studied by Yadav et al. [29,30] and

Chand and Rana [31] with assumption that the value of thenanoparticle fraction at the boundary could manage in thesame way as the temperature. But in due course, it turned

out that physically the above boundary conditions may be dif-ficult to establish and the boundary condition for nanoparti-cles needs to be more realistic.

Very recently, Nield and Kuznetsov [32] suggested that thevalue of the temperature can be imposed on the boundaries,but the nanoparticle fraction adjusts so that the nanoparticleflux is zero on the boundaries. In this respect, this model is

more realistic physically than that employed by previousauthors. With this new boundary conditions, Agarwal [33]studied the effect of rotation on the onset of convection in a

Darcy porous medium. The effect of thermal non-equilibrium was studied by Agarwal et al. [34] and Yadavand Lee [35]. An anisotropic effect on the onset of nanofluid

convection was studied by Shivakumara and Dhananjaya[36]. They obtained that the anisotropy parameter is to hastenthe onset of convection.

In this paper, the problem of thermal instability of rotating

nanofluid is revised with new boundary condition that thenanoparticle flux is assumed to be zero rather than prescribingthe nanoparticle volume fraction on the rigid impermeable

boundaries. This change requires a rescaling of the parametersthat are involved. For this set of boundary conditions, the ana-lytical solution for the eigenvalue problem in exact form is not

possible and hence the resulting eigenvalue problem is solvednumerically using higher order Galerkin method. The

nanofluid layer: A revised model, Ain Shams Eng J (2015), http://dx.doi.org/



Thermal instability in a rotating nanofluid layer 3

numerical computations are presented for water-basednanofluids with alumina and copper nanoparticles. With newboundary conditions, the results are modified quantitatively

than the previous ones.

Problem formulation and analysis

The system considered here is a nanofluid layer which is keptrotating about vertical axis at a constant angular velocityX� ¼ 0; 0;X�ð Þ, and heated from below as shown in Fig.1.

We choose a Cartesian system of coordinate x; y; z in whichzaxis is taken at right angle to the boundaries. The nanofluidis confined between two parallel rigid plates z� ¼ 0 and

z� ¼ L, where temperature is kept constants at the boundaries:

T� ¼ T�0, at z� ¼ 0 and T� ¼ T�1 < T�0

� �, at z� ¼ L. Asterisks are

used to distinguish the dimensional variables from the non-dimensional variables (without asterisks). The governing equa-tions to describe the Boussinesq flow under this model are

[29,32]:

r� �~v� ¼ 0; ð1Þ

q0

@

@t�þ ~v� � r�ð Þ

� �~v� ¼ �r�p� þ lr�2~v�

þ /�qp þ 1� /�ð Þq0 1� b T� � T�0� ��

~gþ 2q0v� � ~X�; ð2Þ

@

@t�þ ~v� � r�ð Þ

� �/� ¼ DBr�

2

/� þDT

T�0r�2T�: ð3Þ

qcð Þ @

@t�þ ~v� � r�ð Þ

� �T� ¼ kr�2T�

þ qcð Þp DBr�/� � r�T� þDT

T�0

�r�T� � r�T�

� �; ð4Þ

Here~v� is the nanofluid velocity, t� is the time, q0 is the nano-fluid density at reference temperature T�0; p

� is the pressure, qp

is the density of the nanoparticles, b is the thermal expansion

coefficient, /� is the volumetric fraction of nanoparticles, c isthe specific heat (at constant pressure), l; q and k are the vis-cosity, density and thermal conductivity of nanofluid, respec-

tively, DB ¼kBT

�0

3pldpis the Brownian diffusion coefficient,

DT ¼ bl/�

q0is the thermophoretic diffusion coefficient of the

nanoparticles, kB is the Boltzmann’s constant and dp is the

diameter of nanoparticles.

z axis

( )* *0, 0, ,= ΩΩr

* *1 ,T T= *z L=

* *(0, 0, )g= −gr

y axis

O * *0 ,T T= x axis * 0z =

Figure 1 Sketch of the problem geometry and coordinates.


In this paper, the nanoparticle flux with the combination ofBrownian motion and thermophoresis is assumed to be zerorather than prescribing the nanoparticle volume fraction on

the rigid impermeable boundaries. This shows that thenanoparticle fraction adjusts itself. In this respect this modelis more realistic physically than previous. Thus the boundary

conditions are:

w� ¼dw�

dz�¼n�z ¼0; T� ¼T�0; DB

d/�

dz�þDT

T�0

dT�

dz�¼0 at z� ¼0; ð5aÞ

w� ¼dw�

dz�¼n�z ¼0; T� ¼T�1; DB

d/�

dz�þDT

T�0

dT�

dz�¼0 at z� ¼L: ð5bÞ

Here n�z is the z component of vorticity due to rotation. The

above boundary conditions mean no slip, fixed temperatureand mass impermeability at the top and bottom boundaries.

Introducing the following non-dimensional parameters:

x;y;zð Þ¼ x�;y�;z�ð ÞL

; t¼ t�

L2a; p¼ p�

q0a2L2; /¼/�

/�0; T¼T� �T�1

T�0�T�1;

ðu;v;wÞ¼ðu�;v�;w�ÞL

a; nz¼

n�zaL2; ð6Þ

where /�0 is a reference scale for the nanoparticle fraction and

a ¼ k=qcð Þ is the thermal diffusivity. Here c is the specific heatof nanofluid. Then, in the non-dimensional form, Eqs. (1)–(4)

can be written as:

r �~v ¼ 0; ð7Þ

@

@tþ ~v �rð Þ

� �~v¼�rpþPrr2~vþPr

ffiffiffiffiffiffiTa

p~v� ezð Þ

þ H �1�bDT�f gþRaPrT�H Rq�1�bDT��

/�0/�RaPr/�0/T

� ez; ð8Þ

@

@tþ ~v � rð Þ

� �/ ¼ 1

Le

r2/þNA

Le

r2T; ð9Þ

@

@tþ ~v � rð Þ

� �T ¼ r2TþNB

Le

r/ � rTþNANB

Le

rT � rT½ �;

ð10Þ

where H ¼ RaPr

bDT� and DT� ¼ T�0 � T�1� �

. In non-dimensional

form, the boundary conditions become:

w¼0;@w

@z¼0; nz¼0; T¼1;

d/dzþNA

dT

dz¼0 at z¼0; ð11aÞ

w¼0;@w

@z¼0; nz¼0; T¼0;

d/dzþNA

dT

dz¼0 at z¼1: ð11bÞ

The non-dimensional parameters that appear in the Eqs.(7)-(11) are as given below:

Ra¼ gL3bDT�

amðRayleigh numberÞ; Le¼

aDB

ðLewis numberÞ;

Rq¼qp

q0

ðdensity ratioÞ; Ta¼4X�

2

m2L4 ðTaylor numberÞ;

Pr¼taðPrantdl numberÞ;

NB¼qcð Þpqcð Þ /�0 ðmodified specific heat incrementÞ;

NA¼DT

DBT�1

T�0�T�1� �

/�0ðmodified diffusivity ratioÞ:

ð12Þ




4 D. Yadav et al.

Basic flow

The basic flow state of the nanofluid is assumed to be timeindependent and is described by v ¼ 0;T ¼ TbðzÞ; p ¼pbðzÞ;/ ¼ /bðzÞ. Then from Eqs. (8)–(10), equations governingto basic flow are:

� @pb@zþH �1� bDT�f g þ RaPrTb �H Rq � 1� bDT�

� �/�0/b

� RaPr/�0/bTb ¼ 0; ð13Þ

d2/b

dz2þNA

d2Tb

dz2¼ 0; ð14Þ

d2Tb

dz2þNB

Le

dTb

dz

�d/b

dzþNA

dTb

dz

� �¼ 0: ð15Þ

The boundary condition for TbðzÞ and /bðzÞ is:

Tb ¼ 1;d/b

dzþNA

dTb

dz¼ 0 at z ¼ 0; ð16aÞ

Tb ¼ 0;d/b

dzþNA

dTb

dz¼ 0 at z ¼ 1: ð16bÞ

Under the boundary conditions (16), the integration of Eq.(14) gives

d/b

dzþNA

dTb

dz¼ 0: ð17Þ

Using Eq. (17) in the Eq. (15), we get

d2Tb

dz2¼ 0; ð18Þ

whose solution satisfying the boundary conditions (16) shall be

TbðzÞ ¼ 1� z: ð19Þ

Eq. (17) then integrates with the boundary conditions (16), wehave

/bðzÞ ¼ NAzþ /�0; ð20Þ

where /�0 is a reference scale for the nanoparticle fraction.

Perturbation solution

We now superimpose small perturbations on the basic state asgiven below:

~v ¼~v0;T ¼ TbðzÞ þ T0; p ¼ pbðzÞ þ p0;/ ¼ /bðzÞ þ /0;

nz ¼ n0b;z þ n0z ð21Þ

substitute into Eqs. (7)–(10), and linearized by neglectingproducts of primed quantities. The following equations are

obtained when Eqs. (19) and (20) are used:

r �~v0 ¼ 0; ð22Þ

@~v0

@t¼�rp0 þPrr2~v0 þPr


p~v0 � ezð Þ

þ RaPrT0 �H Rq�1�bDT�

� �/�0/

0 �RaPr/�0 Tb/

0 þ/bT0ð Þ

� ez;

ð23Þ

@/0

@tþ w0

@/b

@z¼ 1

Le

r2/0 þNA

Le

r2T0; ð24Þ


@T0

@tþ w0

@Tb

@z¼ r2T0 þNB

Le

@Tb

@z

@/0

@zþ @/b

@z

@T0

@z

�þ 2

� NANB

Le

@Tb

@z

@T0

@z: ð25Þ

Taking operator curl on Eq. (23), we get

Prr2 � @

@t

� �~n0 þ Pr


pr� v0 � ezð Þ þ RaPr r� T0ezð Þ

�H Rq � 1� bDT��

/�0 r� /0ezð Þ� RaPr/

�0 r� Tb/

0ezð Þ þ r � /bT0ezð Þf g ¼ 0: ð26Þ

Here ~n0 ¼ r � v0 is vorticity.Operating on Eq. (26) with curl, in turn, after use of the

identity curl curl � grad div�r2 together with Eq. (22) gives

Prr4 � @

@tr2

� �v0 � Pr


p @~n0

@z

þ RaPr r2 T0eZð Þ � r @T0

@z

� ��H Rq � 1� bDT�

� �

� /�0 r2 /0eZð Þ � r @/0

@z

� �

� RaPr/�0 r2 Tb/

0eZð Þ � r @

@zTb/

0ð Þ ��

þ r2 /bT0ezð Þ � r @

@z/bT

0ð Þ ��

¼ 0: ð27Þ

The z-component of Eqs. (26) and (27) are:

Prr2 � @

@t

� �n0z þ Pr


p @w0

@z¼ 0; ð28Þ

Prr4 � @

@tr2

� �w0 � Pr


p @n0z@zþ RaPrr2

pT0

�H Rq � 1�

� bDT�g/�0r2p/0

� RaPr/�0 r2

p Tb/0ð Þ

hþr2

p /bT0ð Þ� ¼ 0: ð29Þ

For simplicity, we restrict our analysis to two dimensionalrolls, so that all physical quantities are independent of y whichallows us to define a stream function w such that

u0 ¼ �@w=@z;w0 ¼ @w=@x. Using the definition of stream func-tion, the governing equations can be derived from Eqs. (24),(25), (28) and (29) as follows:

@/0

@tþNA

@w@x¼ 1

Le

r2/0 þNA

Le

r2T0; ð30Þ

@T0

@t� @w@x¼ r2T0 �NANB

Le

@T0

@z�NB

Le

@/0

@z; ð31Þ

Prr2 � @

@t

� �n0z þ Pr


p @

@z

@w@x

�¼ 0; ð32Þ

Prr4 � @

@tr2

� �@w@x

�� Pr


p @n0z@zþ RaPrr2

pT0

�H Rq � 1� bDT��

/�0r2p/0

� RaPr/�0 r2

p Tb/0ð Þ þ r2

p /bT0ð Þ

h i¼ 0: ð33Þ

Very recently, Nield and Kuznetsov [32] and Agarwal [33]observed that the oscillatory convection is ruled out fornanofluids with this new set of boundary conditions due to





very large nanofluid Lewis number. Hence, we can take that

the perturbation quantities are of the form as [37]:

w ¼ wðzÞ cos ax; T0 ¼ HðzÞ sin ax; /0

¼ UðzÞ sin ax and n0z ¼ ZðzÞ sin ax ð34Þ

where a is the horizontal wave number of the disturbances. Onsubstituting Eq. (34) into the differential Eqs. (30)–(33), thelinearized equations in dimensionless form are as follows:

NAawþ1

Le

D2 � a2� �

UþNA

Le

D2 � a2� �

H ¼ 0; ð35Þ

�awþ D2 � a2 �NANB

Le

D

�H�NB

Le

DU ¼ 0; ð36Þ

D2 � a2� �

Zþ affiffiffiffiffiffiTa

pDw ¼ 0; ð37Þ

� a D2 � a2� �2

w�ffiffiffiffiffiffiTa

pDZ� a2RaH

þ a2Ra

bDT�Rq � 1�

� bDT�g/�0Uþ a2Ra/�0 1� zð ÞU½

þ NAzþ /�0� �

H� ¼ 0: ð38Þ

The boundary conditions become:

w ¼ Dw ¼ Z ¼ H ¼ 0; DUþNADH ¼ 0 at z ¼ 0 and 1:

ð39Þ

Here D � ddz.

Method of solution

Eqs. (35)–(38) together with the boundary conditions Eq. (39)

constitute a linear eigenvalue problem of the system. Theresulting eigenvalue problem is solved numerically using theGalerkin weighted residuals method. In this method, the test

(weighted) functions are the same as the base (trial) functions.Accordingly W;H;U, and Z are taken in the following way:

w ¼XNs¼1

Asws; H ¼XNs¼1

BsHs; U ¼XNs¼1

CsUs and Z ¼XNs¼1

DsZs;

ð40Þ

where As;Bs;Cs and Ds are constants. The base functions

Ws;Hs;Us and Zs are represented by power series as trivialfunctions satisfying the respective boundary conditions andare assumed in the following form:

Ws ¼ zsþ1 � 2zsþ2 þ zsþ3; Hs ¼ Zs ¼ zs � zsþ1;

Us ¼ NA zsþ1 � zs� �

; s ¼ 1; 2; 3; . . . ð41Þ

Using Eq. (41) into Eqs. (35)–(38) and multiplying Eq. (35) by

Us, Eq. (36) by Hs, Eq. (37) by Zs, Eq. (38) by Ws; performingthe integration by parts with respect to z between z ¼ 0 and 1,we obtain a system of 4s linear algebraic equations in the 4sunknowns As;Bs;Cs and Ds. For the existence of non-trivial

solution, the determinant of the coefficient matrix must vanish,which gives the characteristic equation for the system, withRayleigh number Ra as the eigenvalue of the characteristic

equation. The critical Rayleigh number Rc is obtained by min-imizing Ra with respect to the wave number a for different


fixed values of the other parameters. The CPU time of calcula-

tion is 0.235.

Results and discussion

The effect of rotation on the onset of convection in a nanofluidlayer is investigated by considering a more physically realisticboundary condition on the volume fraction of nanoparticles

i.e. the flux of the volume fraction of nanoparticles is zero con-dition is used at the boundaries. The numerical computationsare presented for water-based nanofluids with alumina andcopper nanoparticles. For the boundary conditions consid-

ered, it is not possible to obtain exact analytical solution usingsingle-term Galerkin method, and hence we have to resort 6-term Galerkin method to solve the resulting eigenvalue prob-

lem for different values of Ta;/�0;NA;NB;Le and Rq. The

Newton–Raphson method is used to obtain the Rayleigh num-ber Ra as a function of wave numbera and the bisection

method is built-in to locate the critical stability parametersðRc; acÞ. According to Buongiorno [6], the values of physicalparameters for alumina-water nanofluid and copper–water

nanofluid are provided in Tables 1 and 2.In order to validate the exactness of the present results, first

test computations are carried out in the absence of nanoparti-

cles, i.e. for regular fluid by taking /�0 ¼ 0. The critical

Rayleigh number Rc and the corresponding critical wave num-ber ac for different values of Ta are obtained in the absence of

nanoparticles and compared our results with the results asgiven in Chandrasekhar [38] for regular fluid as shown inTable 3. This table shows that the present results obtainedby considering six terms in the Galerkin expansion match with

the exact results as given in Chandrasekhar [38]. We are, there-fore, confident that our results are accurate.

Effect of rotation parameter Ta

In Fig. 2, we compare the variation of critical Rayleigh num-ber Rc for alumina-water and copper–water nanofluids as a

function of Taylor number Ta. It is observed that thealumina-water nanofluid is more stable than the copper–waternanofluid in terms of higher value of critical Rayleigh number

Rc. From Fig. 2, we also obtained that the critical Rayleigh-number Rc increases with Taylor number Ta and hence rota-tion has a stabilizing effect on the stability of the system.This can be described as follows: rotation acts so as to suppress

the vertical motion, and hence thermal convection, by restrict-ing the motion to the horizontal plane. The corresponding crit-ical wave number ac is plotted in Fig. 3 and indicates that an

increase in the value of Taylor number Ta tends to increaseac. Thus its effect is to reduce the size of convection cells.

Effect of volumetric fraction of nanoparticles /�0

Fig. 4 represents the influence of the volumetric fraction of

nanoparticles /�0 on the critical Rayleigh number Rc for the

fixed values of other parameters. It is observed that the volu-

metric fraction of nanoparticles /�0 has a destabilizing effect

on the stability of nanofluids. This is because, if volumetric

fraction of nanoparticles /�0 is increased then the Brownian

motion and thermophoretic diffusion of the nanoparticles




Table 1 Some properties of Al2O3–water and Cu–water nanofluids at u�0 ¼ 0:01;T�1 ¼ 300 K;DT� ¼ 10 K and d ¼ 10 nm given by

Buongiorno [6].

Nanofluids l (Pa s) k (W/m K) a (m2/s) DB (m2/s) DT (m2/s) b (1/K) qp (kg/m3) ðqcÞp (J/m3) ðqcÞ (J/m3)

Al2O3–water 10�3 1 2:0� 10�7 4� 10�11 6� 10�11 6� 10�3 4� 103 3:1� 106 4� 106

Cu–water 10�3 1 2:0� 10�7 4� 10�11 6� 10�12 6� 10�4 9� 103 3:4� 106 4� 106

Table 2 The representative value of parameters for Al2O3–

water and Cu–water nanofluids.

Nanofluids Le NA NB Rq

Al2O3–water 5000 5 7:75� 10�3 4

Cu–water 5000 0:5 8:5� 10�3 9

Table 3 Comparison of critical Rayleigh number Rc and

critical wave number ac for different values of Ta with rigid–

rigid boundary conditions for the limiting case of nanofluid

(regular fluids).

Ta Chandrasekhar [38] rigid–rigid Present study rigid–rigid

Rc ac Rc ac

0 1707.8 3.13 1707.81 3.12

10 1713.0 3.10 1712.72 3.12

100 1756.60 3.15 1756.39 3.16

500 1940.30 3.30 1940.21 3.32

1000 2151.70 3.50 2151.32 3.48

2000 2530.50 3.75 2530.06 3.75

5000 3469.20 4.25 3468.32 4.27

10,000 4713.10 4.80 4711.68 4.78

Figure 2 Effect of Taylor number Ta on the critical Rayleigh

number Rc for alumina-water and copper–water nanofluids.

Figure 3 Effect of Taylor number Ta on the critical wave

number ac for alumina-water and copper–water nanofluids.


number Rc for different values of volumetric fraction of nanopar-

ticles /�0 with DT� ¼ 10 K;b ¼ 6� 10�31=K;Le ¼ 5000;NA ¼ 5;

NB ¼ 7:75� 10�3;Rq ¼ 4.


number ac for different values of volumetric fraction of nanopar-

ticles /�0 with DT� ¼ 10 K;b ¼ 6� 10�31=K;Le ¼ 5000;NA ¼ 5;

NB ¼ 7:75� 10�3;Rq ¼ 4.

6 D. Yadav et al.

are increased, which cause destabilizing effect. Due to the com-bined behavior of Brownian motion and thermophoresis of

nanoparticles, the nanofluid is too much unstable than thepure fluids. The critical Rayleigh number is shown to be lowerby three to four orders of magnitude than that for regular flu-

ids. The corresponding critical wave number ac is plotted inFig. 5 and indicates that the critical wave number ac does

not depend on volumetric fraction of nanoparticles /�0.


Effect of Lewis number Le and modified diffusivity ratio NA

To assess the effect of the Lewis number Le and the modifieddiffusivity ratio NA on the stability of the system, the variation





of critical Rayleigh number Rc as a function of Taylor numberTa for different values of the Lewis number Le and the modi-fied diffusivity ratio NA are shown in Figs. 6 and 7, respec-

tively. We found that with an increase in the values of theLewis number Le and the modified diffusivity ratio NA, thecritical Rayleigh number Rc decreases, indicating that both

accelerate the onset of convection in a nanofluid layer. Itmay be quality to the fact that thermophoresis at a highervalue of thermophoretic diffusivity is more supportable to

the disturbance in nanofluids, while both thermophoresis andBrownian motion are driving forces in favor of the motionof nanoparticles. The corresponding critical wave number ac


number Rc for different values of Lewis number Le with /�0 ¼ 0:01;

b ¼ 6� 10�31=K;DT� ¼ 10 K;NA ¼ 5; Rq ¼ 4; NB ¼ 7:75� 10�3.


number Rc for different values of modified diffusivity ration NA

with /�0 ¼ 0:01;b ¼ 6� 10�31=K;DT� ¼ 10 K;Le ¼ 5000;Rq ¼ 4;

NB ¼ 7:75� 10�3.


number ac for different values of Lewis number Le and modified

diffusivity ration NA with /�0 ¼ 0:01; b ¼ 6� 10�31=K;

DT� ¼ 10 K;Rq ¼ 4; NB ¼ 7:75� 10�3.


is plotted in Fig. 8 and shows that the critical wave numberac does not depend on both the Lewis number Le and the mod-ified diffusivity ratio NA.


number Rc for different values of modified specific heat increment

NB with /�0 ¼ 0:01; b ¼ 6� 10�31=K;DT� ¼ 10 K;Le ¼ 5000;

Rq ¼ 4; NA ¼ 5.


number ac for different values of modified specific heat increment

NB with /�0 ¼ 0:01; b ¼ 6� 10�31=K;DT� ¼ 10 K;Le ¼ 5000;

Rq ¼ 4;NA ¼ 5.


number Rc for different values of density ratio Rq with /�0 ¼ 0:01;

b¼ 6� 10�31=K;DT� ¼ 10K;Le ¼ 5000;NB ¼ 7:75� 10�3; NA ¼ 5.





number ac for different values of density ratio Rq with /�0 ¼ 0:01;

b¼ 6� 10�31=K;DT� ¼ 10K;Le ¼ 5000;NB ¼ 7:75� 10�3; NA ¼ 5.

8 D. Yadav et al.

Effect of modified specific heat increment NB

The effect of modified specific heat increment NB on the onsetof convection is made clear in Figs. 9 and 10. From these fig-ures, we observed that the modified specific heat increment NB

has no significant effect on the stability of nanofluids convec-tion. This is because the terms containing NB involves as a

function of NB

Leand the value of NB

Leis too small of order

10�2 � 10�5, pointing to the zero contribution of the nanopar-

ticle flux in the thermal energy conversation.

Figure 13 Variation in streamlines, isotherms and isonanocon

Rq ¼ 4;DT� ¼ 10 K;NA ¼ 5;b ¼ 6� 10�31=K;Le ¼ 5000;NB ¼ 7:75�


Effect of density ratio Rq

Fig. 11 shows the effect of density ratio Rq on the criticalRayleigh number Rc as a function of Taylor number Ta withfixed values of others parameters. From Fig. 11, we observed

that with an increase in the value of the density ratio Rq, the

critical Rayleigh number Rc decreases, indicating that it accel-

erates the onset of convection in a nanofluid layer. This hap-pens because the heavier nanoparticles moving through thebase fluid make more strong disturbances as compared with

the lighter nanoparticles. The corresponding critical wavenumber ac is plotted in Fig. 12 and shows that the critical wavenumber ac does not depend on the density ratio Rq.

Streamlines, isotherms and isonanoconcentrations

Fig. 13 shows the variations in the streamlines, isotherms andisonanoconcentrations of the disturbance shapes,w cos ax;H sin ax and U sin ax, for two values of Ta (0 and

100) with Rq ¼ 4;DT� ¼ 10 K;NA ¼ 5; b ¼ 6� 10�31=K;Le ¼5000; NB ¼ 7:75� 10�3 at their critical state. The streamlines,the isotherms and the isonanoconcentrations are displayed in

x; zð Þ space. From these figures, it is evident that with theincrease of rotation, flow profile remains same but magnitudesof the stream function, isotherm and isonanoconcentrations

decrease implying a delay in the onset of instability with an

centrations at different values of Taylor number Ta with

10�3 at their critical state.




Table 4 Comparative results of the critical Rayleigh number Rc and the critical wave number ac for different values of Ta for Al2O3–

water and Cu–water nanofluids with the (i) zero flux and (ii) the constant nanoparticle boundary conditions.

Ta ac Al2O3–water nanofluid Cu–water nanofluid

(i) Rc (ii) Rc (i) Rc (ii) Rc

0 3.12 0.138 0.689 0.051 0.025

100 3.16 0.142 0.709 0.053 0.026

500 3.32 0.157 0.783 0.058 0.029

1000 3.49 0.174 0.868 0.065 0.032

2000 3.75 0.204 1.021 0.076 0.038

5000 4.27 0.280 1.340 0.104 0.052

10,000 4.79 0.381 1.901 0.141 0.071


increase in the value of the rotation parameter Ta. Also thestreamlines, isotherms and isonanoconcentrations in the subse-quent cells are alternately identical with and opposite to that of

the adjoining cell, showing the symmetry in the formation ofconvective cells.

Comparison between different boundaries

To know clearly the similarities as well as differences betweenthe zero flux and the constant nanoparticle boundary condi-tions on the stability characteristics of the system, the critical

Rayleigh number for different values of Taylor number Ta

for these two boundary conditions is compared in Table 4for alumina-water and copper–water nanofluid. For

Alumina-water nanofluid, zero flux nanoparticle boundarycondition has more destabilizing effect than the constantnanoparticle boundary conditions, while reverse for copper–

water nanofluid. This is because for zero flux nanoparticleboundary conditions, the modified diffusivity ratio NA isimportant and this is higher for alumina-water nanofluid thancopper–water nanofluid.

Conclusions

In this paper, we have re-examined the problem of convective

instability for a rotating nanofluid layer with the assumptionthat the nanoparticle flux is zero at the boundaries. The choiceof this form of boundary conditions is more realistic than the

previous ones [30]. The model used for the nanofluid incorpo-rated the effects of Brownian motion and thermophoresis. Theeigenvalue problem is solved numerically using 6-term

Galerkin method for alumina-water and copper–waternanofluids. For Alumina-water nanofluid, zero flux nanoparti-cle boundary condition has more destabilizing effect than the

constant nanoparticle boundary conditions, while reverse forcopper–water nanofluid. The results also shows that the volu-

metric fraction of nanoparticles /�0, the Lewis number Le, the

modified diffusivity ratio NA and the density ratio Rq acceler-

ate the onset of convection, whereas Taylor number Ta delays

the onset of convection.

Acknowledgment

This work was supported by the Yonsei University ResearchFund of 2014.


References

[1] Xuan Y, Li Q. Heat transfer enhancement of nano-fluids. Int J

Heat Fluid Flow 2000;21:58–64.

[2] Das SK, Choi SUS, Patel HE. Heat transfer in nanofluids – a

review. Heat Transfer Eng 2006;27:3–19.

[3] Aybar H, Sharifpur M, Azizian MR, Mehrabi M, Meyer JP. A

review of thermal conductivity models for nanofluids. Heat

Transfer Eng 2014. http://dx.doi.org/10.1080/01457632.2015.

987586.

[4] Manna I. Synthesis, characterization and application of nano-

fluid: an overview. J Indian Inst Sci 2000;89:21–33.

[5] Yu W, Xie H. A review on nanofluids: preparation, stability

mechanisms, and applications. J Nanomater 2012;2012:1–17.

[6] Buongiorno J. Convective transport in nanofluids. J Heat

Transfer 2006;128:240–50.

[7] Tzou DY. Instability of nanofluids in natural convection. ASME J

Heat Transfer 2008;130:372–401.

[8] Tzou DY. Thermal instability of nanofluids in natural convection.

Int J Heat Mass Transfer 2008;51:2967–79.

[9] Nield DA, Kuznetsov AV. The onset of convection in a horizontal

nanofluid layer of finite depth. Eur J Mech B Fluids

2010;29:217–23.

[10] Nield DA, Kuznetsov AV. Thermal instability in a porous

medium layer saturated by a nanofluid. Int J Heat Mass

Transfer 2009;52:5796–801.

[11] Kuznetsov AV, Nield DA. Thermal instability in a porous

medium saturated by a nanofluid: Brinkman model. Transp

Porous Media 2010;81:409–22.

[12] Chand R, Rana GC. Thermal Instability of Rivlin–Ericksen

elastico-viscous nanofluid saturated by a porous medium. J Fluids

Eng 2012;134:121203.

[13] Chand R, Rana GC. Thermal instability in a Brinkman porous

medium saturated by nanofluid with no nanoparticle flux on

boundaries. Special Top Rev Porous Media: An Int J

2014;5:277–86.

[14] Yadav D, Bhargava R, Agrawal GS. Boundary and internal heat

source effects on the onset of Darcy–Brinkman convection in a

porous layer saturated by nanofluid. Int J Therm Sci

2012;60:244–54.

[15] Yadav D, Bhargava R, Agrawal GS. Thermal instability in a

nanofluid layer with vertical magnetic field. J Eng Math

2013;80:147–64.

[16] Yadav D, Bhargava R, Agrawal GS, Hwang GS, Lee J, Kim MC.

Magneto-convection in a rotating layer of nanofluid. Asia–Pacific

J Chem Eng 2014;9:663–77.

[17] Chand R, Rana GC. Magneto convection in a layer of nanofluid

in porous medium – a more realistic approach. J Nanofluids

2015;4:196–202.

[18] Nield DA, Kuznetsov AV. The onset of double-diffusive convec-

tion in a nanofluid layer. Int J Heat Fluid Flow 2011;32:771–6.


http://refhub.elsevier.com/S2090-4479(15)00080-5/h0005




http://dx.doi.org/10.1080/01457632.2015.987586

http://dx.doi.org/10.1080/01457632.2015.987586












































10 D. Yadav et al.

[19] Kuznetsov AV, Nield DA. The onset of double-diffusive

nanofluid convection in a layer of a saturated porous medium.

Transp Porous Media 2010;85:941–51.

[20] Yadav D, Agrawal GS, Bhargava R. The onset of convection in a

binary nanofluid saturated porous layer. Int J Theor Appl

Multiscale Mech 2012;2:198–224.

[21] Rana GC, Thakur RC, Kango SK. On the onset of double-

diffusive convection in a layer of nanofluid under rotation

saturating a porous medium. J Porous Media 2014;17:657–67.

[22] Rana P, Agarwal S. Convection in a binary nanofluid saturated

rotating porous layer. J Nanofluids 2015;4:59–65.

[23] Nield DA, Kuznetsov AV. The onset of convection in a layer of a

porous medium saturated by a nanofluid: effects of conductivity

and viscosity variation and cross-diffusion. Transp Porous Med

2012;92:837–46.

[24] Yadav D, Agrawal GS, Bhargava R. The onset of double diffusive

nanofluid convection in a layer of a saturated porous medium

with thermal conductivity and viscosity variation. J Porous Media

2013;16:105–21.

[25] Yadav D, Bhargava R, Agrawal GS, Yadav N, Lee J, Kim MC.

Thermal instability in a rotating porous layer saturated by a non-

Newtonian nanofluid with thermal conductivity and viscosity

variation. Microfluid Nanofluid 2014;16:425–40.

[26] Umavathi JC, Yadav D, Mohite MB. Linear and nonlinear

stability analyses of double-diffusive convection in a porous

medium layer saturated in a Maxwell nanofluid with variable

viscosity and conductivity. Elixir Mech Eng 2015;79:30407–26.

[27] Bhadauria BS, Agarwal S. Natural convection in a nanofluid

saturated rotating porous layer: a nonlinear study. Transp Porous

Media 2011;87:585–602.

[28] Vadasz P. Flow and thermal convection in rotating porous media.

In: Vafai K, editor. Handbook of porous media. New York:

Marcel Dekker, Inc.; 2000. p. 395–440.

[29] Yadav D, Agrawal GS, Bhargava R. Thermal instability of

rotating nanofluid layer. Int J Eng Sci 2011;49:1171–84.

[30] Yadav D, Bhargava R, Agrawal GS. Numerical solution of a

thermal instability problem in a rotating nanofluid layer. Int J

Heat Mass Transfer 2013;63:313–22.

[31] Chand R, Rana CG. On the onset of thermal convection in

rotating nanofluid layer saturating a Darcy–Brinkman porous

medium. Int J Heat Mass Transfer 2012;55:5417–24.

[32] Nield DA, Kuznetsov AV. Thermal instability in a porous

medium layer saturated by a nanofluid: a revised model. Int J

Heat Mass Transfer 2014;68:211–4.

[33] Agarwal S. Natural convection in a nanofluid-saturated rotating

porous layer: a more realistic approach. Transp Porous Media

2014;104:581–92.

[34] Agarwal S, Rana P, Bhadauria BS. Rayleigh–Benard convection

in a nanofluid layer using a thermal nonequilibrium model. J Heat

Transfer 2014;136:122501–14.

[35] Yadav D, Lee J. The effect of local thermal non-equilibrium on

the onset of Brinkman convection in a nanofluid saturated

rotating porous layer. J Nanofluid 2015;4:335–42.


[36] Shivakumara IS, Dhananjaya M. Penetrative Brinkman convec-

tion in an anisotropic porous layer saturated by a nanofluid. Ain

Shams Eng J 2015. http://dx.doi.org/10.1016/j.asej.2014.12.005.

[37] Borujerdi AN, Noghrehabadi AR, Rees DAS. Influence of Darcy

number on the onset of convection in a porous layer with a

uniform heat source. Int J Therm Sci 2008;47:1020–5.

[38] Chandrasekhar S. Hydrodynamic and hydromagnetic stability.

Oxford: Oxford University Press; 1961.

Dr. Dhananjay Yadav is a Post-Doctoral

Research Scientist in the School of

Mechanical Engineering, Yonsei University,

South Korea. He received his PhD degree

from Department of Mathematics, Indian

Institute of Technology (IIT) Roorkee, India,

in 2013 and post graduation (MSc) in

Mathematics from DDU University

Gorakhpur, India, in 2007. His primary

research interests are Hydrodynamic and

Hydromagnetic Stability, Convection during

CO2 dissolution process, Transient Stability, Fluid flow in porous

media, Thermal instability, Magnetic Field Effects, Rotation, Double-

diffusive convection, Soret-driven buoyancy convection, Spectral

Methods, Direct Numerical Simulation (DNS).

Dr. G.S. Agrawal obtained his PhD degree

from the Indian Institute of Technology,

Kanpur, India, in 1970. Subsequently, he

joined the Indian Institute of Technology,

Roorkee, where he served as Lecturer, Reader

Assistant and Associate Professor in the

Department of Mathematics. Currently, he is

COE and Head of the Institute of Computer

Applications, Manglayatan University,

Aligarh, India.

Dr. Jinho Lee is presently holding the post of

Professor, School of Mechanical Engineering,

Yonsei University, South Korea. He received

his BS and MS in Mechanical Engineering

from Yonsei University South Korea in 1974

and 1976, respectively and PhD in Mechanical

Engineering from Case Western Reserve

University, USA, in 1982. He has more than

30 years of teaching and research experience.

He has guided five students for PhD. His

about 100 research articles in various inter-

national journals and conferences were published. His primary

research interest includes computational fluid dynamics; heat and mass

transfer; heat pump; hydrodynamic and hydromagnetic stability.































































Thermal instability in a rotating nanofluid layer: A revised model

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