Diff Eqns Dyn Sys

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Differ Equ Dyn Syst

DOI 10.1007/s12591-015-0268-4

O R I G I NA L R E S E A R C H

Numerical Study on Binary Nanofluid Convection

in a Rotating Porous Layer

Jyoti Sharma1 Urvashi Gupta2 R. K. Wanchoo2

Foundation for Scientific Research and Technological Innovation 2016

Abstract The present paper investigates the convection in a binary nanofluid layer in porous

medium under the influence of rotation using DarcyBrinkman model. A set of partial differ-

ential equations based on conservation laws for binary nanofluid convection are solved using

Normal mode technique and one term weighted residual method. The problem is analyzed

for both stationary as well as oscillatory convection for free-free boundaries of the layer.

The oscillatory motions come into existence for bottom heavy configuration of nanoparticles

in the fluid layer. As far as thermal Rayleigh number is concerned, it does not show muchvariation with respect to different nanoparticles (alumina, copper, titanium oxide, silver) for

bottom heavy configuration. Rotation parameter is found to stabilize the system significantly.

Keywords Binary convectionNanofluidDarcyBrinkman modelRotationBrownianmotion Thermophoresis

Introduction

Nanofluid is a highly influential term which is being discussed within the heat transfer

community over a wide spectrum. Nanofluid represents the suspension of nanometer-sized

particles (oxides, nitrides, ceramics, metals and semiconductors) in base fluids (water, ethyl-

ene glycol, oil). The idea of introducing nanofluids first came into the mind of Choi [1] who

claimed that the heat transfer can be enhanced with the addition of nanoparticles in the fluid.

Buongiorno [2] developed a system of conservation equations for nanofluids incorporating

the effect of Brownian motion and thermophoresis. Using this model, Tzou [3] studied the

thermal convection in nanofluids analytically using eigenfunction expansion and found that

B Urvashi Gupta

[email protected]; [email protected]

1 Energy Research Centre, Panjab University, Chandigarh 160014, India

2 Dr. S.S. Bhatnagar University Institute of Chemical Engineering and Technology, Panjab University,

Chandigarh 160014, India

1 3
http://crossmark.crossref.org/dialog/?doi=10.1007/s12591-015-0268-4&domain=pdf


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Differ Equ Dyn Syst

critical Rayleigh number is reduced with the addition of nanoparticles. By using single term

Galerkin approximation Kuznetsov and Nield [4] found the expression for thermal Rayleigh

number and the condition of overstability in porous medium using DarcyBrinkman model.

Further, Bhadauria et al.[5] and Yadav et al. [6] made an extension of the thermal instability

problem by introducing Coriolis force term due to rotation in momentum equation for porousand non-porous medium respectively and concluded that rotation increases the stability of

the system. Gupta et al. [7] studied the nanofluid convection under vertical magnetic field

and found that stability rises with rise in magnetic field parameter.

When the nanoparticles are added in a binary fluid such as salty water, it is known as a

binary nanofluid. Double diffusive convection in nanofluid/convection in binary nanofluid is

like a triple diffusion process in which variations are caused by three different components

heat, nanoparticles and solute which have different rates of diffusion. Very few investigations

have been done until now on the convection in binary nanofluids. Double diffusive convection

in a nanofluid layer for flow in a porous medium was first studied by Kuznetsov and Nield

[8]. The complex expressions for Rayleigh number have been approximated to get simplified

expressions. Further, Yadav et al. [9] and Gupta et al. [10] investigated the convection in

a binary nanofluid layer independent of the restrictions on parameters. The influence of

magnetic field on binary nanofluid convection was considered by Gupta et al. [ 11]. They

have made the valid approximations in the complex expressions for analytical study and

alumina-water nanofluid is used for numerical investigation of the problem.

The influence of rotation on binary nanofluid convection in porous medium using Darcy

Brinkman model for free-free boundaries is investigated in this work. Due to the presence of

rotation, Coriolis force term is added in the momentum equation and DarcyBrinkman model

is used to write conservation equations for the system as given by Kuznetsov and Nield [8].These equations are made non-dimensional by making the variables dimensionless and small

perturbations are imposed on initial solution to get perturbed equations. Further normal mode

technique and one term weighted residual methods are used to find the solution of perturbed

equations. We have three diffusing components; heat, solute and nanoparticles and hence

the problem becomes much more complex for oscillatory convection and cannot be studied

analytically. Thus to study the problem we take numerical values of various parameters

involved for alumina, copper, titanium oxide and silver in water based nanofluids and use

the software Mathematica to solve equations for thermal Rayleigh number for each value

of wave number for the two types of convection . To plot the stability curves for oscillatory

convection, interpolation is used. Tabulated values are also provided wherever needed toanalyze the problem completely.

Formulation of Problem and Relevant Partial Differential Equations

A rotating binary nanofluid layer with angular velocity is heated and soluted from below

as shown in Fig.1.The temperatures T1 and T0, the volume fractions of nanoparticles 1and0 and the solute concentrationsC1and C0 are taken at the bottom and top of the layer,

respectively. The basic equations which express the convection in a rotating binary nanofluidlayer in the light of DarcyBrinkman model (Buongiorno[2], Yadav et al. [6] and Kuznetsov

and Nield[4,8]) are

uD=0, (1)f

uD

t= p+ 2uD

uD

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Fig. 1 Sktech of the physical

system

+ p+(1) {(1T(TT0)C(CC0))}g+ 2

(uD), (2)

(c)m T

t+(c)f uD T= km2T+ (c)p

DB T+ DT T T

T0

+c DT C2C, (3)

t

+ uD

=

DB+DT TT0

, (4)

C

t+ uD

C= DS2C+DC T2 T. (5)

Here, Eqs. (1)-(5) are the conservation equations for mass, momentum, thermal energy,

nanoparticles and solute, respectively.The physical parameters are: fluid velocity uD =(u,v,w)(m/s), volume fraction of nanoparticles , timet(s), coefficient of diffusion due to

Brownian motionDB (m2/s), coefficient of diffusion due to thermophoresisDT(m

2/s), fluid

temperatureT(K), density of the fluid at upper boundary 0

(kg/m3), fluid pressure p(Pa),

fluid viscosity(N s/m2), thermal volumetric coefficientT(K1), solutal volumetric coef-

ficientC, acceleration due to gravity g(m/s2), medium effective viscosity(N s/m2), fluid

density (kg/m3), fluid specific heat c(J/kg K), medium conductivitykm (W/m K), medium

porosity, medium permeability (m2), nanoparticles densityp (kg/m3), nanoparticle spe-

cific heatcp(J/kg K), solute concentrationC, diffusivity of solute DS, diffusivity of Dufour

typeDT Cand diffusivity of SorettypeDC T. We introducedimensionless variables as follows:

(x,y,z)= (x, y, z)/d, t= tm /d2, u= uDd/m ,p= p/m ,

=

1

10 , C= C

C0

C1C0 , T= T

T0

T1T0 ,with m= km

(c)f, = (c)m

(c)f, (6)

where the superscript denotes the dimensionless variable. Thus using Eqs. (6),Eqs.(1)(5)(after removing the superscript ) are

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u=0, (7)Dn

PR

u

t= p+Dn2uuRm k Rnk+ RD Tk+ Rs

L sCk+ T a(uk),

(8)

T

t+u T= 2T+ NB

LN T+ NDNB

LNT T+ST C2C, (9)

1

t+ 1

u= 1

LN2+ ND

LN2T, (10)

1

C

t+ 1

u C= 1

Ls2C+SC T2 T, (11)

where the non-dimensional parameters are: Prandtl number PR = fm ; Darcy num-ber Dn

=

d

2 ; Nanofluid Lewis number LN

= mDB

; Solute Lewis number L S

= m

DS;

Thermal Darcy-Rayleigh number RD = gTd(T1T0)m ; Solute Rayleigh number Rs =gCd(C1C0)

Ds; Nanoparticle Rayleigh number Rn = (p)(10)gd

m; Basic-density

Rayleigh number Rm = [p 0+(10)]gdm

; Taylor number T a=

2d2

2; Diffusivity

ratioND= DT(T1T0)DB T0(10) ; Particle density increment NB= (c)P

(c) f(10);

Dufour parameter ST C= DT C(C1C0)m (T1T0) ;

Soret parameter SC T= DC T(T1T0)m (C1C0) . (12)

Initial Flow and Disturbance Equations

Initially, the fluid layer is assumed to be in a state of rest, so the physical quantities: tempera-

ture, concentration of solute and nanoparticles volume fraction vary in the vertical direction

only i.e.

u=0, T= TI(z) , =I(z) , C=CI(z) , p= pI(z) , (13)where the subscript I denotes the initial flow. Let us apply Eqs. (13) to Eqs.(7)(11) and

use the fact that for most nanofluids Lewis number is large and diffusivity ratio is small

(Buongiorno[2]), we get

TI= I= CI= 1z . (14)Let us apply small perturbations on the initial flow and write

u

=0

+u, p

= pI

+p, T

=TI

+T, C

=CI

+C,

=I

+, (15)

where the superscript - denotes the perturbed variable. Using Eqs. (15) on the set of Eqs.

(7)(11), linearizing and using Eqs. (14) (after dropping the superscript -), we get

u=0, (16)Dn

PR

u

t= p+ Dn2uu Rnk+ RD Tk+ Rs

L SCk+

T a(uk), (17)

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Differ Equ Dyn Syst

T

tw= 2T ND

LN

T

z+

z

2NDNB

LN

T

z+ST C2C, (18)

1

t 1

w= 1

LN2+ ND

LN2 T, (19)

1

C

t 1

w= 1

L S2C+SC T2T. (20)

Using the identity F

= ( F) 2Ffor any vector fieldF, on Eq.(17) and

using Eq. (16), we getDn

PR

tDn2 +1

Dn

PR

t2 Dn4 + 2

w+ Rn2H RD2HT

Rs

Ls 2

HC+T a

2w

z2

=0, (21)

where2 = 2x2

+ 2y2

+ 2z2

and2H= 2

x2+ 2

y2.

Results and Discussions

The differential equations (21) and (18)(20) form an eigenvalue problem which will be

solved using the normal mode technique. Let

(w, T, C, )=(W(z), T(z),(z),(z)) exp(i kxx+i kyy+st). (22)Using (22), in the above mentioned equations, we get

s Dn

PR Dn D2 a2+11+ s Dn

PR

D2 a2 D n D2 a22+T a D2W

s Dn

PR Dn D2 a2+1RDa2T Rs

L Sa2Rna2

=0, (23)

1

W+

1

L S D2 a2

s

+SC T

D2 a2

T=0, (24)

WsT+ D2 a2T+ NBLN

(DT D )2NDNBLN

DT+ST CD2 a2=0, (25)

1

W

1

LN

D2 a2 s

ND

LN

D2 a2T=0, (26)

with D ddz

,a=(k2x+k2y )1/2.We write s= i , where is real and is a dimensionless frequency. The conditions for

free- free boundaries are

W= D2W= T== =0 at z= 0 and z=1. (27)Let the trial functions satisfying the conditions (27) as

(W, T, , )=(A,B, C,D) sin z. (28)By making use of orthogonality of the trial functions we obtain four equations in four

unknowns A, B, C and D. Elimination of these unknowns from the obtained set of equa-

tions produces the eigenvalue equation as

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Rn

1+ D n J+ Dns

PR

J ND

JLs

J ST C

+ s

LN

+J2 SC TST C+

(J+s)

J+s Ls

Ls

a2

J

LN+ s

. J SC TRs 1+ D n J+

Dns

PR a2

Ls

RD J+ s Ls

1+ D n J+ Dns

PR a2

L s

+ J ST C JLN + s

J SC T

J

1+ D n J+ Dns

PR

2+ 2 T a

RD

1+ D n J+ Dns

PR

a2

(J+s) J

LN+ s

J+ s Ls

J

1+ D n J+ DnsPR

2 + 2 T aL s

Rs

1+ D n J+ Dns

PR

a2

L s

=0, (29)

where J= 2

+a2

.

Stationary Convection

At the state of marginal stability, when the amplitudes of small disturbances grow or damped

aperiodically then the transition from stability to instability takes place via a stationary pattern

of motions which is described bys=i =0. Then the eigenvalue equation(29)reduces to

RD=

11 Ls ST C

Dn J3

a2 + J2

a2+ J 2 T a

a2(1+ Dn J) + LN

Rn (1SC TST CLs )

Rs

1

SC T

Rn ND.

(30)

For nanofluid convection without rotation, Eq.(30)reduces to

RD= Dn J3

a2 + J

2

a2+ LN

Rn Rn ND. (31)

which is in confirmation with the results of Brinkman model of Kuznetsov and Nield[4].Note that the expression(30)for thermal Rayleigh number is independent of Prandtl number

and heat capacity ratio.

Oscillatory Convection

For oscillatory convections= i = 0,we separate real and imaginary parts of eigenvalueEq.(29)by puttings=i .

J4

L sLN+ 2Dn J5

LsLN+ Dn2J6

L sLN J4 SC TST C

LN 2Dn J5 SC TST C

LN Dn2J6 SC TST C

LN

+J3 2 T a

LsLN J

3 SC TST C2T a

LN J

22

2

J22

L s 2Dn J

32

2 2Dn J

32

L s Dn

2J42

2 Dn

2J42

L s J

22

LN

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Differ Equ Dyn Syst

2Dn J32

LN Dn

2J42

LN Dn

2J42

LsLNP2R

+Dn2J4 SC TST C

2

LN

P2

R

2Dn J32

Ls PR 2Dn

2J42

Ls PR 2Dn J

32

L sLNPR

2Dn J32

LNPR 2Dn

2J42

LsLNPR 2Dn

2J42

LNPR

+2Dn J3 SC TST C

2

PR+ 2Dn

2J4 SC TST C2

PR J

2T a2

2

J2 T a2

L s J

2T a2

LN+ Dn

2J24

2P2R+ Dn

2J24

Ls P2R

+

Dn 2J24

LNP2

R +

2Dn J4

2

PR +

2Dn2J24

2

PR

J2RDa2

LsLN

Dn J3RDa2

L sLN

+J2 ST CRDa

2

LN+ Dn J

3 ST CRDa2

LN J

2NDRna2

LsLN

Dn J3NDRna

2

L sLN+ J

2Rna2

L s + Dn J

3Rna2

L s + J

2ND ST CRna2

LN

+Dn J3ND ST CRna

2

LN J

2 SC TST CRna2

Dn J3 SC TST CRna

2

+ J2 SC TRsa

2

LsLN + Dn J3 SC TRsa

2

L sLN

J2Rs a2

LsLN Dn J

3Rs a2

LsLN+ RD

2a2

2 + DnJ RD

2a2

2

+DnJ RD 2a2

L s PR+ DnJ RD

2a2

PRLN Dn J S T CRD

2a2

PR

Rn22

DnJ Rn

2a2

+ Dn J N DRn

2a2

LNPR

Dn J Rn2a2

L s PR

DnJ Rn2a2

PR +

Rs 2a2

Ls +

DnJ Rs2a2

L s

Dn J S C TRs2a2

L s PR+ DnJ Rs

2a2

Ls PR+ DnJ Rs

2a2

L sLN=0, (32)

a2J Rn

L S+ a

2Dn J2Rn

L S J

3

2 2Dn J

23

2 Dn

2J33

2

2 T a3

2 + J

3

L S+ 2Dn J

4

L S+ Dn

2J5

L S+ J

2 2 T a

L S

+a2J Rn

+ a2Dn J2Rn

a2J RsL S

a2Dn J2Rs L S

+ J3LNL S

+2Dn J4

LNL S+ Dn

2J5

LNL S+ J

2 2T a

LNL S a

2J Rs

LNL S

a2Dn J2Rs

LNL S+ J

3

LN+ 2Dn J

4

LN+ Dn

2J5

LN+ J

2 2 T a

LN

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Differ Equ Dyn Syst

a2J RnND

LN a

2Dn J2RnND

LN+ Dn

2J5

2 P2R Dn

2J33

L SP2

R

Dn2J33

LN

LS

P 2

R

Dn2J33

LN

P 2

R

+ a2Dn J2Rn

L SPR 2Dn J

23

2 PR

2Dn2J33

2 PR 2Dn J

23

L SPR 2Dn

2J33

L SPR a

2Dn Rn3

PR

+a2Dn Rs 3

L SPR+ 2Dn J

4

LNL SPR+ 2Dn

2J5

LNL SPR a

2Dn J2Rs

LNL SPR

2Dn J23

LNPR 2Dn

2J33

LNPR a

2Dn J2RnND

LNL SPR a

2JRD

L S

a2Dn J2RD

LS

a2JRD

LN

a2Dn J2RD

LN

+ a2Dn3RD

2 PR

a2Dn J2RD

LNL SPR+ a

2J RsSC T

L S+ a

2Dn J2Rs SC T

L S

+a2Dn J2Rs SC T

LNL SPR+ a

2Dn J2RnND ST C

LNPR+ a

2JRD ST C

+a2Dn J2RD ST C

+ a

2Dn J2RD ST C

LNPR J

3SC TST C

2Dn J4SC TST C

Dn

2J5SC TST C

J

2 2 T aSC TST C

+Dn2J33 SC TST C

P 2R a

2Dn J2Rn SC TST C

PR

2Dn J4SC TST C

LNPR 2Dn

2J5SC TST C

LNPR=0. (33)

For convection through oscillations, we solve Eqs. (32) and (33) to determine critical

Rayleigh number for which is real. Thermal Darcy Rayleigh number given by Eq. (30)

contains four parameters depending on nanofluid properties namely nanofluid Lewis num-

ber, diffusivity ratio, concentration Rayleigh number and Prandtl number which strongly

influences the stability of the system. For most of the nanofluids, Lewis number is largeand diffusivity ratio is small and hence coefficient of Rn is large and positive in Eq. (30).

Thus bottom heavy distribution of nanoparticles must stabilize the system appreciably. Also

coefficient ofT a is positive in Eq. (30) meaning thereby that rotation parameter has a sta-

bilizing effect for stationary convection. It is necessary to note that Eqs. ( 32) and(33) for

oscillatory motions are complex and to find results, some approximations may be made such

as Lewis number and Prandtl number approach to infinity as used by Kuznetsov and Nield

[8]. In order to study the present problem completely without using any approximation on

the variables, let us consider the same numerically for alumina, titanium oxide, copper and

silver nanoparticles in water based nanofluid.

Numerical Results and Discussion

Equation(30) for stationary convection and Eqs. (32) and(33)for oscillatory convection

are analyzed numerically. Figures25show the stability curves for bottom heavy binary

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Differ Equ Dyn Syst

Table 1 Physical properties of water and nanoparticles under consideration

Physical properties Water Alumina Copper Silver Titanium oxide

(kg/m3) 997.1 3970 8933 10500 4250

k(W/m K) 0.613 40 401 429 8.9

Fig. 2 Effect of different

nanoparticles on Thermal

DarcyRayleigh number

Fig. 3 Effect of porosity on

Thermal DarcyRayleigh number

nanofluid convection. The values of various parameters for water based binary nanofluids

(using results of Buongiorno [2] and Table1in Eqs.(12)) are:For alumina nanoparticles LN= 5000,ND=5,Rn=0.1,PR=5;For titanium oxide nanoparticles LN= 5000,ND=19,Rn=0.107, PR=5;For copper nanoparticles LN= 5000,ND=0.5,Rn=0.26, PR=5;For silver nanoparticles:LN= 5000,ND= 0.5,Rn= 0.31, PR= 5. We fix the other

parameters as: Ls= 2; T a= 200;Rs= 200; ST C= 0.001;SC T= 1;= 2.5; =0.4;Dn= 0.5. The parameter values for alumina-water nanofluid are used to study theeffects of porosity, Taylor number and solute Rayleigh number on the system.

Here it is worthwhile to mention that to draw stability curves for oscillatory convection,

various parameter values in Eqs. (32) and(33) for different wave numbers are used to get acubic equation in2 using the software Mathematica. Thereafter the positive value of2 is

used (that makes the system unstable) to find the values RD for different wave numbers. For

graphical representation, interpolation is used with a polynomial of degree seven.

Figure 2 illustrates the effect of differentnanoparticles on thestability of waterbased binary

nanofluids. Silver-water is found to be more stable than copperwater which is more stable

than titanium oxide-water which in turn is more stable than alumina- water for stationary

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Differ Equ Dyn Syst

Table 2 Tabulated values of

RD (oscillatory) for water based

binary nanofluid with alumina

nanoparticles for different values

of=0.4, 0.6, 0.8

a =0.4 =0.6 =0.8RD (oscillatory)) RD (oscillatory) RD (oscillatory)

1 3807.737 3968.428 4048.571

2 947.853 1112.892 1195.439

3 510.973 677.326 760.266

4 513.422 679.783 762.727

5 748.382 914.309 997.049

6 1208.109 1373.236 1455.584

7 1937.308 2101.198 2182.932

Fig. 4 Effect of Taylor number

on Thermal DarcyRayleighnumber

Fig. 5 Effect of solute Rayleigh

number on Thermal

DarcyRayleigh number

convection while the effect for oscillatory motions is so small that it is not reflected in graph.

The destabilizing effect of porosity on the system for stationary mode of convection and

stabilizing effect for oscillatory motions is shown in Fig. 3. To give the clear picture of

variation for oscillatory motions, tabulated values of Thermal DarcyRayleigh number for

different values of porosity are also given in Table 2.The rotation parameter (Taylor number) has a strong stabilizing influence on the layer as

illustrated in Fig.4while the destabilizing effect of solute Rayleigh number on the system is

shown in Fig.5for both stationary as well as oscillatory convection. It is noteworthy that the

mode of convection remains oscillatory as depicted in Figs. 25 and the critical wave number

rises with the rise in Taylor number [Refer Fig. 4] while it doesnt get much influenced by

other parameters [Refer Figs.2,3,5].

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Differ Equ Dyn Syst

Conclusions

The binary convection problem is studied numerically to consider the impact of rotation/

Coriolis force and porosity using DarcyBrinkman model. The relevant partial differential

equations are solved using the methodologies of superposition of basic possible modes andsingle term Galerkin approximation. The impact of alumina, titanium oxide, silver and copper

nanoparticles is studied on water based binary nanofluids using the software Mathematica.

The mode of convection is found to be oscillatory for bottom heavy configuration of nanopar-

ticles and oscillatory motions are notmuch influenced by nanoparticle properties. Therotation

parameter is found to stabilize the layer significantly while solute Rayleigh number has a

destabilizing influence. Porosity destabilizes the system for stationary convection while it has

a stabilizing impact for oscillatory convection. The critical wave number (where convection

starts) rises with rise in rotation parameter and remains almost unaffected with the presence

of nanoparticles, solute and porosity of the system.

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4. Kuznetsov, A.V., Nield, D.A.: Thermal instability in a porous medium layer saturated by a nanofluid:

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