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7/25/2019 Diff Eqns Dyn Sys
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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
dr_urvashi_gupta@yahoo.com; urvashi@pu.ac.in
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
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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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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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Differ Equ Dyn Syst
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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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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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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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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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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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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