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Transp Porous Med (2011) 88:107–131DOI 10.1007/s11242-011-9727-8

Convective Transport in a Nanofluid Saturated PorousLayer With Thermal Non Equilibrium Model

B. S. Bhadauria · Shilpi Agarwal

Received: 25 October 2010 / Accepted: 11 January 2011 / Published online: 29 January 2011© Springer Science+Business Media B.V. 2011

Abstract The effect of local thermal non-equilibrium on linear and non-linear thermalinstability in a horizontal porous medium saturated by a nanofluid has been investigatedanalytically. The Brinkman Model has been used for porous medium, while nanofluid incor-porates the effect of Brownian motion along with thermophoresis. A three-temperature modelhas been used for the effect of local thermal non-equilibrium among the particle, fluid, andsolid-matrix phases. The linear stability is based on normal mode technique, while for non-linear analysis, a minimal representation of the truncated Fourier series analysis involvingonly two terms has been used. The critical conditions for the onset of convection and the heatand mass transfer across the porous layer have been obtained numerically.

Keywords Local thermal non-equilibrium · Nanofluid · Porous medium ·Instability · Natural convection · Horton–Roger–Lapwood problem

Latin SymbolsDB Brownian Diffusion coefficientDT Thermophoretic diffusion coefficientDa Darcy numberPr Prandtl numberd Dimensional layer depthkf Effective thermal conductivity of porous mediumkT Thermal diffusivity of porous mediumLe Lewis number

B. S. Bhadauria (B) · S. AgarwalDepartment of Mathematics, Faculty of Science, DST-Centre for Interdisciplinary Mathematical Sciences,Banaras Hindu University, Varanasi 221005, Indiae-mail: [email protected]

S. Agarwale-mail: [email protected]

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108 B. S. Bhadauria, S. Agarwal

NA Modified diffusivity ratioNB Modified particle-density incrementNHP Nield number for the fluid/particle interfaceNHS Nield number for the fluid/solid-matrix interfacep Pressureg Gravitational accelerationRa Thermal Rayleigh–Darcy numberRm Basic density Rayleigh numberRn Concentration Rayleigh numbert TimeT Nanofluid temperatureTc Temperature at the upper wallTh Temperature at the lower wallv Nanofluid velocityvD Darcy velocity εv(x, y, z) Cartesian coordinates

Greek symbols

αf Thermal diffusivity of the fluid defined askf

(ρc)fβ Proportionality factorγP Modified thermal capacity ratioγS Modified thermal capacity ratioε PorosityεP Modified thermal diffusivity ratioεS Modified thermal diffusivity ratioμ Viscosity of the fluidρf Fluid densityρp Nanoparticle mass density(ρc)f Heat capacity of the fluid(ρc)s Heat capacity of the solid-matrix material(ρc)p Heat capacity of the nanoparticle materialφ Nanoparticle volume fractionψ Stream function

Subscriptsb Basic solutionf Fluid phasep Particle phases Solid-matrix phase

Superscripts* Dimensional variable′ Perturbation variable

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Convective Transport in a Nanofluid Saturated Porous Layer 109

1 Introduction

Natural Convection, or buoyancy driven convection, is the heat removal strategy adopted ina wide variety of industries ranging from transportation, HVAC, and energy production andsupply to electronics, textiles and paper production, geophysical problems, nuclear reactorsto name a few (Choi 1999). Conventional heat transfer liquids have low thermal conductiv-ity. Nanofluids are mixtures of base fluid such as water or ethylene-glycol with a very smallamount of nanoparticles such as metallic or metallic oxide particles (Cu, Cuo, Al2O3, SiO,TiO), having dimensions from 1 to 100 nm, with very high thermal conductivities. It was Choi(1995) who christened the term “nanofluid”. A significant feature of nanofluids is thermalconductivity enhancement, a phenomenon which was first reported by Masuda et al. (1993).Many modern industries deal with heat transfer in some or the other way, and thus have astrong need for improved heat transfer mediums. This could possibly be nanofluids—becauseof some potential benefits over normal fluids—large surface area provided by nanoparticlesfor heat exchange, reduced pumping power due to enhanced heat transfer, minimal clogging,innovation of miniaturized systems leading to savings of energy and cost. Eastman et al.(2001) reported an increase of 40% in the effective thermal conductivity of ethylene-glycolwith 0.3% volume of copper nanoparticles of 10-nm diameter. Das et al. (2003) reporteda 10–30% increase of the effective thermal conductivity in alumina/water nanofluids with1–4% of alumina. These reports led Buongiorno and Hu (2005) to suggest the possibility ofusing nanofluids in advanced nuclear systems. Another recent application of the nanofluidflow is in the delivery of nano-drug as suggested by Kleinstreuer et al. (2008).

Eastman et al. (2004) conducted a comprehensive review on thermal transport in nano-fluids to conclude that a satisfactory explanation for the abnormal enhancement in thermalconductivity and viscosity of nanofluids needs further studies. Buongiorno (2006) conducteda comprehensive study to account for the unusual behavior of nanofluids based on Iner-tia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effects, fluid drainageand gravity settling, and proposed a model incorporating the effects of Brownian diffusionand the thermophoresis. With the help of these equations, studies were conducted by Tzou(2008a, b), Kim et al. (2004, 2006, 2007) and more recently by Nield and Kuznetsov (2009,2010).

Convection in porous media is of practical applications in modern science and engi-neering, including food and chemical processes, rotating machineries like nuclear reactors,petroleum industry, biomechanics, and geophysical problems. Convection in porous mediumhas been studied by many authors including Horton and Rogers (1945), Lapwood (1948),Nield (1968), Rudraiah and Malashetty (1986), Murray and Chen (1989), Malashetty (1993)Pop and Ingham (2001), Vafai (2005, 2010), Nield and Bejan (2006), and Vadasz (2008). Theabove studies have been conducted assuming local thermal equilibrium (LTE) between thefluid and solid-matrix phases, i.e., it is assumed that the temperature difference at any locationbetween the two phases is absent. But for many practical applications, involving high-speedflows or large temperature differences between the fluid and solid phases, the assumption ofLTE is inadequate and it is important to take account of the local thermal non-equilibrium(LTNE) effects. The LTNE model of convective heat transfer in porous medium has beendealt by many authors including Kuznetsov (1998), Ingham and Pop (1998, 2005), Rees andPop (2005), Baytas and Pop (2002), Baytas (2003), Saeid (2004), Malashetty et al. (2005a, b,2008), and Straughan (2006).

Because of their unique properties as heat transfer fluids, nanofluids are being lookedupon as great coolants of the future. Thus studies need to be conducted involving nanofluidsin porous media and without it. Recently, Kuznetsov and Nield (2010a) studied the onset

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of thermal instability in a porous medium saturated by a nanofluid using Brinkman modeland incorporating the effects of Brownian motion and thermophoresis of nanoparticles. Theyfound that the critical thermal Rayleigh number can be reduced or increased by a substantialamount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy by the presence of the nanoparticles. The same Horton–Rogers–Lapwood Problem wasinvestigated by Nield and Kuznetsov (2009) for the Darcy Model. Agarwal and Bhadauria(2010) studied thermal instability in a rotating porous layer saturated by a nanofluid fortop-heavy and bottom-heavy suspension considering Darcy Model. These studies dealt withthermal equilibrium condition between fluid-particle phases and fluid-solid matrix phases.However, here we need to recollect that thermal lagging between the particle and fluid phaseshas been proposed by Vadasz (2005, 2006) as an explanation for the observed increase inthe thermal conductivity of nanofluids. Due to applications of nanofluids and porous mediatheory in drying, freezing of foods, and applications in everyday technology such as micro-wave heating, rapid heat transfer from computer chips via use of porous metal foams andtheir use in heat pipes, study of LTNE turns important. Kuznetsov and Nield (2010b) andNield and Kuznetsov (2010) investigated the effect of LTNE on the onset of convection in ananofluid saturated porous medium and in a nanofluid layer. They found that in case of linearnon-oscillatory instability, the effect of LTNE can be significant for some circumstances butremains insignificant for a typical dilute nanofluids.

Apart from the above studies on thermal instability in nanofluids, no other study is avail-able, therefore, we intend to investigate this problem further. Assuming that the nanoparticlesbeing suspended in the nanofluid using either surfactant or surface charge technology, pre-venting the agglomeration and deposition of these on the porous matrix. In this article, westudy the linear and non-linear thermal instability in a porous medium saturated by nanofluidusing Horton–Roger–Lapwood problem based on the Brinkman’s Model, considering LTNEbetween the fluid/particle and fluid/solid-matrix interphases.

2 Governing Equations

We consider a porous layer saturated by a nanofluid, confined between two horizontal bound-aries at z = 0 and z = d, heated from below and cooled from above. The boundaries are imper-meable and perfectly thermally conducting. The porous layer is extended infinitely in x andy-directions, and z-axis is taken vertically upward with the origin at the lower boundary. TheLTNE between the fluid and solid-matrix and fluid and particle phase have been considered,thus heat flow has been described using three temperature model. Th and Tc are temperatureat the lower and upper walls, respectively, the former being greater. The conservation equa-tions for the total mass, momentum, thermal energy in the fluid phase, thermal energy in theparticle phase, thermal energy in the solid-matrix phase, and nanoparticles, come out to be asbelow. A detailed derivation of these has been dealt by Buongiorno (2006), Tzou (2008a, b)Nield and Kuznetsov (2009), and Kuznetsov and Nield (2010b);

∇ · vD = 0 (1)

0 = −∇p + μ∇2v − μ

KvD + [

φρp + (1 − φ){ρ(1 − β(Tf − Tc))}]

g (2)

ε(1 − φ0)(ρc)f

[∂Tf

∂t+ 1

εvD · ∇Tf

]= ε(1 − φ0)kf∇2Tf + ε(1 − φ0)(ρc)p

×[

DB∇φ · ∇Tf + DT∇Tf · ∇Tf

Tf

]+ hfp(Tp − Tf )+ hfs(Ts − Tf ) (3)

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εφ0(ρc)p

[∂Tp

∂t+ 1

εvD · ∇Tp

]= εφ0kp∇2Tp + hfp(Tf − Tp) (4)

(1 − ε)(ρc)s∂Ts

∂t= (1 − ε)ks∇2Ts + hfs(Tf − Ts) (5)

∂φ

∂t+ 1

εvD · ∇φ = DB∇2φ + DT

Tc∇2Tf (6)

where vD = (u, v, w) is the Darcy velocity. In these equations, ρ is the fluid density, K isporosity, (ρc)f , (ρc)p, (ρc)s the effective heat capacities, and kf , kp, ks, the effective ther-mal conductivities of the fluid, particle, and solid phases respectively. DB and DT denotethe Brownian diffusion coefficient and thermophoretic diffusion, respectively. The interfaceheat transfer coefficients between the fluid/particle phases and the fluid/solid-matrix phaseshave been designated by hfp and hfs, namely. We assume the flow to be slow to neglect anadvective term and a Forchheimer quadratic drag term from the momentum equation. In theabove equations, both Brownian transport and thermophoresis coefficients are taken to betime independent, in tune with the recent studies that neglect the effect of thermal transportattributed to the small size of the nanoparticles [as per recent arguments by Keblinski andCahil (2005)]. Further, Thermophoresis and Brownian transport coefficients are assumedto be temperature-independent due to the fact that the temperature ranges under consider-ation are not far away from the critical value, and the volume averages over a representativeelementary volume.

Assuming the temperature and volumetric fraction of the nanoparticles to be constant atthe boundaries, and that LTNE is present there, we get the boundary conditions to be

v = 0, Tf = Th, Tp = Th, Ts = Th, φ = φ1 at z = 0, (7)

v = 0, T = Tc, Tp = Tc, Ts = Tc, φ = φ0 at z = d. (8)

where φ1 is greater than φ0. To non-dimensionalize the variables, we write

(x∗, y∗, z∗) = (x, y, z)/d, (u∗, v∗, w∗) = (u, v, w)d/αf , t∗ = tα f /d2, p∗ = pK/μαf ,

φ∗ = φ − φ0

φ1 − φ0, T ∗

f = Tf − Tc

Th − Tc, T ∗

p = Tp − Tc

Th − Tc, T ∗

s = Ts − Tc

Th − Tc

where αf = kf(ρc)f

. Then Eqs. 1–8 take the form (after dropping the asterisk)

∇ · v = 0 (9)

0 = −∇p + Da∇2v − v − Rmez + RaT ez − Rnφez (10)∂Tf

∂t+ vε

· ∇Tf = ∇2Tf + NB

Le∇φ · ∇Tf + NA NB

Le∇Tf · ∇Tf

+NHP(Tp − Tf )+ NHS(Ts − Tf ) (11)

∂Tp

∂t+ vε

· ∇Tp = εp∇2Tp + γp NHP(Tf − Tp) (12)

∂Ts

∂t= εs∇2Ts + NHS(Tf − Ts) (13)

∂φ

∂t+ 1

εv · ∇φ = 1

Le∇2φ + NA

Le∇2Tf (14)

v = 0, Tf = 1, Tp = 1, Ts = 1, φ = 1 at z = 0, (15)

v = 0, Tf = 0, Tp = 0, Ts = 0, φ = 0 at z = 1. (16)

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Here Da = μKμd2 is the Darcy’s number,

Le = αf

DBis the Lewis number,

Ra = ρgβK d(Th − Tc)

μαfis the Rayleigh − −Darcy number,

Rm = [ρpφ0 + ρ(1 − φ0)]gK d

μαfis the basic density Rayleigh number,

Rn = (ρp − ρ)(φ1 − φ0)gK d

μαfis the concentration Rayleigh number,

NB = ε(ρc)p(φ1 − φ0)

(ρc)fis the modified particle density increment,

NA = DT(Th − Tc)

DBTc(φ1 − φ0)is the modified diffusivity ratio which is similar

to the Soret parameter that arises in cross diffusion in thermal instability.

NHP = hfpd2

ε(1 − φ0)kf, NHS = hfsd2

ε(1 − φ0)kf

are the interface heat transfer parameters called as Nield number by Vadasz (2006).

γp = (1 − φ0)

φ0

(ρc)f(ρc)p

, γs = ε(1 − φ0)

(1 − ε)

(ρc)f(ρc)s

are modified thermal capacity ratios.

εp = kp/(ρc)pkf/(ρc)f

, εs = ks/(ρc)skf/(ρc)f

are modified thermal diffusivity ratios.

3 Basic Solution

At the basic state, the nanofluid is assumed to be at rest, therefore, the quantities at the basicstate will vary only in the z-direction, and will be given by

v = 0, p = pb(z), Tf = Tfb(z), Tp = Tpb(z), Ts = Tsb(z), φ = φb(z) (17)

Substituting Eq. 17 in Eqs. 11–14 to obtain

d2Tfb

dz2 + NB

Le

dφb

dz

dTfb

dz+ NA NB

Le

(dTfb

dz

)2

+ NHP(Tp − Tf )+ NHS(Ts − Tf ) = 0 (18)

εpd2Tpb

dz2 + γp NHP(Tf − Tp) = 0 (19)

εsd2Tsb

dz2 + γs NHS(Tf − Ts) = 0 (20)

d2φb

dz2 + NAd2Tfb

dz2 = 0 (21)

and using order magnitude analysis, (Kuznetsov and Nield 2010b) we get:

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d2Tpb

dz2 = 0,d2Tfb

dz2 = 0,d2Tsb

dz2 = 0,d2φb

dz2 = 0, (22)

The boundary conditions for solving Eq. 22 can be obtained from Eq. 15 and 16 as:

Tfb = Tpb = Tsb = 1, φb = 1, at z = 0, (23)

Tfb = Tpb = Tsb = 0, φb = 0, at z = 1. (24)

The remaining solution pb(z) at the basic state can easily be obtained by substitutingTfb, Tpb, Tsb from Eq. 22, and then integrating Eq. 10 for pb. Solving Eq. 22, subject toconditions (23) and (24), we obtain:

Tfb = Tpb = Tsb = 1 − z (25)

φb = 1 − z. (26)

4 Stability Analysis

To perturb the basic state of the system, we write

v = v′, p = pb + p′, Tf = Tfb + T ′f ,

Tp = Tpb + T ′p, Ts = Tsb + T ′

s , φ = φb + φ′. (27)

Substituting the above expression (27) in Eqs. 9–14, and using the expressions (25) and (26),eliminating the pressure and introducing the stream function, to obtain equations correspond-ing to two dimensional rolls with all physical quantities independent of y, as

(Da∇2 − 1)∇2(

−∂ψ∂x

)= Rn

∂2φ

∂x2 − Ra∂2Tf

∂x2 (28)

∂Tf

∂t+ 1

ε

∂ψ

∂t= ∇2Tf + NHP(Tp − Tf )+ NHS(Ts − Tf )+ 1

ε

(∂ (ψ, Tf )

∂(x, z)

)(29)

∂Tp

∂t+ 1

ε

∂ψ

∂t= εp∇2Tp + γp NHP(Tf − Tp)+ 1

ε

(∂(ψ, Tp)

∂(x, z)

)(30)

∂Ts

∂t= εs∇2Ts + γs NHS(Tf − Ts) (31)

∂φ

∂t+ 1

ε

∂ψ

∂t= 1

Le∇2φ + NA

Le∇2Tf + 1

ε

(∂(ψ, φ)

∂(x, z)

)(32)

The Eqs. 28–32 are subject to stress-free, isothermal, iso-nanoconcentration boundary con-ditions:

ψ = ∂2ψ

∂z2 = Tf = Tp = Ts = φ = 0, at z = 0, 1 (33)

The linear stability analysis is well studied and reported by Kuznetsov and Nield (2010a).The critical Rayleigh numbers for stationary mode of convection is given by

Rast =(ε

α2c

) {δ2 (

1 + Daδ2) + RnLeα2

δ2ε

}{εpδ

2 + γp NHP

εpδ2 + (1 + γp)NHP

}

×{

(δ2 + NHP + NHS

) − γp(NHP)2

εpδ2 + γp NHP− γs(NHS)

2

εsδ2 + γs NHS

}

− RnNA (34)

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where δ2 = π2 + α2.

To perform a nonlinear stability analysis, we take the modes (1,1) for stream function,and (0,2) and (1,1) for temperature and nanoparticle concentration, respectively, in Fourierseries expression, we have

ψ = A11(t)sin(αx)sin(π z) (35)

Tf = B11(t)cos(παx)sin(π z)+ B02(t)sin(2π z) (36)

Tp = C11(t)cos(παx)sin(π z)+ C02(t)sin(2π z) (37)

Ts = D11(t)cos(παx)sin(π z)+ D02(t)sin(2π z) (38)

φ = E11(t)cos(παx)sin(π z)+ E02(t)sin(2π z) (39)

where the coefficients A11(t), B11(t), B02(t),C11(t),C02(t), D11, D02, E11, and E02 arefunctions of time. Substituting Eqs. 35–39 in Eqs. 28–32, and using the orthogonality con-dition with the eigenfunctions associated with the considered minimal model, we get

A11(t) = (RnE11(t)− RaB11(t))α2

α(δ4 Da + δ2)(40)

dB11(t)

dt= NHP(C11(t)− B11(t))+ NHS(D11(t)− B11(t))− δ2 B11(t)

−αA11(t)

ε− πα

εA11(t)B02(t) (41)

dB02(t)

dt= 1

2

[παε

A11(t)B11(t)− 8π2 B02(t)+ 2NHP(C02(t)− B02(t))

+2NHS(D02(t)− B02(t))] (42)

dC11(t)

dt= γp NHP(B11(t)− C02(t))− αA11(t)

ε− εpC11(t)δ

2 − παA11(t)C02(t) (43)

dC02(t)

dt= 1

2

[παε

A11(t)C11(t)− 8π2εpC02(t)+ 2γp NHP(B02(t)− C02(t))]

(44)

dD11(t)

dt= γs NHS(B11(t)− D11(t))− εsδ

2 D11(t) (45)

dD02(t)

dt= γs NHS(B02(t)− D02(t))− 4π2εs D02(t) (46)

dE11(t)

dt= αA11(t)

ε− δ2

LeE11(t)− NAδ

2

LeB11(t)− πα

εA11(t)E02(t) (47)

dE02(t)

dt= −

[4π2

Le(E02(t)− NA B02(t))− πα

εA11(t)E11(t)

](48)

The above system of simultaneous autonomous ordinary differential equations will be solvednumerically using Runge–Kutta–Gill method.

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5 Heat and Nanoparticle Concentration Transport

We define the thermal Nusselt number Nuf (t) for fluid phase as

Nuf (t) = Heat transport by (conduction + convection)

Heat transport by conduction

= 1 +⎡

⎣

∫ 2π/αc0

(∂Tf∂z

)dx

∫ 2π/αc0

(∂Tb∂z

)dx

⎤

⎦

z=0

(49)

Substituting the expressions (25) and (36) in Eq. 49, we get

Nuf (t) = 1 − 2πB02(t). (50)

The thermal Nusselt numbers for particle and solid-matrix phases, nanoparticle concentra-tion Nusselt number, Nuφ(t) can also be defined similar to the thermal Nusselt number forfluids, and can be obtained as:

Nup(t) = 1 − 2πC02(t) (51)

Nus(t) = 1 − 2πD02(t) (52)

Nuφ(t) = (1 − 2πE02(t))+ NA(1 − 2πB02(t)). (53)

6 Results and Discussion

6.1 Linear Stability Analysis

The expression for stationary convection in (34), is

Rast =(ε

α2c

) {δ2 (

1 + Daδ2) + RnLeα2

δ2ε

}{εpδ

2 + γp NHP

εpδ2 + (1 + γp

)NHP

}

×{

(δ2 + NHP + NHS

) − γp(NHP)2

εpδ2 + γp NHP− γs(NHS)

2

εsδ2 + γs NHS

}

− RnNA (54)

For thermal equilibrium system, we have NHS = NHP = 0, thus we get

Rast = Rn(Le − NA)+ ε[1 + Daδ2]δ4

α2 , (55)

which is similar to the result obtained by Kuznetsov and Nield (2010a).In case of ordinary fluid, Le = NA = 0, NHP = 0 and εs = 1. Putting these in the Eq. 54,

we obtain

Rast =(ε

α2c

) {δ4(1 + Daδ2)

} {1 + NHS

δ2 + γs NHS

}(56)

This result is in agreement with the result obtained by Malashetty et al. (2005a) in their workwith ordinary fluids under LTNE conditions. For LTE, in the above case NHS = 0, so weget

Rast =(ε

α2c

) {δ4(1 + Daδ2)

}(57)

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which is a well known result for an ordinary fluid flowing through porous medium. WhenDa = 0, the minimum is attained at α = π , with the minimum value becoming 4π2. Onthe other hand, when Da is very large in comparison with unity, the minimum is attained atα = π/

√2, and the value becomes 27π4/4.

In Fig. 1a–k, we present neutral stability curves for Rast versus the wavenumber α forthe fixed values of Rn, Le, NA, Da, ε, εp, εs, γp, γs, NHP, and NHS with variation in one ofthese parameters. In all these plots, it is interesting to note that the value of Ra starts from ahigher note, falls rapidly with increasing α, and then increases steadily. The Fig. 1c, i, andj correspond to the variation of Ra with respect to α at different values of modified diffu-sivity ratio NA, modified thermal capacity ratio γs, and inter phase heat transfer parameterNHP. These plots reveal that on increasing the value of these parameters, the value of Ratends to decrease, i.e., the system tends to remain unstable. In all the other plots, depictingthe variation of Ra with α for Rn, Le, Da, ε, εp, εs, γp, and NHS, we conclude that theseparameters have a stabilizing effect on the system as, when we consider higher values ofthese parameters, the value of Racr increases.

In the Fig. 2, we draw the neutral stability curve for LTNE and LTE. We see that the valueof Rayleigh number Ra is less in case of LTNE than LTE. This implies that convection startsearlier in the case of LTNE than LTE. The observed phenomenon may be attributed to the factthat because of temperature difference between the fluid, particle and solid matrix phases,there occurs transfer of energy between them. This leads to a chaotic state and enhances theonset of convection in case of LTNE.

In the Fig. 3, we compare the Rayleigh number Ra for nanofluids with ordinary fluids.It is to be noted that the value of Rayleigh number is less in the case of ordinary fluid thannanofluid, or to say convection sets in earlier in ordinary fluids than nanofluids. This impliesthat the thermal conductivity of nanofluids is higher than ordinary fluids.

The nature of critical values of Rayleigh number Ra and the critical values of wave numberα as functions of inter phase heat transfer parameters or Nield numbers for fluid/nanoparti-cle inter phase, NHP, and fluid/solid-matrix inter phase NHS, for Rn = 4, Le = 10, NA =4, Da = 0.01, ε = 0.04, εP = 0.4, εS = 0.7, γP = 5, γS = 5, with a variation in the valueof one of these parameters, are shown in Figs. 4, 5 and Figs. 6, 7, respectively. For very smalland large values of NHP and NHS, we observe that the stability criterion is independent oftheir value, and that the value of NHP and NHS play a significant role in the stability criteriononly in the intermediate range. The reason behind this state being, that at NHP → 0 andNHS → 0, there occurs almost zero heat transfer between fluid/nanoparticle and fluid/solid-matrix inter phases, and the properties of nanoparticle or solid-matrix do not interfere in theonset of convection. While, when NHP → ∞ and NHS → ∞, the three have almost the equaltemperatures and behave as a single phase. Between these two extremes, a LTNE effect isobserved being attributed to NHP and NHS.

Figure 4 presents the variation of critical Rayleigh number Racr with Nield number forthe fluid/solid-matrix inter phase NHS for different parameters. The figures indicate that thevalue of Racr increases from its LTE value for very small NHS to LTNE value for large NHS.The system tends to stabilize for the intermediate values of NHS. The effect of the parameters,concentration Rayleigh number Rn, Lewis number Le, Darcy number Da, porosity ε, mod-ified thermal diffusivity ratios εP, εS and modified thermal capacity ratio γp, on the systemis to increase the value of the critical Rayleigh number Racr as their values increase, thuspreventing the system from destabilization. While on increasing the value of other param-eters, modified diffusivity ratio NA, Nield number for the fluid/particle inter phase NHP

and modified thermal capacity ratio γs, the value of Racr falls trending the system towardsdestabilization. Thus, the effect of these parameters is to reduce the stabilizing effect of the

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(a) (b) (c)

(f)(e)(d)

(g)

(j) (k)

(h) (i)

Fig. 1 Neutral stability curves for different values of a Rn, b Le, c NA , d Da, e ε, f εP , g εS , h γP , i γS ,j NHP, and k NHS

inter-phase heat transfer coefficient. Also from Figs. 4g and i, we conclude that for smallNHS the Racr remains independent of γs and εS, and their effect enters the scenario only forlarge values of NHS.

Figure 5 envisages the effect of NHS on the value of critical wavenumber αc. We observethat value of critical wavenumber αc increases with increasing NHS from LTE value whenNHS is small, to its maximum LTNE value for intermediate NHS, and finally bounces back toits LTE value for large NHS. This implies that the value of critical wavenumber αc approachesto its LTE value when NHS → 0 and NHS → ∞. For the parameters, Rn, Le, γp, εp, εs andNHP, as we increase the value of these the maximum value of αc also increases. While forDa, ε and γs, an increase in their value decreases the minimum value of αc. Also from Fig. 5gand i, we conclude that αc remains independent of the values of γs and εS for small and largevalues of NHS and they have a significant effect only for intermediate values of NHS.

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Fig. 2 Comparison of the value of Rayleigh number Ra for LTNE and LTE

Fig. 3 Comparison of the value of Rayleigh number Ra for Nanofluid with Ordinary fluid

In Fig. 6, we present the variation of critical Rayleigh number Racr with Nield number forthe fluid/particle inter phase NHP for different parameters. The trend observed in the figuresis just opposite to that observed for Nield number for the fluid/particle inter phase NHS. Thefigure indicates that the value of Racr decreases from its LTE value for very small NHP toLTNE value for large NHP. The system tends to destabilize for the intermediate values ofNHP. The effect of the parameters concentration Rayleigh number Rn, Lewis number Le,Darcy number Da, porosity ε, modified thermal diffusivity ratios εP, εS and Nield numberfor fluid/solid-matrix inter phase NHS, on the system is to inhibit the decrease in the value ofthe critical Rayleigh number Racr, thus preventing the system from destabilization. While onincreasing the values of, modified diffusivity ratio NA and modified thermal capacity ratioγs, the value of Racr falls further trending the system towards destabilization.

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(a) (b)

(d)

(f)

(h)(g)

(e)

(c)

Fig. 4 Variation of critical Rayleigh number Racr with NHS for different values of a Rn, b Le, c NA , d γP ,e γS , f εP , g εS , and h NHP

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(a) (b)

(d)(c)

(e) (f)

(h)(g)

Fig. 5 Variation of critical wavenumber αc with NHS for different values of a Rn, b Le, c NA , d γP , e γS ,f εP , g εS , and h NHP

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(a) (b)

(d)(c)

(g) (h)

(e) (f)

Fig. 6 Variation of critical Rayleigh number Racr with NHP for different values of a Rn, b Le, c NA,d Da, e γS , f εP , g εS and h NHS

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(a) (b)

(d)(c)

(e) (f)

(h)(g)

Fig. 7 Variation of critical wavenumber αc with NHP for different values of a Rn, b Le, c NA,d Da, e γS , f εP , g εS , and h NHS

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(a) (b)

Fig. 8 Variation of a critical Rayleigh number, Racr , b critical wave number αc with concentration Rayleighnumber, Rn, for different values of Le, NA , and Da

In the Fig. 7, we have exhibited the critical wavenumber αc as a function of NHP. Weobserve that the value of critical wavenumber αc decreases with increasing NHP from LTEvalue when NHP is small, to its minimum LTNE value for intermediate NHP, and finallybounces back to its LTE value for large NHP. This implies that the value of critical wavenum-ber αc approaches to its LTE value when NHP → 0 and NHP → ∞. This is quite obvious asthe corresponding physical situation are anonymous. At NHP → 0, the particle phase doesnot interfere with the thermal field of the fluid, which is free to act independently, while asNHP → ∞, the particle/fluid phases have attained the identical temperatures, and behaveas single phase only. We conclude that as time passes, and heat intensifies, the nanofluidsbehave more like a single phase fluid rather than like a conventional solid–liquid mixture.For the parameters, Rn, Le, NA, and ε, we see that an increase in their values has effect onthe critical value of wave number αc only when NHP → 0 and NHP → ∞. For intermedi-ate values of NHP, i.e., when αc falls to its minimum LTNE value, the value of αc remainsindependent of these parameters. However, for Da, γs and NHS, as we increase their value,minimum value of αc decreases, whereas, an increase in the values of εp and εs, increases theminimum value of αc. Also from Figs. 6g and 7g, we see that the values of critical Rayleighnumber Racr and critical wavenumber αc are independent of the value of modified thermaldiffusivity ratio εp for small NHP.

In Fig. 8a and b, the variation of critical thermal Rayleigh number Racr for stationary con-vection and the corresponding critical wave number αc with the nanoparticle concentrationRayleigh number Rn is depicted. From Fig. 8a, it is worth noting that for small values ofRn, we have small values of Racr which increases on increasing nanoparticle concentrationRayleigh number. Similarly, from Fig. 8b we make out that the corresponding critical wavenumber is also small for small Rn and increases with increasing Rn.

In Fig. 8a, we observe that on increasing the values of Rayleigh–Darcy number Da andLewis number Le, the value of Racr increases, while on increasing the value of modifieddiffusivity ratio NA, the Racr decreases. From Fig. 8b, it is to be noted that with an increasein the value of Lewis number Le, value of critical wave number αc increases, while withan increase in Rayleigh–Darcy number Da, it decreases. However, for modified diffusivityratio NA, there seems to be no effect on the value of critical wave number αc.

Figure 9a and b display the variation of critical thermal Rayleigh number Racr and the cor-responding critical wave number αc with the nanofluid Lewis number Le. It is to be observed

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(a) (b)

Fig. 9 Variation of a critical Rayleigh number, Racr , b critical wave number αc with Lewis number, Le, fordifferent values of Rn, NA , and Da

from Fig. 9a that for small values of Le, we have smaller values of Racr which increase withincreasing nanofluid Lewis number Le. Similar trend is also observed for the correspondingcritical wave number in Fig. 9b.

It is to be noted in these figures that on increasing concentration Rayleigh number Rn,value of critical Rayleigh number Racr as well as critical wave number αc increases. How-ever, for Darcy number Da, the value of Racr increases while αc decreases. With an increasein modified diffusivity ratio NA, Racr is seen to decrease, but no effect of it is observed onthe value of critical wave number αc.

6.2 Non-Linear Unsteady Stability Analysis

The linear solutions exhibit a considerable variety of behavior of the system, and the tran-sition from linear to non-linear convection can be quite complicated, but interesting to dealwith. We need to study the time dependent results to analyze the same. This transition can bewell understood by the analysis of (40–48) whose solutions give a detailed description of thetwo-dimensional problem. We solve the Eqs. 40–48 numerically, using Runge–Kutta–GillMethod, and calculate various Nusselt numbers as function of time t .

The nature of Nusselt numbers, Nuφ, Nu(fluid), Nu(particle) and Nu(solid-matrix), as afunction of time t , for Rn = 4, Le = 10, NA = 4, Ra = 28, Da = 0.02, ε = 0.04, NHP =0.001, NHS = 10, γp = 5, γs = 5, εp = 0.4, and εs = 0.7 with a variation in the valueof one of these parameters, is shown in Figs. 10, 11, 12, and 13, respectively. These figuresindicate that initially, when time is small, there occurs large scale oscillations in the values ofNuφ, Nu(fluid),Nu(particle) and Nu(solid-matrix), indicating an unsteady rate of mass andheat transfer in the fluid, particle and solid-matrix phases. As time passes by, these valuesapproach to steady state, corresponding to a near conduction instead of convection stage.

In the Fig. 10, the transient nature of concentration Nusselt number or Sherwood num-ber(as some researchers name it) is visible. We can observe that the effect of increasing thevalue of modified thermal capacity ratio γs, on the amplitude of oscillations is to increaseit, i.e., an increase in the value of the parameter brings about an increase in the rate of masstransfer across the porous medium layer. While an increase in the value of porosity Nieldnumber for fluid/particle interphase NHP is to decrease the rate of mass transfer.

Figure 11 depicts the transient nature of Nusselt number for the fluid phase. There occurslarge amount of heat transfer in the fluid phase initially, and with large time the amount of

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(a)

(c)

(e)

(g) (h)

(f)

(b)

(d)

Fig. 10 Variation of Concentration Nusselt Number Nuφ with time t for different values of a Rn, b Le,c NHP, d NHS, e γP , f γS , g εP , and h εS

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(a) (b)

(d)(c)

(e) (f)

(h)(g)

Fig. 11 Variation of Nusselt Number Nu(fluid) with time t for different values of a Rn, bLe, c NHP,d NHS, e γP , f γS , g εP , and h εS

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(a) (b)

(d)(c)

(e) (f)

(h)(g)

Fig. 12 Variation of Nusselt Number Nu(particle) with time t for different values of a Rn, b Le, c NHP,d NHS, e γP , f γS , g εP , and h εS

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(a) (b)

(d)(c)

(e) (f)

(h)(g)

Fig. 13 Variation of Nusselt Number Nu(solid-matrix) with time t for different values of a Rn, b Le, c NHP,d NHS, e γP , f γS , g NHP, εP , and h εS

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heat transfer approaches a near constant value. The effect of increasing porosity on the heattransfer in fluid phase is to decrease the amount of heat transfer.

The transient nature of Nusselt number for particle phase has been shown in Fig. 12. Wesee that the amplitude of heat transfer in particle phase is small initially, but increases rapidlywith time t to approach to a steady value for large values of time t . The effect of variousparameters is also well pronounced for Nu(particle) as compared to other phases. We can seethat the effect of increasing the values of concentration Rayleigh number Rn, Nield numberfor fluid/particle interphase NHP and modified thermal diffusivity ratio εp on the value ofNusselt number Nu is to decrease it, thus decreasing rate of heat transfer. While on increasingthe values of modified thermal capacity ratio γs, the rate of heat transferred is increased.

The Nusselt number for solid-matrix phase follows the trend demonstrated by its coun-terparts, and shows an unsteady rate of heat transfer in solid-matrix phase initially, whichapproaches to a constant value as time passes by. A comparative study of the Figs. 11, 12,and 13, i.e., in the values of thermal Nusselt numbers for fluid, particle and solid-matrixphases, reveals that the amplitude of heat transfer is maximum for the particle phase, indicat-ing that the amount of heat transfer is maximum in nanoparticles then fluid or solid-matrixphases. This gain of energy by the nanoparticles may be the possible cause of observedenhanced thermal conductivity of nanofluids.

7 Conclusions

We considered linear stability analysis in a horizontal porous medium saturated by a nano-fluid, heated from below and cooled from above, using Brinkman model which incorporatesthe effect of Brownian motion along with thermophoresis, under non-equilibrium conditions.Further bottom heavy suspension of nanoparticles has been considered. Linear analysis hasbeen made using normal mode technique, and the effect of various parameters on the onsetof thermal instability has been found.The results have been presented graphically. We drawthe following conclusions:

1. The effect of Rn, Le, Da, ε is to stabilize the system.2. Convection sets in earlier for LTNE as compared to LTE.3. The effect of time on Nusselt numbers is found to be oscillatory, when t is small. However

when time t becomes very large Nusselt number approaches the steady value.4. On increasing the value of thermal Rayleigh number Ra, the rate of mass and heat transfer

is increased.5. The effect of modified diffusivity ratio NA on the rate of mass transferred and heat trans-

ferred in particle phase is just opposite. While an increase in its value increases the rateof mass transferred, it decreases the amount of heat transferred in the particle phase.

Acknowlegments Author Shilpi gratefully acknowledges the financial assistance from Banaras HinduUniversity as a research fellowship.

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