RESEARCH OpenAccess ... · assuming local thermal equilibrium(LTE) between the fluid and particle phases as well as fluid and solid-matrix phases, i.e., the temperature gradient at

Agarwal and Bhadauria Nano Convergence (2015) 2:6 DOI 10.1186/s40580-014-0037-z

RESEARCH Open Access

Thermal instability of a nanofluid layer underlocal thermal non-equilibriumShilpi Agarwal1* and Beer Singh Bhadauria2

Abstract

In this paper, we study the effect of local thermal non-equilibrium on the linear thermal instability in a horizontal layerof a Newtonian nanofluid. The nanofluid layer incorporates the effect of Brownian motion along with thermophoresis.A two-temperature model has been used for the effect of local thermal non-equilibrium among the particle and fluidphases. The linear stability is based on normal mode technique and for nonlinear analysis, a minimal representation ofthe truncated Fourier series analysis involving only two terms has been used. We observe that for linear instability, thevalue of Rayleigh number can be increased by a substantial amount on considering a bottom heavy suspension ofnano particles. The effect of various parameters on Rayleigh number has been presented graphically. A weaknonlinear theory based on the truncated representation of Fourier series method has been used to find theconcentration and the thermal Nusselt numbers. The behavior of the concentration and thermal Nusselt numbers isalso investigated by solving the finite amplitude equations using a numerical method.

Keywords: Nanofluid; Instability; Natural convection

1 BackgroundNatural convection in fluids has been a topic of interestfor researchers like Nield and Bejan [1], Pop and Ingham[2], Ingham and Pop [3], Vafai [4,5], Vadasz [6], due to itsappearance in industry and machineries where heat trans-fer is encountered. In the present scenario where the focushas shifted fromordinary fluids to nanofluids to be used asheat transfer mediums, the phenomena needs to be criti-cally studied for them [7]. Accordingly, natural convectionhas been studied in nanofluids by Buongiorno [8], Tzou[9,10], Kim et al. [11-13], and based on these results in thecurrent decade by Nield and Kuznetsov [14], Kuznetsovand Nield [15] for the Horton-Rogers-Lapwood Prob-lem of onset of thermal instability in a porous mediumsaturated by a nanofluid, using Darcy and Brinkmanmodels, respectively, and incorporating the effects ofBrownian motion and thermophoresis of Nanoparticles.They found that the critical thermal Rayleigh numbercan be reduced or increased by a substantial amount,depending on whether the basic nanoparticle distribu-tion is top-heavy or bottom-heavy, by the presence of

*Correspondence: [email protected] of Mathematics, Amity Institute of Applied Sciences, AmityUniversity, Noida, Uttar Pradesh, IndiaFull list of author information is available at the end of the article

the nanoparticles. Based on these conservation equations,some recent studies have been performed by Agarwalet al. [16,17], Bhadauria and Agarwal [18,19], Agarwal andBhadauria [20], Rana and Agarwal [21], Agarwal [22]. Forthe case of a simple nanofluid layer, studies have beenperformed by Yadav et al. [23] for fluid layer, Bhadauriaand Agarwal [24], Agarwal and Bhadauria [25-27], for arotating nanofluid layer.In the above studies, investigations have been done

assuming local thermal equilibrium(LTE) between thefluid and particle phases as well as fluid and solid-matrixphases, i.e., the temperature gradient at any locationbetween the two phases is assumed to be negligible.But, as thermal lagging between the particle and fluidphases has been proposed by Vadasz as an explanationfor the observed increase in the thermal conductivity ofnanofluids, we need to study local thermal non equilib-rium(LTNE) model. The LTNE model of convective heattransfer in porous medium has been dealt by Kuznetsovand Nield [15], Agarwal and Bhadauria [20], Bhadauriaand Agarwal [18,19] to claim that the effect of LTNEcan be significant for some circumstances but remainsinsignificant for typical dilute nanofluids.Apart from the above few studies on thermal insta-

bility in nanofluids, no other study is available on this

© 2015 Agarwal and Bhadauria; licensee Springer. This is an Open Access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly credited.

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Agarwal and Bhadauria Nano Convergence (2015) 2:6 Page 2 of 16

problem, therefore we intend to investigate this prob-lem further. Assuming that the nanoparticles being sus-pended in the nanofluid using either surfactant or surfacecharge technology Nield and Kuznetsov [14], preventingthe agglomeration and deposition of these on the porousmatrix, in the present article, we study the linear andnon-linear thermal instability in a nanofluid layer, usingRayleigh Bénard problem, considering LTNE between thefluid/particle interphase.

2 MethodsWe consider a horizontal layer of nanofluid, confinedbetween two horizontal boundaries at z=0 and z=d,heated from below and cooled from above. The bound-aries are impermeable and perfectly thermally conduct-ing. The fluid layer is extended infinitely in x andy-directions, and z-axis is taken vertically upward withthe origin at the lower boundary. The local thermal non-equilibrium between the fluid and particle phase has beenconsidered, thus heat flow has been described using twotemperature model. Th and Tc are the temperatures atthe lower and upper walls respectively, the former beinggreater. The conservation equations for the total mass,momentum, thermal energy in the fluid phase,thermalenergy in the particle phase, and nanoparticles, come outto be as below. A detailed derivation of the conservationequations has been dealt by Buongiorno, Tzou, and Nieldand Kuznetsov:

∇ · v = 0 (1)

ρf

(∂v∂t

+v ·∇v)

= −∇p + μ∇2v

+ [φρp+(1−φ){ρf (1−β(Tf −Tc))

}]g

(2)

(ρc)f[∂Tf

∂t+ v · ∇Tf

]= kf ∇2Tf + (ρc)p

×[DB∇φ ·∇Tf +DT

∇Tf · ∇Tf

Tf

]

+ hfp1 − φ0

(Tp − Tf

)(3)

φ0(ρc)p[∂Tp

∂t+ vD · ∇Tp

]= φ0kp∇2Tp

+ hfp(Tf − Tp

) (4)

∂φ

∂t+ v · ∇φ = DB∇2φ + DT

Tc∇2Tf (5)

where v = (u, v,w) is the fluid velocity. In these equations,ρ is the fluid density, (ρc)f , (ρc)p, the effective heatcapacities of the fluid and particle phases respectively,and kf the effective thermal conductivity of fluid phase.DB and DT denote the Brownian diffusion coefficientand thermophoretic diffusion respectively. In the aboveequations, both Brownian transport and thermophore-sis coefficients are taken to be time independent, in tunewith the recent studies that neglect the effect of thermaltransport attributed to the small size of the nanoparticles(as per recent arguments by Keblinski and Cahill [28]).Further, Thermophoresis and Brownian transport coeffi-cients are assumed to be temperature - independent dueto the fact that the temperature ranges under consider-ation are not far away from the critical value, and thevolume averages over a representative elementary volume.Assuming the temperature and volumetric fraction of

the nanoparticles to be constant at the stress-free bound-aries, we may assume the boundary conditions on T andφ to be:

v = 0, T = Th, φ = φ1 at z = 0, (6)

v = 0, T = Tc, φ = φ0 at z = d, (7)where φ1 is greater than φ0. To non-dimensionalize thevariables we take(

x∗, y∗, z∗) = (x, y, z)/d, t∗ = tαf /d2, αf = kf(ρc)f(

u∗, v∗,w∗) = (u, v,w)d/αf , p∗ = pd2/μαf ,

φ∗ = φ − φ0φ1 − φ0

, T∗ = T − TcTh − Tc

,

Equations (1)–(7), then take the form (after droppingthe asterisk):

∇ · v = 0, (8)

1Pr

(∂v∂t

+ v · ∇v)

= −∇p + ∇2v − Rmez

+ RaTez − Rnφez ,(9)

∂Tf

∂t+ v · ∇Tf = ∇2Tf + NB

Le∇φ · ∇Tf

+ NANBLe

∇Tf · ∇Tf + NH(Tp − Tf

),

(10)

∂Tp

∂t+ v · ∇Tp = ε∇2Tp + γNH

(Tf − Tp

)(11)

∂φ

∂t+ v · ∇φ = 1

Le∇2φ + NA

Le∇2Tf , (12)

v = 0, T = 1, φ = 1 at z = 0 (13)


v = 0, T = 0, φ = 0 at z = 1 (14)

Here

Pr = μ

ραf, is the Prandtl number,

Le = αf

DB, is the Lewis number,

Ra = ρgβd3 (Th − Tc)

μαf, is the Rayleigh

−Darcy number,

Rm =[ρpφ0 + ρ (1 − φ0)

]gd3

μαf, is the basic density

Rayleigh number,

Rn =(ρp − ρ

)(φ1 − φ0) gd3

μαf, is the concentration

Rayleigh number,

NB = (ρc)p (φ1 − φ0)

(ρc)f, is the modified particle

density increment,

and

NA = DT (Th − Tc)

DBTc(φ1 − φ0), is the modified diffusivity

ratio which is similar to the Soret parameter thatarises in cross diffusion in thermal instability,

NH = hfpd2

(1 − φ0) kf, is the interphase heat transfer

parameter or the Nield number,

γ = (1 − φ0) (ρc)fφ0(ρc)p

, is the modified thermal capacity

ratio,

ε = kp/(ρc)pkf (ρc)f

, is the thermal diffusivity ratio.

3 Basic solutionThe basic state of the nanofluid layer is assumed to be atrest, therefore we have

v=0, p=pb(z), Tf =Tfb(z), Tp=Tpb(z), φ=φb(z) (15)

Substituting eq. (15) in Eqs. (10)–(12), we get

d2Tfb

dz2+ NB

Ledφbdz

dTfb

dz+ NANB

Le

(dTfb

dz

)2

+ NH(Tp − Tf ) = 0 (16)

The second and third terms in equation (16) are small[Nield and Kuznetsov [29]], so we have:

d2Tpb

dz2= 0,

d2Tfb

dz2= 0,

d2φbdz2

= 0, (17)

The boundary conditions for solving the above equationare found as

Tfb = Tpb = 1, φb = 1, at z = 0, (18)

Tfb = Tpb = 0, φb = 0, at z = 1. (19)

On solving eq. (17), subject to conditions (18) and (19),we obtain the basic temperature and concentration fieldsas

Tfb = Tpb = 1 − z (20)

φb = 1 − z. (21)

4 Stability analysisTo superimpose the perturbations on the basic state wewrite

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

Tp = Tpb + T ′p, φ = φb + φ′. (22)

We substitute the expression (22) in Eqs. (9)–(12), anduse the expressions (20) and (21). For simplicity, we con-sider the case of two dimensional rolls, assuming allphysical quantities to be independent of y. The reduceddimensionless governing equations after eliminating thepressure term are

1Pr

∂

∂t(∇2

1ψ) = ∇4

1ψ−Ra∂Tf

∂x+Rn

∂φ

∂x+ 1Pr∂(ψ ,∇2

1ψ)

∂(x, z)(23)

∂Tf

∂t+ ∂ψ

∂x= ∇2

1Tf + ∂(ψ ,Tf

)∂(x, z)

+NH(Tp − Tf

)(24)

∂Tp

∂t+ ∂ψ

∂x= ε∇2

1Tp + ∂(ψ ,Tp

)∂(x, z)

+ γNH(Tf − Tp

)(25)

∂φ

∂t+ ∂ψ

∂x= 1

Le∇21φ + NA

Le∇21Tf + ∂ (ψ , φ)

∂(x, z)(26)

The equations (23)–(26) are solved subject to stress-free,isothermal, iso-nanoconcentration boundary conditions:

ψ = ∂2ψ

∂z2= Tf = Tp = φ = 0 at z = 0, 1 (27)

For linear stability analysis we use the normal modetechnique and then critical Rayleigh numbers for


stationary and oscillatory onset of convection and thefrequency of oscillations, ω, are given by

Rast = δ6

α2c

(1 + NH(ε − 1)

εδ2 + (1 + γ )NH

)

− Rn{NA − Le − NHLe(ε − 1)

εδ2 + (1 + γ )NH

} (28)

Raosc = δ2

α2

[δ4

(1+NH (ε − 1)P1

P21 + ω2

)−ω

2

Pr

(1−NH(ε − 1)δ2

P21 + ω2

)]

− Rn(δ2/Le

)2 + ω2

[δ2

Le

(P2 − NH (ε − 1) δ2P1

P21 + ω2

)

−ω2

(1 − NH(ε − 1)δ2

P21 + ω2

)]

(29)

ω2c =

−X2 +√X22 − 4X1X3

2X1(30)

where

δ2 = π2 + α2c and αc = π√2.

Also

X1 = P3 + δ4

α2

X2 = P3δ4

Le2+ P3P21 + P3NHP1(ε − 1)+ δ8

α2Le2+ δ4P21

α2

− NH(ε − 1)δ6

α2+ RnP2 + Rnδ2

Le

X3 =P3P21δ4

Le2+ P3P1NH(ε − 1)

δ4

Le2+ δ8P21α2Le2

− δ10NH(ε − 1)α2Le2

+ RnP2P21 − RnP1NHδ2(ε − 1)

+ RnP21δ2

Le− RnNHδ4(ε − 1)

Le

These expressions were obtained from Nield andKuznetsov by dropping the terms pertaining to porousmedia.For local nonlinear stability analysis we take the follow-

ing Fourier expressions:

ψ =∞∑n=1

∞∑m=1

Amnsin (mαx) sin (nπz) (31)

Tf =∞∑n=1

∞∑m=1

Bmn(t)cos(mαx)sin(nπz) (32)

Tp =∞∑n=1

∞∑m=1

Cmn(t)cos(mαx)sin(nπz) (33)

φ =∞∑n=1

∞∑m=1

Dmn(t)cos(mαx)sin(nπz) (34)

In what followswe take themodes (1, 1) for stream func-tion, and (0, 2) and (1, 1) for temperature and nanoparticleconcentration

ψ = A11(t)sin(αx)sin(πz) (35)

Tf = B11(t)cos(αx)sin(πz) + B02(t)sin(2πz) (36)

Tp = C11(t)cos(αx)sin(πz) + C02(t)sin(2πz) (37)

φ = D11(t)cos(αx)sin(πz) + D02(t)sin(2πz) (38)

where the amplitudes A11(t),B11(t),B02(t),C11(t),C02(t),D11(t) and D02(t) are functions of time and areto be determined. Substituting equations (35)–(38) inequations (23)–(26) taking the orthogonality conditionwith the eigenfunctions associated with the consideredminimal model, we get

dA11(t)dt

= Prδ2{α [RnD11(t)− RaB11(t)] − δ4A11(t)

}(39)

dB11(t)dt

=NH [C11(t)− B11(t)] − [αA11(t)

+ δ2B11(t)+ παA11(t)B02(t)] (40)

dB02(t)dt

= 12{παA11(t)B11(t)− 8π2B02(t)

+ 2NH [C02 − B02]}(41)

dC11(t)dt

=−{αA11(t)+εδ2C11(t)−γNH[B11(t)− C11(t)]

+ παA11(t)C02(t)}

(42)

dC02(t)dt

=12{παA11(t)C11(t)− 8π2εC02(t)

+ 2γNH [B02(t)− C02(t)]} (43)

dD11(t)dt

= − αA11(t)− δ2

LeD11(t)− NAδ

2

LeB11(t)

− παA11(t)D02(t)(44)

dD02(t)dt

= − 4π2

Le[D02(t)+ NAB02(t)]

+ παA11(t)D11(t)(45)

The above system of simultaneous autonomous ordi-nary differential equations be subsequently solved numer-ically using Runge-Kutta-Gill method.


5 Heat and nanoparticle concentration transportThe thermal Nusselt number for fluid phase, Nuf (t) isdefined as

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

= 1 +

⎡⎢⎢⎢⎣

2π/αc∫0

(∂Tf∂z

)dx

2π/αc∫0

(∂Tb∂z

)dx

⎤⎥⎥⎥⎦z=0

(46)

substituting equations (20) and (36) in eq. (46), we get

Nuf (t) = 1 − 2πB02(t) (47)

The thermal Nusselt number for particle phase, Nup(t),and the nanoparticle concentration Nusselt number,Nuφ(t), are defined similar to the thermal Nusselt number.Following the procedure adopted for arriving at Nuf (t),one can obtain the expression for Nup(t) and Nuφ(t) inthe form:

Nup(t) = 1 − 2πC02(t) (48)

Nuφ(t) = (1 − 2πD02(t))+ NA (1 − 2πB02(t)) (49)

6 Results and discussionThe expressions for the Rayleigh number for bothstationary and oscillatory convection have been pre-sented analytically in equations (28) and (29) respectively.

Considering the expression for stationary convection ineq. (28), we have

Rast = δ6

α2c

(1 + NH(ε − 1)

εδ2 + (1 + γ )NH

)

− Rn{NA − Le − NH(ε − 1)δ2

εδ2 + (1 + γ )NH

} (50)

For thermal equilibrium condition NH = 0, so we get

Rast = δ6

α2c− Rn (NA − Le) (51)

This result agrees with the result obtained by Nield andKuznetsov under LTNE.In Figure 1, we present linear stability curves showing

the stationary and oscillatory modes of convection. In thefigure we see that the region of over stability under-lies theregion of damped oscillations, i.e. at the start of instability,oscillatory or periodic pattern of motion prevails. There-fore we can say that the Principle of Exchange of stabilitiesis invalid [30,31]. This can be explained as since oscilla-tory motion is prevalent at the onset of convection, therestoring forces provoked are strong enough to preventthe system from tending away from equilibrium.In Figure 2(a-f ) we present neutral stability curves for

Rast versus the wavenumber α for the fixed values ofRn, Le,NA, ε, γ and NH , respectively, with variation in oneof these parameters. In all these plots, it is interesting tonote that the value of Ra starts from a higher note, fallsrapidly with increasing α, and then increases steadily. TheFigure 2(a), (b), (d) and (e) correspond to the variation ofRa with respect to α at different values of concentration

Figure 1 Linear stability curve showing stationary and oscillatorymodes of convection.


(a) (b)

(c) (d)

(e) (f)

Figure 2 Neutral stability curves for different values of (a) Rn, (b) Le, (c) NA, (d) ε, (e) γ , (f) NH .

Rayleigh number Rn, Lewis number Le, thermal diffusivityratio ε, and modified thermal capacity ratio γ . These plotsreveal that on increasing the value of these parameters, thevalue of Racr increases, i.e. the system tends to stabilize.

However, in the Figure 2(c) and (f ), it is to be noted thatthe effect of the parameters NA and NH is to destabilizethe system. As we increase their value, the value of Racrdecreases, indicating that the convection starts earlier.


In the Figure 3, we draw the neutral stability curve forLTNE and LTE. We see that the value of Rayleigh numberRa is less in case of LTNE than LTE. This implies that con-vections starts earlier in the case of LTNE than LTE. Theobserved phenomenon may be attributed to the fact thatbecause of temperature difference between the fluid andparticle phases, there occurs transfer of energy betweenthem. This leads to a chaotic state and enhances the onsetof convection in case of LTNE.In the Figure 4, we compare the Rayleigh number Ra for

nanofluids with ordinary fluids under thermal nonequi-librium conditions. It is to be noted that the value ofRayleigh number is less in the case of ordinary fluidthan nanofluid, or to say convection sets in earlier inordinary fluids than nanofluids. This implies that the ther-mal conductivity of nanofluids is higher than ordinaryfluids.The nature of critical values of Rayleigh number Ra

and the critical values of wave number α as functionsof inter phase heat transfer parameter or Nield numberfor fluid/nanoparticle inter phase, NH , for Rn = 4, Le =900,NA = 1, ε = 0.04, γ = 5, with a variation in thevalue of one of these parameters, are shown in Figures 5and 6 respectively. For very small and large values of NH ,we observe that the stability criterion is independent of itsvalue, and that the value ofNH play a significant role in thestability criterion only in the intermediate range. The rea-son behind this state being, that at NH → 0, there occursalmost zero heat transfer between fluid/nanoparticle interphase, and the properties of nanoparticle do not interferein the onset of convection.While, whenNH → ∞, the two

have attained almost the equal temperatures and behaveas a single phase. Between these two extremes, a LTNEeffect is observed being attributed to NH .In the Figure 5, we present the variation of criti-

cal Rayleigh number Racr with Nield number for thefluid/particle inter phaseNH for different parameters. Thefigure indicates that the value of Racr decreases from highvalues for very small NHP to small LTNE value for largeNHP . The system tends to destabilize for the intermediatevalues of NHP . The effect of the parameters concentrationRayleigh number Rn, Lewis number Le, thermal diffu-sivity ratio ε, and modified thermal capacity ratio γ , onthe system is to inhibit the decrease in the value of thecritical Rayleigh number Racr , thus preventing the sys-tem from destabilization. While for the other parameter,modified diffusivity ratio NA, on increasing its values, thevalue of Racr falls further trending the system towardsdestabilization.In the Figure 6, we have exhibited the critical wavenum-

ber αc as a function of NH for both stationary andoscillatory convection. We observe that value of criticalwavenumber αc decreases with increasing NH from highvalues when NH is small, to its minimum LTNE valuefor intermediate NH , and finally bounces back to highervalues for large NH . This implies that the the value ofcritical wavenumber αc approaches to its LTE value whenNH → 0 and NH → ∞. This is quite obvious as the cor-responding physical situation are anonymous. At NH →0, the particle phase does not interfere with the ther-mal field of the fluid, which is free to act independently,while as NH → ∞, the particle/fluid phase have attained

Figure 3 Comparison of the value of Rayleigh number Ra for LTNE and LTE.


Figure 4 Comparison of the value of Rayleigh number Ra for Nanofluid with Ordinary fluid.

the identical temperatures, and behave as single phaseonly. We conclude that as time passes, and heat intensi-fies, the nanofluids behave more like a single phase fluidrather than like a conventional solid-liquid mixture. Forthe parameters, Rn, Le,NA, and ε, we see that an increasein their values has effect on the critical value of wavenumber αc only when NH → 0 and NH → ∞. For inter-mediate values of N, i.e. when αc falls to its minimumLTNE value, the value of αc remains independent of theseparameters. But for γ , as we increase its value, minimumvalue of αc decreases. Also, we notice that the minimumvalue of αc is less for oscillatory convection as comparedto stationary convection.The variation of critical frequency ω2

c for the oscillatorymode of convection with Nield number NH for differentparameter values is shown in Figure 7(a)–(f ). It is clearfrom these figures that the critical frequency decreasesfrom a constant value, when NH is very small to its min-imum value, and then with further increase in NH , itbounces back to another constant value for large valuesof NH . Figure 7(a) displays the effect of concentrationRayleigh number Rn on the value of critical frequency ω2

c .We can observe that an increase in the value of Rn inhibitsthe decrease in the value of critical frequency ω2

c . A sim-ilar effect on critical frequency ω2

c has been observed inFigure 7(d), 7(e) and 7(f ) with modified thermal capacityratio γ , thermal diffusivity ratio ε, and Prandtl number Pr.The parameters, Lewis number Le, and modified diffusiv-ity ratio NA do not have a very pronounced effect on thevalue of critical frequencyω2

c . In general, the value of criti-cal frequency ω2

c decreases from its LTE value to its LTNEvalue, and then increases back to attain another constantvalue in the intermediate range.

6.1 Non-linear unsteady stability analysisThe linear solutions exhibit a considerable variety ofbehavior of the system and the transition from linear tonon-linear convection can be quite complicated, but inter-esting to deal with. We need to study the time dependentresults to analyze the same. This transition can be wellunderstood by the analysis of the seven coupled ordinarydifferential eqs. (39)–(45) whose solutions give a detaileddescription of the two-dimensional problem. We solvethe eqs. (39)–(45) numerically, using Runge−Kutta−GillMethod, and calculate Nusselt numbers as function oftime t. Also the seven-mode Differential eqs. (39)–(45)have an interesting property in phase-space:

∂ ˙A11∂A11

+ ∂ ˙B11∂B11

+ ∂ ˙B02∂B02

+ ∂C11∂C11

+ ∂ ˙C02∂C02

+ ∂D11∂D11

+ ∂D02∂D02

= −[δ2Pr + NH + δ2 + 4π2 + NH + γpNH

+ εδ2 + 4π2ε + γNH + δ2

Le+ 4π2

Le

]< 0

(52)

which indicates that the system is dissipative andbounded. This implies that the trajectories of theeqs. (39)–(45) will be attracted to a set of measure zeroin the phase space, in other words they will approach to afixed point, a limit cycle or to a unknown attractor.The nature of Nusselt numbers, Nuφ ,Nu(fluid), and

Nu(particle), as a function of time t, for Rn = 4, Le =10,NA = 5,Ra = 5000, Pr = 0.75, ε = 0.04,NH = 20,


(a) (b)

(c)

(e)

(d)

Figure 5 Variation of critical Rayleigh number Racr with NH for different values of (a) Rn, (b) Le, (c) NA, (d) γ and (e) ε.

and γ = 0.5 with a variation in the value of one ofthese parameters, is shown in Figures 8 and 9 respec-tively. These figures indicate that initially, when time is

small, there occurs large scale oscillations in the values ofNuφ ,Nu(fluid), and Nu(particle), indicating an unsteadyrate of mass and heat transfer in the fluid and particle


(a) (b)

(c)

(e)

(d)

Figure 6 Variation of critical wave number αcwith NH for different values of (a) Rn, (b) Le, (c)NA , (d) γ and (e) ε.

phases. As time passes by, these approach to steady values,corresponding to a near conduction instead of convectionstage.

In the Figure 8, the transient nature of concentra-tion Nusselt number or Sherwood number (as someresearchers name it), is visible. We can observe that


(a) (b)

(c) (d)

(e) (f)

Figure 7 Variation of critical frequency ω2c with NH for different values of (a) Rn, (b) Le, (c) NA , (d) γ , (e) ε and Pr.

the effect of increasing the value of modified diffusiv-ity ratio or Soret parameter NA, thermal Rayleigh num-ber Ra, Prandtl number Pr, modified thermal capacity

ratio γ , and the Nield number NH , on the amplitudeof oscillations is to increase it, i.e., an increase in thevalue of these parameters brings about an increase


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 8 Variation of Concentration Nusselt Number Nuφ with time t for different values of (a) Rn, (b) Le, (c) NA, (d) Ra, (e) ε, (f) Pr, (g) γ ,and (h) NH .


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 9 Variation of Nusselt Number Nu(fluid) with time t for different values of (a) Rn, (b) Le, (c) NA , (d) Ra, (e) ε, (f) Pr, (g) γ , and (h) NH .


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 10 Variation of Nusselt Number Nu(particle) with time t for different values of (a) Rn, (b) Le, (c) NA , (d) Ra, (e) ε, (f) Pr, (g) γ , and(h) NH .


in the rate of mass transfer across the nanofluidlayer.Figure 9 depicts the transient nature of Nusselt num-

ber for the fluid phase. There occurs large amount of heattransfer in the fluid phase initially, and with large timethe amount of heat transfer approaches a near constantvalue. We can observe that the effect of increasing thevalue of thermal Rayleigh number Ra, Prandtl number Pr,modified thermal capacity ratio γ , and the Nield num-ber NH , on the amplitude of oscillations is to increase it,i.e., an increase in the value of these parameters bringsabout an increase in the rate of heat transfer across thefluid phase.The transient nature of Nusselt number for particle

phase has been shown in Figure 10.We see that the ampli-tude of heat transfer in particle phase is small initially, butincreases rapidly with time t to approach a steady valuefor large values of time t. The effect of the parametersRa, Pr, γ , and NH on the amplitude of oscillations is toincrease them.

7 ConclusionsWe considered linear stability analysis in a horizontalnanofluid layer, heated from below and cooled fromabove, incorporating the effect of Brownian motion alongwith thermophoresis, under non-equilibrium conditions.Further bottom heavy suspension of nanoparticles hasbeen considered. Linear analysis has been made usingnormal mode technique, while for weakly nonlinear anal-ysis, we have used a minimal representation of truncatedFourier Series involving only two terms. Then the effect ofvarious parameters on the onset of thermal instability hasbeen found.The results have been presented graphically.We draw the following conclusions:

1. The effect of the concentration Rayleigh number Rn,Lewis number Le, modified thermal capacity ratio γ ,and thermal diffusivity ratio ε is to stabilize thesystem.

2. Convection sets in earlier for LTNE as compared toLTE.

3. The value of critical wave number αc is lower foroscillatory convection then for stationary convection.

4. The effect of time on Nusselt numbers is found to beoscillatory, when t is small. However when time tbecomes very large Nusselt number approaches thesteady value.

5. On increasing the value of thermal Rayleigh numberRa, the rate of mass and heat transfer is increased.

NomenclatureLatin SymbolsDB Brownian Diffusion coefficient.DT thermophoretic diffusion coefficient.

Pr Pradtl number.d dimensional layer depth.Le Lewis number.NA modified diffusivity ratio.NB modified particle-density increment.NH Nield Number.p pressure.g Gravitational acceleration.Ra thermal Rayleigh-Darcy number,Rm basic density Rayleigh number,Rn concentration Rayleigh number,t time.T nanofluid temperature.Tc temperature at the upper wall.Th temperature at the lower wall.v nanofluid velocity.(x, y, z) Cartesian coordinates.

Greek symbolsαf thermal diffusivity of the fluid.β proportionality factor.γ modified thermal capacity ratio.ε thermal diffusivity ratio.ψ stream function.μ viscosity of the fluid.ρf fluid density.ρp nanoparticle mass density(ρc)f Heat capacity of the fluid.(ρc)p Heat capacity of the nanoparticle material.φ nanoparticle volume fraction.α wave number.ω frequency of oscillations.

Subscriptsb basic solution.f Fluid phase.p Particle phase.c critical.

Superscripts∗ dimensional variable’ perturbation variable

Operators

∇2 ∂2

∂x2+ ∂2

∂y2+ ∂2

∂z2.

∇21

∂2

∂x2+ ∂2

∂z2.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsSA and BSB completed the analytic and numeric analysis of the problem anddrafted the manuscript. All authors read and approved the final manuscript.


Author details1Department of Mathematics, Amity Institute of Applied Sciences, AmityUniversity, Noida, Uttar Pradesh, India. 2DST-Centre for InterdisciplinaryMathematical Sciences, Department of Mathematics, Faculty of Science,Banaras Hindu University, Varanasi-221005, India.

Received: 9 July 2014 Accepted: 28 October 2014

References1. DA Nield, A Bejan, Convection in Porous Media (3rd edition). (Springer, New

York, 2006)2. I Pop, DB Ingham, Convective Heat Transfer: Mathematical and

Computational Modeling of Viscous Fluids and Porous Media. (Oxford,Pergamon, 2001)

3. DB Ingham, I Pop, Transport Phenomena in Porous Media III. (Elsevier,Oxford, 2005)

4. K Vafai, Handbook of Porous Media (2nd edition). (Taylor & Francis,New York, 2005)

5. K Vafai, Porous Media: Applications in Biological Systems and Biotechnology.(CRC Press, Hindawi Publications, New York, 2010)

6. P Vadasz, Emerging Topics in Heat andMass Transfer in PorousMedia.(Springer, New York, 2008)

7. L Godson, B Raja, DM Lal, S Wongwises, Enhancement of Heat TransferUsing Nanofluids - An Overview. Renew. Sustain. Energ. Rev. 14, 629–641(2010)

8. J Buongiorn, Convective transport in nanofluids. ASME Jr. Heat Transfer.128, 240–250 (2006)

9. DY Tzou, Instability of nanofluids in natural convection. ASME, Jr. HeatTransfer. 130, 072401 (2008a)

10. DY Tzou, Thermal instability of nanofluids in natural convection. Int. Jr.Heat Mass Transfer. 51, 2967–2979 (2008b)

11. J Kim, YT Kang, CK Choi, Analysis of convective instability and heattransfer characteristics of nanofluids. Phys. Fluids. 16, 2395–2401 (2004)

12. J Kim, CK Choi, YT Kang, MG Kim, Effects of thermodiffusion andnanoparticles on convective instabilities in binary nanofluids. NanoscaleMicroscale Thermophys. Eng. 10, 29–39 (2006)

13. J Kim, YT Kang, CK Choi, Analysis of convective instability and heattransfer characteristics of nanofluids. Int. J. Refrig. 30, 323–328 (2007)

14. DA Nield, AV Kuznetsov, Thermal instability in a porous medium layersaturated by nonofluid. Int. Jr. Heat Mass Transfer. 52, 5796–5801 (2009)

15. AV Kuznetsov, DA Nield, Thermal instability in a porous medium layersaturated by a nanofluid: Brinkman Model. Trans. Porous Med.81, 409–422 (2010)

16. S Agarwal, BS Bhadauria, PG Siddheshwar, Thermal instability of ananofluid saturating a rotating anisotropic porous medium. Spec. Top.Rev. Porous. Media Int. J. 2(1), 53–64 (2011). Begell House, USA

17. S Agarwal, NC Sacheti, P Chandran, BS Bhadauria, AK Singh, Non-linearconvective transport in a binary nanofluid saturated porous layer.Transport Porous Medium. 93(1), 29–49 (2012)

18. BS Bhadauria, S Agarwal, Natural convection in a nanofluid saturatedrotating porous layer: a nonlinear study. Transport Porous Media.87(2), 585–602 (2011)

19. BS Bhadauria, S Agarwal, Convective transport in a nanofluid saturatedporous layer with thermal non equilibrium model. Transport PorousMedia. 88(1), 107–131 (2011)

20. S Agarwal, BS Bhadauria, Natural convection in a nanofluid saturatedrotating porous layer with thermal non equilibrium model. TransportPorous Media. 90, 627–654 (2011). Springer

21. P Rana, S Agarwal, Convection in a binary nanofluid saturated rotatingporous layer. J. Nanofluids. 4(1), pp1-7 (2014 in press). American ScientificPublishers

22. S Agarwal, Thermal instability of a nanofluid saturating an anisotropicporous medium layer under local thermal non-equilibrium condition,J. Nano Energy Power Res. (2014, in press). American ScientificPublishers

23. Y Dhananjay, GS Agrawal, R Bhargava, Rayleigh Bénard convection innanofluids. Int. J. Appl. Math Mech. 7(2), 61–76 (2011)

24. BS Bhadauria, S Agarwal, Natural Convection in a Rotating NanofluidLayer, MATECWeb of Conferences, EDP Sciences. 1, 06001 (2012)

25. S Agarwal, BS Bhadauria, Convective heat transport by longitudinal rollsin dilute Nanoliquids. J. Nanofluids. 3(4), pp 1-11 (2014). AmericanScientific Publishers

26. S Agarwal, BS Bhadauria, Unsteady heat and mass transfer in a rotatingnanofluid layer. Continuum Mech. Therm. 26, 437–445 (2014)

27. S Agarwal, BS Bhadauria, Flow patterns in linear state of Rayleigh-Bénardconvection in a rotating nanofluid layer. Appl. Nanosci. 4(8), pp 935-941(2013). doi:10.1007/s13204-013-0273-2

28. P Keblinski, DG Cahil, Comments on model for heat conduction innanofluids. Phy. Rev Lett. 95(209401) (2005)

29. DA Nield, AV Kuznetsov, The effect of local thermal nonequilibrium onthe onset of convection in a nanofluid. J. Heat. Tran. 132, 052405 (2010)

30. S Chandrashekhar, Hydrodynamic and Hydromagnetic Stability. (OxfordUniversity Press, Oxford, 1961)

31. PG Drazin, DH Reid, Hydrodynamic Stability. (Cambridge University Press,Cambridge, 1981)

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RESEARCH OpenAccess ... · assuming local thermal equilibrium(LTE) between the fluid and particle phases as well as fluid and solid-matrix phases, i.e., the temperature gradient at