Transp Porous Med (2014) 104:581–592 DOI 10.1007/s11242-014-0351-2

Natural Convection in a Nanofluid-Saturated Rotating Porous Layer: A More Realistic Approach Shilpi Agarwal

Received: 22 May 2014 / Accepted: 5 June 2014 / Published online: 25 June 2014 © Springer Science+Business Media Dordrecht 2014

Abstract The present work aims at studying the thermal instability in a rotating porous layer saturated by a nanofluid based on a new boundary condition for the nanoparticle fraction, which is physically more realistic. The model used for nanofluid combines the effect of Brownian motion along with thermophoresis, while for a porous medium Brinkman model has been used. A more realistic set of boundary conditions where the nanoparticle volume fraction adjusts itself including the contributions of the effect of thermophoresis so that the nanoparticle flux is zero at the boundaries has been considered. Using linear stability analysis, the expression for critical Rayleigh number has been obtained in terms of various non-dimensional parameters. The effect of various parameters on the onset of instability has been presented graphically and discussed in detail. Keywords Nanofluid · Porous medium · Instability · Natural convection · Rotation · Taylor number Nomenclature Latin symbols DB DT Da Pr d kT km Le NA NB

Brownian diffusion coefficient Thermophoretic diffusion coefficient Darcy number Pradtl number Dimensional layer depth Effective thermal conductivity of porous medium Thermal diffusivity of porous medium Lewis number Modified diffusivity ratio Modified particle-density increment

S. Agarwal (B) Department of Mathematics, Galgotias University, Greater Noida, Uttar Pradesh, India e-mail: [email protected]

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p g Ra Rm Rn t T Tc Th v vD (x ∗ , y ∗ , z ∗ ) Ta

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Pressure Gravitational acceleration Thermal Rayleigh–Darcy number Basic density Rayleigh number Concentration Rayleigh number Time Temperature Temperature at the upper wall Temperature t at the lower wall Nanofluid velocity Darcy velocity εv Cartesian coordinates Taylor number

Greek symbols α β ε μ μ¯ ρf ρp (ρc)f (ρc)m (ρc)p γ φ ν ψ α ω

Horizontal wave number Proportionality factor Porosity Viscosity of the fluid Effective viscosity of the porous medium Fluid density Nanoparticle mass density Heat capacity of the fluid Effective heat capacity of the porous medium Effective heat capacity of the nanoparticle material (ρc)m Parameter defined as (ρc)f Nanoparticle volume fraction Kinematic viscosity μ/ρf Stream function Wave number Frequency of oscillation

Subscripts b

Basic solution

Superscripts ∗



Dimensional variable Perturbation variable

Operators ∇2 ∇12

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∂2 ∂2 ∂2 + 2 + 2 2 ∂x ∂y ∂z ∂2 ∂2 + 2 ∂x2 ∂z

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1 Introduction With the modern day industries facing the massive problem of cooling of the high-energy devices, researchers are focusing their attention toward a relatively new class of heat transfer media, the nanofluids, from the conventional heat transfer mediums with limited cooling capacities. Nanofluids are engineered by suspending the nanoparticles (metal particles or metallic oxides or nonmetallic oxide particles) in the common base fluids like water, ethylene glycol, and engine oils. Most of these heat exchanging situations in modern science and engineering, including food and chemical processes and rotating machineries like nuclear reactors, petroleum industry, biomechanics, and geophysical problems, correspond to the classical problem of the onset of convection in a horizontal layer occupied by a porous medium uniformly heated from below, commonly known as the Horton–Rogers–Lapwood problem or B´enard–Darcy problem. A comprehensive review on thermal transport in nanofluids was made by Eastman et al. (2004), who concluded that despite several attempts, a satisfactory explanation for the abnormal enhancement in thermal conductivity and viscosity in nanofluids is yet to be found. It was Buongiorno (2006), who came out with a comparatively satisfactory model including the effects of Brownian motion and thermophoresis of the nanoparticles suspended. An extension to this was made by Nield and Kuznetsov (2009, 2011), Kuznetsov and Nield (2010a, b), Bhadauria et al. (2011) with the assumption that the value of the nanoparticle fraction at the boundary can be controlled as the temperature thereat. Since porous medium is one of the key participants in many heat transfer situations, based on the same assumptions, studies were carried out by Agarwal et al. (2011), Bhadauria and Agarwal (2011), Agarwal and Bhadauria (2011), and Chand and Rana (2012) pertaining to rotating porous medium saturated by a nanofluid. But in the due course, it turned out that physically the above boundary conditions may be hard to establish and the boundary conditions need to be more realistic. In their very recent work, Nield and Kuznetsov (2014) came out with a new set of boundary condition that assumes that there is no nanoparticle flux at the plate and that the particle fraction value there adjusts accordingly. With such an assumption, the presence of oscillatory convection turns oblivious due to the absence of the two opposing agencies affecting instability. Assuming that the nanoparticles being suspended in the base fluid, using either surfactant or surface charge technology, prevent the agglomeration and deposition of these on the porous matrix (Nield and Kuznetsov 2009), in the present article, we study the effect of rotation on thermal instability in a porous medium saturated by a nanofluid. We use the Horton–Rogers– Lapwood model based on the Brinkman model for the fluid flow, with corrections to include effects due to nanoparticles.

2 Governing Equations We consider a densely packed porous layer saturated by a nanofluid, confined between two horizontal boundaries at z = 0 and z = H , heated from below and cooled from above. The boundaries are impermeable and perfectly thermally conducting. The porous layer is extended infinitely in x and y directions, with the z-axis extending vertically upward with the origin at the lower boundary. The porous layer is rotating uniformly about z-axis. The Coriolis effect has been taken into account by including the Coriolis force term in the momentum equation, whereas the centrifugal force term can be realized as a gradient of a scalar and hence has been absorbed into the pressure term. In addition, the local thermal equilibrium between the fluid and solid has been considered, and thus, the heat flow has been described using one

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equation model. Th and Tc are the temperatures at the lower and upper walls, respectively, the former being greater. Employing the Oberbeck–Boussinesq approximation, the governing equations to study the thermal instability in a nanofluid-saturated rotating porous medium are (Buongiorno 2006; Tzou 2008a, b; Nield and Kuznetsov 2014): ∇ · vD = 0 ρf ∂vD μ 2 = −∇ p + μ∇ ¯ 2 vD − vD + (vD × Ω) ε ∂t K δ + [φρp + (1 − φ){ρ(1 − β(T − Tc ))}]g ∂T DT (ρc)m ∇T · ∇T ] + (ρc)f vD · ∇T = km ∇ 2 T + ε(ρc)p [DB ∇φ · ∇T + ∂t Tc ∂φ DT 2 1 ∇ T + vD · ∇φ = DB ∇ 2 φ + ∂t ε Tc

(1)

(2) (3) (4)

where vD = (u, v, w) is the Darcy velocity. In the above equations, both Brownian transport and thermophoresis coefficients are taken to be time-independent, in tune with the studies that neglect the effect of thermal transport attributed to the small size of the nanoparticles (as per arguments by Keblinski and Cahil 2005). Further, thermophoresis and Brownian transport coefficients are assumed to be temperature-independent due to the fact that the temperature ranges under consideration are not far away from the critical value. Assuming the temperature and volumetric fraction of the nanoparticles to be constant at the boundaries, we take the boundary conditions to be DB ∂ T ∂φ + = 0 at z = 0, ∂z T∞ ∂z DB ∂ T ∂φ w = 0, T = Tc , DB + = 0 at z = d. ∂z T∞ ∂z

w = 0, T = Th , DB

(5) (6)

The dimensionless variables are considered as given below: (x ∗ , y ∗ , z ∗ ) = (x, y, z)/d, t ∗ = tkT /γ d 2 (u ∗ , v ∗ , w ∗ ) = (u, v, w)d/kT , p ∗ = pK /μkT T − Tc φ ∗ = (φ − φ0 )/(φ0 ) T ∗ = Th − Tc (ρcp )m km , γ= . The proceeding non-dimensionalized governing equations (ρc)f (ρcp )f are (after dropping the asterisk) where kT =

∇ ·v=0 √ Da ∂v ˆ − Rm eˆz + RaT eˆz − Rnφ eˆz = −∇ p + Da∇ 2 v − v + T a(v × k) Pr ∂t NA NB NB ∂T + v · ∇T = ∇ 2 T + + ∇φ · ∇T + ∇T · ∇T γ ∂t Le Le 1 1 2 NA 2 ∂φ + v · ∇φ = ∇ φ+ ∇ T ∂t ε Le Le

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(7) (8) (9) (10)

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The other non-dimensional parameters that appear in the above equations are as given below:   2Ω K z 2 , is the Taylor−Vadasz number, Ta = νε kT z , is the Lewis number, Le = DB ρgβ K z H (Th − Tc ) Ra = , is the Rayleigh−Darcy number, μk T z [ρp φ0 + ρ(1 − φ0 )]gK z H Rm = , is the basic density Rayleigh−Darcy number, μk T z (ρp − ρ)φ0 gK z H , is the concentration Rayleigh−Darcy number, Rn = μk T z ε(ρc)p φ0 , is the modified particle density increment NB = (ρc)f and NA =

DT (Th − Tc ) , is the modified diffusivity ratio, which is similar DB Tc φ0

to the Soret parameter that arises in cross diffusion problems in thermal instability. Also the boundary conditions become w = 0, T = Th ,

∂T ∂φ + NA = 0 at z = 0, ∂z ∂z

(11)

w = 0, T = Tc ,

∂T ∂φ + NA = 0 at z = d. ∂z ∂z

(12)

3 Basic Solution At the basic state, the nanofluid is assumed to be at rest; therefore, the quantities at the basic state will vary only in z direction and are given by v = 0, T = Tb (z), φ = φb (z), p = pb (z).

(13)

Substituting Eq. (13) in the Eqs. (9) and (10), we get N B dφb dTb NA NB d2 Tb + + dz 2 Le dz dz Le



dTb dz

2 =0

d 2 φb d2 Tb + N A 2 = 0. 2 dz dz

(14) (15)

Under the boundary conditions (11) and (12), the integration of Eq. (15) gives dφb dTb + NA = 0. dz dz

(16)

d2 Tb = 0, dz 2

(17)

Using this in the Eq. (14), we get

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whose solution satisfying the boundary conditions shall be Tb = 1 − z.

(18)

Equation 16, now on integration, along with the boundary conditions assumed, results to φb = φ0 + N A z.

(19)

4 Perturbation Solution We now superimpose perturbations on the basic state as given below: v = v , p = pb + p  , T = Tb + T  , φ = φb + φ  .

(20)

For the sake of simplicity of calculations, we consider the case of two-dimensional rolls, assuming all physical quantities to be independent of y. Substituting the expression (20) in Eqs. (8)–(10), and using the expressions (18) and (19), we obtain the following equations in linear form after neglecting the nonlinear terms 



∇ · v = 0

√ Da ∂ ˆ − Da∇ 2 + 1 v = −∇ p + RaT kˆ − Rnφ kˆ + T a(v × k) Pr ∂t   ∂φ 2N A N B ∂ T NB ∂ T ∂T − w = ∇2T + − − γ ∂t Le ∂z ∂z Le ∂z ∂φ NA 2 NA 1 2 + w= ∇ φ+ ∇ T ∂t ε Le Le ∂T ∂φ w = 0, T = 0, + NA = 0 at z = 0, and at z = 1. ∂z ∂z

(21) (22) (23) (24) (25)

The parameter Rm is not carried over in the subsequent equations since it is part of the hydrostatic equilibrium. Now, taking the curl of Eq. (22), we get   √ Da ∂ ∂w  − Da∇ 2 + 1 ∇ × v = T a . (26) Pr ∂t ∂z Taking curl of Eq. (22) twice, and considering the vertical component, we obtain   √ ∂ Da ∂ 2 − Da∇ + 1 ∇ 2 w = Ra∇12 T − Rn∇12 φ − T a (∇ × v). Pr ∂t ∂z

(27)

The operator, ∇ 2 , is the two-dimensional Laplacian operator on the horizontal plane. Combining the above two equations, we obtain 2    Da ∂ Da ∂ ∂ 2w 2 2 2 2 2 − Da∇ + 1 ∇ w = (Ra∇1 T − Rn∇1 φ) − Da∇ + 1 − T a 2 . Pr ∂t Pr ∂t ∂z (28) To find the solution for the unknown fields in Eqs. (28), (23), and (24), we use the normal mode technique and thus write (w  , T  , φ) = [W (z), Θ(z), Φ(z)] exp [st + i(lx + my)],

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(29)

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where l and m are the wave numbers in x and y directions, respectively, and s = s1 + iω, s1 is the growth rate, and ω is the frequency of oscillations. Substituting these expressions into the differential equations (28), (23), and (24), we get 2  Da ∂ − Da(D 2 − α 2 ) + 1 (D 2 − α 2 )W Pr ∂t   Da ∂ ∂2W − Da∇ 2 + 1 − T a 2 , (30) = [Ra(−α 2 )Θ − Rn(−α 2 )Φ] Pr ∂t ∂z   ∂Θ N B ∂Θ ∂Φ 2N A N B ∂Θ γ − W = (D 2 − α 2 )Θ + − − (31) ∂t Le ∂z ∂z Le ∂z ∂Φ NA 2 NA 1 (32) + W = (D 2 − α 2 )Φ + (D − α 2 )Θ ∂t ε Le Le with ∂Θ ∂Φ W = 0, Θ = 0, + NA = 0 at z = 0 and at z = 1, (33) ∂z ∂z 1

where D = d/dz and α = (l 2 + m 2 ) 2 , is the dimensionless horizontal wave number. We use the Galerkin-type method and obtain an approximate solution of the system (30)–(33). For this, we write    W = A p Wp , Θ = Bp Θp , Φ = Cp Φp , (34) where Wp = Θp = sin pπ z; Φp = −N A sin pπ z p = 1, 2, 3, . . . ,

(35)

satisfying the boundary conditions (33). Substituting the expressions (34) in Eqs. (30)–(32), and seeking the orthogonality of the equations to each trial function, we obtain a system of 3N linear algebraic equations in the unknowns, namely Ap , Bp , Cp ; p = 1, 2, . . . , N . To obtain a non-trivial solution for the equations in the system, we need to equate to zero the determinants of the coefficients of the linear equations. Regarding Ra as the eigenvalue, its value is found in terms of the other parameters.

5 Onset of Instability 5.1 Stationary Convection In the case of neutral stability, we take s1 = 0 in our analysis. Further, for stationary convection to occur, we must have ω = 0. We take N = 1 and get the value of Ra for stationary case as   1 δ4 T aπ 2 δ 2 1 st 2 − Rn N A Le + (36) Ra = 2 (1 + Daδ ) + 2 α α (1 + Daδ 2 ) ε Le where δ 2 = α 2 + π 2 . In the absence of nanoparticles and their rotational component in the above equation, we get the critical Rayleigh number as Ra st =

δ4 (1 + Daδ 2 ) α2

(37)

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which is a well-known result for fluids. We know that for a typical nanofluid, the value of Le, N A and Rn are all positive. Thus, the coefficient of the third term on the right-hand side of Eq. (36) is negative. Therefore, we can say that the critical value of Ra decreases by a substantial amount with the presence of nanoparticle. 5.2 Oscillatory Convection Now, for the oscillatory convection to occur, we must have s1 = iω. We take N = 1 and get the value of Ra for oscillatory case as    4  δ2 2 Da Rn N A osc 2 2 δ (Le + ε) + ω2 Le2 γ Ra = 2 δ (1 + Daδ ) − γω − α Pr ε(δ 4 + ω2 Le2 )   Da T aπ 2 2 2 2  δ (1 + Daδ ) + ω γ (38) +  Da 2 Pr α 2 (1 + Daδ 2 )2 + ω2 ( ) Pr where the frequency of oscillations is given by −X 2 + X 22 − 4X 1 X 3 ω2 = 2X 1

(39)

where

   4   Da 2 LeDa 2 2 2 2 4 Da , X 2 = y1 Le 1 + Daδ +δ + y3 Le2 X1 = − y2 Pr Pr Pr  2  2 X 3 = y1 δ 4 1 + Daδ 2 − y2 1 + Daδ 2 + y3 δ 4    δ 2 Da 2  Rn N A  2 y1 = 2 y2 = δ + 1 + Daδ 2 γ δ Leγ − δ 2 Le (Le + ε) α Pr ε    Da T aπ 2 γ 1 + Daδ 2 − δ 2 y3 = α2 Pr 

For oscillatory convection to occur, ω2 > 0, which does not seem possible in this case, ruling out the possibility of occurrence of oscillatory convection.

6 Results and Discussion The nanoparticle flux being zero at the boundaries implies the absence of the two opposing forces responsible for the occurrence of overstability or the oscillatory mode of convection. Thus, we are left only with the analysis of the non-oscillatory mode or the damped phase of convection. In Eq. (36), when T a = 0, we obtain   1 δ4 1 Ra st = 2 (1 + Daδ 2 ) − Rn N A Le + (40) α ε Le Now, when Da = 0, the minimum is attained at α = π, with   1 1 Ra st = 4π 2 − Rn N A Le + ε Le

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(41)

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Fig. 1 Neutral stability curves for different values of concentration Rayleigh number Rn

Fig. 2 Neutral stability curves for different values of Lewis number Le

On the other √ hand , when Da is very large in comparison with unity, the minimum is attained at α = π/ 2, and the value becomes   1 27π 4 1 st Da − Rn N A Le + (42) Ra = 4 ε Le So, we can say that in the absence of nanoparticles, we get an already established result that the critical Rayleigh–Darcy number is equal to 4π 2 when Da = 0, and when Da is very large, it comes out to be 27π 4 /4. Thus, we see that on applying a single term Galerkin approximation, we get the same standard results without any overestimate. It is important to observe here that we get an upper bound of the value from the above two expressions. Also, it is worth observing here that the parameter N B disappears from the subsequent equations due to the orthogonality of the first-order trial functions and their first derivatives in the approximation theory employed, pointing to the zero contribution of the nanoparticle flux in the thermal energy conservation.

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Fig. 3 Neutral stability curves for different values of modified diffusivity ration N A

Fig. 4 Neutral stability curves for different values of Darcy number Da

Now for the “Principle of Exchange of Stabilities” to be valid, as Chandrashekhar (1961) and Drazin and Reid (1981) put it, stationary should be the mode of onset of convection which should transmit to the case of overstability. They explain it as even a slight disturbance in the system provokes restoring forces tending the system away from the equilibrium indicating the formation of prominent B´enard cells in the liquid layer. For the case in question, convection grows only aperiodically ruling out the possibility of overstability. This points that the restoring forces provoked prevent the system from reaching the point where equilibrium ceases and prominent B´enard cells form. In the Figs. 1, 2, 3, 4, 5, and 6, we draw the neutral stability curves depicting the variation in thermal Rayleigh number Ra with the wave number α for Rn = 4, Le = 10, N A = 1, Da = 0.1, T a = 100, ε = 0.7 with a variation in one of these parameters. By studying the Figs. 1, 2, and 3, it can easily be said that the concentration Rayleigh number Rn, Lewis number Le, and the Modified diffusivity ratio N A have a destabilizing effect on the system. An increase in the value of any of these parameters leads to the decrease in the value on

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Fig. 5 Neutral stability curves for different values of Taylor’s number T a

Fig. 6 Neutral stability curves for different values of porosity ε

thermal Rayleigh number Racr , thus indicating an advancement in the onset of convection. The effect of these graphically supports what can be concluded by looking at the expression of the thermal Rayleigh number in Eq. (36). The next figure, Fig. 4, has an interesting result to display. An increase in the value of the Darcy number Da not only enhances the onset of convection, but also has an effect on the critical wave number αcr . The value of αcr decreases with increasing Darcy number. The next two figures, Figs. 5 and 6, depict the influence of Taylor’s number, i.e., rotation and porosity ε, respectively. Both these parameters have a stabilizing effect on the conformity. Increase in their value delays the onset of convection just as expected.

7 Conclusions We studied the problem of convective instability for a nanofluid-saturated rotating porous layer complying with the Brinkman model, with the assumption that the nanoparticle flux

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is zero at the boundaries. The choice of this form of boundary conditions is more realistic than the previous ones. The expression for aperiodic instability has been obtained, while the possibility of overstability gets neglected due to the absence of two opposing forces that could lead to the latter. The expression for stationary convection indicates that the parameters Rn, Le, N A destabilize the system since none of them can be negative. The fact also gets supported from the figures obtained. The other parameters T a and ε stabilize the system. Acknowledgments The author SA greatly acknowledges the inputs provided by Prof. B. S. Bhadauria, Head Department of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow, India, for carrying out this work. The author is also thankful to the referees for their useful comments that helped in improving the manuscript.

References Agarwal, S., Bhadauria, B.S., Siddheshwar, P.G.: Thermal instability of a nanofluid saturating a rotating anisotropic porous medium. Spec. Top. Rev. Porous Media 2(1), 53–64 (2011) Agarwal, S., Bhadauria, B.S.: Natural convection in a nanofluid saturated rotating porous layer with thermal non equilibrium model. Transp. Porous Media 90, 627–654 (2011) Bhadauria, B.S., Agarwal, S., Kumar, A.: Non-linear two-dimensional convection in a nanofluid saturated porous medium. Transp. Porous Media 90(2), 605–625 (2011) Bhadauria, B.S., Agarwal, S.: Natural convection in a nanofluid saturated rotating porous layer: a nonlinear study. Transp. Porous Media 87(2), 585–602 (2011) Buongiorno, J.: Convective transport in nanofluids. ASME J. Heat Transf. 128, 240–250 (2006) Chand, R., Rana, G.C.: On the onset of thermal convection in rotating nanofluid layer saturating a Darcy– Brinkman porous medium. Int. J. Heat Mass Transf. 55, 5417–5424 (2012) Chandrashekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961) Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981) Eastman, J.A., Choi, S.U.S., Yu, W., Thompson, L.J.: Thermal transport in nanofluids. Annu. Rev. Mater. Res. 34, 219–246 (2004) Keblinski, P., Cahil, D.G.: Comments on model for heat conduction in nanofluids. Phy. Rev Lett. 95, 209401 (2005) Kuznetsov, A.V., Nield, D.A.: Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model. Transp. Porous Media 81, 409–422 (2010a) Kuznetsov, A.V., Nield, D.A.: Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid. Transp. Porous Media 83, 425–436 (2010b) Nield, D.A., Kuznetsov, A.V.: Thermal instability in a porous medium layer saturated by nanofluid. Int. J. Heat Mass Transf. 52, 5796–5801 (2009) Nield, D.A., Kuznetsov, A.V.: The effect of vertical throughflow on thermal instability in a porous medium layer saturated by a nanofluid. Transp. Porous Media 87, 765–775 (2011) Nield, D.A., Kuznetsov, A.V.: Thermal instability in a porous medium layer saturated by a nanofluid: a revised model. Int. J. Heat Mass Transf. 68, 211–214 (2014) Tzou, D.Y.: Instability of nanofluids in natural convection. ASME. J. Heat Transf. 130, 072401 (2008a) Tzou, D.Y.: Thermal instability of nanofluids in natural convection. Int. J. Heat Mass Transf. 51, 2967–2979 (2008b)

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