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A revised model for the effect of nanoparticle mass flux on the thermal instability of a nanofluid layer

  • Ozwah S. Alharbi and Abdullah A. Abdullah EMAIL logo
Published/Copyright: December 31, 2021
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 54 Issue 1

Abstract

A revised model of the nanoparticle mass flux is introduced and used to study the thermal instability of the Rayleigh-Benard problem for a horizontal layer of nanofluid heated from below. The motion of nanoparticles is characterized by the effects of thermophoresis and Brownian diffusion. The nanofluid layer is confined between two rigid boundaries. Both boundaries are assumed to be impenetrable to nanoparticles with their distribution being determined from a conservation condition. The material properties of the nanofluid are allowed to depend on the local volume fraction of nanoparticles and are modelled by non-constant constitutive expressions developed by Kanafer and Vafai based on experimental data. The results show that the profile of the nanoparticle volume fraction is of exponential type in the steady-state solution. The resulting equations of the problem constitute an eigenvalue problem which is solved using the Chebyshev tau method. The critical values of the thermal Rayleigh number are calculated for several values of the parameters of the problem. Moreover, the critical eigenvalues obtained were real-valued, which indicates that the mode of instability is via a stationary mode.

Keywords: linear stability; Rayleigh-Benard convection; nanofluid; thermophoresis; Brownian; Chebyshev tau method
MSC 2010: 76E06

1 Introduction

Recently, there has been widespread interest in studying nanofluids and their potential applications. A nanofluid is a mixture of a base fluid such as water or an organic solvent and nanoparticles such as copper, copper oxide, alumina, or multi-walled carbon nanotubes. Many researchers [1,2,3, 4,5] showed that the property of heat transfer in this type of fluid increases substantially compared to ordinary fluids. One of the important investigation of nanofluids is the studying of the thermal instability of this type of fluid. These types of investigations attracted the interests of many researchers. Kim et al. [6] analyzed the thermal instability driven by buoyancy and heat transfer characteristics of nanofluids. Buongiorno [7] proposed a model for convective transport in nanofluids in which the effects of Brownian diffusion and thermophoresis are taken into consideration and showed that only these mechanisms are the most significant drivers of nanoparticle motion. Tzou [8,9] used the model proposed by Buongiorno [7] to study the thermal instability of the Rayleigh-Benard problem for different values of temperatures and nanoparticle volume fractions and showed that for nanofluids, the critical values of the Rayleigh numbers are less by one or two orders of magnitude compared to the corresponding values of base fluids. Kuznetsov and Nield [10] studied analytically the onset of thermal instability in a horizontal layer of a porous medium saturated by a nanofluid using the model proposed by Buongiorno [7]. The same model is used by Nield and Kuznetsov [11] to study the thermal instability of a horizontal layer of nanofluid with a finite depth and concluded that in the case of a typical nanofluid, the main contribution of the nanoparticles is through the effect of buoyancy forces associated with the conservation of the nanoparticles.

The effect of rotation on the thermal instability of nanofluid layer is studied by Yadav et al. [12] analytically. They showed that rotation has a stabilizing effect depending on the values of the parameters of the problem. Yadav et al. [13] studied the effect of a vertical magnetic field on the thermal instability of a horizontal layer of nanofluid and showed that the magnetic field has a stabilizing effect. The joint effect of both rotation and magnetic field on the thermal instability of a nanofluid layer is investigated by Mahajan and Arora [14]. Nield and Kuznetsov [15] revised their paper [11] and changed the boundary conditions of the nanoparticle volume fraction to boundary conditions on the diffusive mass flux of nanoparticles which is physically more realistic. They showed that with the new boundary conditions, oscillatory convection cannot occur and the effect of nanoparticles on non-oscillatory convection is destabilizing. According to this study, most investigations focusing on the thermal instability of a horizontal nanofluid layer employed this revised model to analyze several problems (Agarwal et al. [16], Yadav et al. [17], Agarwal and Rana [18], Abdullah and Lindsay [19], Abdullah et al. [20], Rana et al. [21], Ahuja and Gupta [22]).

It is noted that all the studies [11,12,13, 14,15,16, 17,18,19, 20,21,22] assumed that Brownian and thermophoresis diffusions are constants in the expression of the diffusive mass flux of the nanoparticles and that the material properties of nanofluid such as effective thermal conductivity and effective viscosity behave as parameters across the nanofluid layer. As a result of these assumptions the gradient steady-state distributions of temperature and volume fraction arising in the analysis are constants. In the present study, a revised model is proposed assuming that Brownian and thermophoresis diffusions are functions of temperature and nanoparticle volume fraction in the expression of the diffusive mass flux of the nanoparticles. Moreover, the present study assumes that the material properties of nanofluid are allowed to depend on the local volume fraction of nanoparticles. These materials are modelled by non-constant constitutive expressions which were developed by Kanafer and Vafai [23] using experimental data.

The work is set out as follows. Section 2 formulates the problem and constructs the steady-state solution. Section 3 constructs the linearized non-dimensional equations. Section 4 formulates the normal mode analysis. Section 5 presents results and Section 6 concludes.

2 Problem formulation

Consider an infinite horizontal layer of a viscous incompressible nanofluid confined between the planes x 3 ∗ = 0 and x 3 ∗ = d . The boundaries in these planes are assumed to be rigid and kept at constant Kelvin temperatures T 0 and T c , respectively.

2.1 Field equations

The field equations for a layer of nanofluid as formulated by Buongiorno [7] and Tzou [8,9] have the form

(1) ∂ V j ∗ ∂ x j ∗ = 0 ,

(2) ρ eff ( ϕ ∗ ) ∂ V j ∗ ∂ t ∗ + V k ∗ ∂ V j ∗ ∂ x k ∗ = − ∂ P ∗ ∂ x j ∗ + ∂ ∂ x k ∗ μ eff ( ϕ ∗ ) ∂ V j ∗ ∂ x k ∗ + ∂ V k ∗ ∂ x j ∗ + [ ϕ ∗ ρ p + ρ 0 ( 1 − ϕ ∗ ) ( 1 − β ( T ∗ − T 0 ) ) ] g j ,

(3) ∂ ϕ ∗ ∂ t ∗ + V k ∗ ∂ ϕ ∗ ∂ x k ∗ = − 1 ρ p ∂ J k ∗ ∂ x k ∗ ,

(4) ( ρ c ) eff ( ϕ ∗ ) ∂ T ∗ ∂ t ∗ + V k ∗ ∂ T ∗ ∂ x k ∗ = − ∂ q k ∗ ∂ x k ∗ + h p ( T ∗ ) ∂ J k ∗ ∂ x k ∗ ,

where V j ∗ is the nanofluid velocity; P ∗ is the hydrostatic pressure; ϕ ∗ is the volume fraction of the nanoparticles; T ∗ is the Kelvin temperature of the nanofluid; ρ 0 is the base fluid density at reference temperature; g j is the gravitational acceleration; h p ( T ∗ ) is the specific enthalpy of nanofluid material; and ρ eff , μ eff , and c eff , are respectively, the effective density, effective viscosity, and effective specific heat at constant pressure. The quantities ρ eff and c eff are related to ( ρ f , c f ) and ( ρ p , c p ) , the density and specific heat of the base fluid and nanoparticle material, respectively, by the expressions

(5) ρ eff ( ϕ ∗ ) = ϕ ∗ ρ p + ( 1 − ϕ ∗ ) ρ f ,

(6) ( ρ c ) eff ( ϕ ∗ ) = ϕ ∗ ρ p c p + ( 1 − ϕ ∗ ) ρ f c f .

The heat flux vector q k ∗ in equation (4) has constitutive expression

(7) q k ∗ = − κ eff ( ϕ ∗ ) ∂ T ∗ ∂ x k ∗ + h p ( T ∗ ) J k ∗ ,

where κ eff ( ϕ ∗ ) denotes the effective thermal conductivity of the nanofluid. Buongiorno’s contribution to equation (3) resides in the specification of the nanoparticle mass flux J j ∗ with constitutive formulation

(8) J j ∗ = − ρ p D B ∂ ϕ ∗ ∂ x j ∗ + D T T ∗ ∂ T ∗ ∂ x j ∗ ,

where the first component describes the Brownian diffusion of nanoparticles and the second component describes the influence of thermophoresis. The parameters D B and D T represent, respectively, the Brownian and thermophoretic diffusion coefficients and have the parameters C B and C T constitutive specifications

(9) D B = k T ∗ 3 π μ f d p = C B T ∗ , D T = μ f ρ f 0.26 κ f 2 κ f + κ p ϕ ∗ = C T ϕ ∗ ,

where k is the Boltzmann constant, μ f is the viscosity of the base fluid, d p is the nanoparticle diameter, κ f and κ p are, respectively, the thermal conductivities of the base fluid and the nanoparticle material.

All previous research have assumed that local temperature and local nanoparticle volume fraction in D B and D T are constants. In the present study, we propose a revised model in which the local temperature and local nanoparticle volume fraction in D B and D T are not constants. Moreover, effective dynamic viscosity and effective thermal conductivity are assumed to be functions of nanoparticle volume fraction which are modelled by non-constant constitutive expressions developed by Brinkman [24], Hamilton and Crosser [25], and Kanafer and Vafai [23] based on experimental results. These non-constant parameters contribute extra terms to the linear stability analysis that are not present in previous works. The presence of these extra terms largely changes the stability boundary of the current problem.

Khanafer and Vafai [23] suggested the following models for the effective dynamic viscosity for distilled water (DW)/alumina and DW/cupric oxide nanofluids.

DW/alumina

(10) μ eff = a 0 + a 1 ϕ + a 2 ϕ 2 + a 3 ϕ 3 + a 4 T + a 5 ϕ 2 T 2 + a 6 ϕ T 3 + a 7 ϕ 2 d p 2 + a 8 ϕ 3 d p 2 .

DW/cupric oxide

(11) μ eff = b 0 + b 1 ϕ + b 2 ϕ 2 + b 3 ϕ 3 + b 4 T + b 5 T 2 + b 6 ϕ T + b 7 ϕ 2 T 2 + b 8 ϕ T 3 ,

where d p is the nanoparticle diameter and a 0 , … , a 8 , b 0 , … , b 8 are constants. In these expressions, the Celsius temperature T ∈ [ 20 , 70 ] and the nanoparticle volume fraction ϕ ∈ [ 0.01 , 0.09 ] . For the case of the DW/alumina nanofluid, d p ∈ [ 13 , 133 ] . Earlier Brinkman [24] suggested the following form for the effective dynamic viscosity:

(12) μ eff = μ f ( 1 − ϕ ) − 5 / 2 .

For the effective thermal conductivity, Khanafer and Vafai [23] suggested the following models for DW/alumina and DW/cupric oxide nanofluids

(13) κ eff κ f = 1 + ϕ 1.0112 + 114.5625 d p + 0.040457 κ p ,

where κ p is the thermal conductivity of the nanoparticle material. The efficacy of Khanafer and Vafai model of thermal conductivity will be compared by the following model suggested by Hamilton and Crosser [25]

(14) κ eff κ f = ( n − 1 ) ( 1 − ϕ ) κ f + ( n ϕ + 1 − ϕ ) κ p κ p ( 1 − ϕ ) + ( n − 1 + ϕ ) κ f ,

where n = 3 / χ denotes the “shape factor” of the nanoparticles and χ denotes particle sphericity. Typical values of n satisfy 3 < n < 6 with values near n = 3 being associated with nanoparticles of approximately spherical shape and those near n = 6 being associated with elongated (or rod-like) nanoparticles.

2.2 Boundary conditions

Mechanical, thermodynamic, and nanoparticle conditions must be imposed on the boundaries x 3 ∗ = 0 and x 3 ∗ = d . Here we assume that the temperature is fixed and the normal component of the nanoparticle flux is zero at both boundaries. These conditions are embodied in the equations

(15) V k ∗ = 0 , ∂ V 3 ∗ ∂ x k ∗ = 0 , T ∗ = T 0 , J 3 ∗ = 0 at x 3 ∗ = 0 , V k ∗ = 0 , ∂ V 3 ∗ ∂ x k ∗ = 0 , T ∗ = T c , q 3 ∗ = 0 , J 3 ∗ = 0 at x 3 ∗ = d ,

subject to the constraint that

(16) lim ∣ A ∣ → ∞ 1 ∣ A ∣ ∫ A 1 d ∫ 0 d ϕ ∗ d x 3 ∗ d x δ ∗ = ϕ ¯ ,

where δ = 1 , 2 and A is an arbitrary region of the ( x 1 ∗ , x 2 ∗ ) plane enclosing area ∣ A ∣ and ϕ ¯ is the average volume fraction of nanoparticles across the layer.

2.3 Steady-state solution

The set of governing equations (1)–(4) together with the boundary conditions (15) have a steady-state solution in which the fluid is at rest, the diffusive mass flux is zero, and all other variables are functions of x 3 ∗ alone. Let T ∗ = T ( x 3 ∗ ) , ϕ ∗ = Φ ( x 3 ∗ ) , P ∗ = P ( x 3 ∗ ) denote the solution of the steady-state problem, then it follows that T ( x 3 ∗ ) and Φ ( x 3 ∗ ) satisfy

(17) κ eff ( Φ ) d T d x 3 ∗ = 0 , J 3 = − ρ p C B T d Φ d x 3 ∗ + C T Φ T d T d x 3 ∗ = 0 .

Using equations (15), (17), and the constraint condition

(18) 1 d ∫ 0 d Φ ( x 3 ∗ ) d x 3 ∗ = ϕ ¯ ,

we can show that

(19) Φ = C 1 exp C T C B T ,

where C 1 is a constant. Applying the boundary conditions Φ = Φ 0 and T = T 0 at x 3 ∗ = 0 , we obtain

(20) Φ 0 = C 1 exp C T C B T 0 .

Substituting C 1 in (19), we obtain

(21) Φ = Φ 0 exp η ( T 0 − T ) T ,

where Φ 0 is the volume fraction of nanoparticles on the lower boundary x 3 ∗ = 0 and η = C T C B T 0 . Clearly in this problem the steady-state solution is well approximated by an exponential distribution of nanoparticle volume fraction and a linear distribution of temperature.

3 Non-dimensional linearized equations

Following standard procedures, equations (1)–(4) and their associated boundary conditions (15), (16) are linearized about the steady-state solution and then non-dimensionlized by introducing the following non-dimensional variables:

(22) x k = x k ∗ d , v k = v k ˆ d α , t = t ∗ α d 2 , p = p ˆ d 2 μ f α , ψ = ϕ ˆ ∣ Φ d − Φ 0 ∣ , θ = T ˆ ∣ T d − T 0 ∣ , J k = J k ˆ d ρ p C B T 0 ∣ Φ d − Φ 0 ∣ ,

where α = κ f / ρ f c f is the diffusivity, T d = T ( d ) , Φ d = Φ ( d ) . Now substitute (22) into equations (1)–(4) to obtain

(23) ∂ v k ∂ x k = 0 ,

(24) f 1 Pr ∂ v j ∂ t = − ∂ p ∂ x j + ∂ ∂ x k f 2 ∂ v j ∂ x k + ∂ v k ∂ x j − Rn ψ δ j 3 + ( 1 − Φ ) Ra θ δ j 3 ,

(25) Le ∂ ψ ∂ t + v 3 ∣ Φ d − Φ 0 ∣ d Φ d x 3 = − ∂ J k ∂ x k ,

(26) f 3 ∂ θ ∂ t + v 3 ∣ T d − T 0 ∣ d T d x 3 = ∂ ∂ x k f 4 ∂ θ ∂ x k + ∂ ∂ x k 1 ∣ T d − T 0 ∣ d T d x 3 f 5 ψ − 1 ∣ T d − T 0 ∣ d T d x 3 N B Le J k ,

where f 1 , … , f 5 are auxiliary functions of x 3 with expressions

(27) f 1 ( x 3 ) = ρ eff ( Φ ) ρ f , f 2 ( x 3 ) = μ eff ( Φ ) μ f , f 3 ( x 3 ) = ( ρ c ) eff ( Φ ) ( ρ c ) f , f 4 ( x 3 ) = κ eff ( Φ ) κ f , f 5 ( x 3 ) = ∣ Φ d − Φ 0 ∣ ∂ ∂ Φ κ eff κ f .

In equations (23)–(26) the Prandtl number Pr, the concentration Rayleigh number Rn , the thermal Rayleigh number Ra, the Lewis number Le, and the modified particle density increment N B have definitions

(28) Pr = ( μ c ) f κ f , Rn = ( ρ p − ρ 0 ) ∣ Φ d − Φ 0 ∣ g d 3 α μ f , Ra = ρ 0 β ∣ T − T 0 ∣ g d 3 α μ f , Le = α C B T 0 , N B = ( ρ c ) p ( ρ c ) f ∣ ϕ d − ϕ 0 ∣ .

To complete the non-dimensional process it remains to specify the linearized non-dimensional form of the nanoparticle mass flux J k given in (8). The details are omitted, but the final form is

(29) J k = − T T 0 ∂ ψ ∂ x k + T 0 T Φ ϕ ¯ N A ∂ θ ∂ x k + 2 N A η θ ϕ ¯ d Φ d x 3 + η T ψ d T d x 3 δ k 3 ,

where the modified diffusivity ratio, N A , is given by

(30) N A = η T d − T 0 Φ d − Φ 0 ϕ ¯ T 0 .

The linearized non-dimensional boundary conditions on the lower boundary have the form

(31) v 3 ( 0 ) = ∂ v 3 ( 0 ) ∂ x 3 = 0 ,

(32) θ ( 0 ) = 0 ,

(33) ∂ ψ ( 0 ) ∂ x k + Φ 0 ϕ ¯ N A ∂ θ ( 0 ) ∂ x k + η T 0 ψ ( 0 ) d T d x 3 = 0 ,

and on the upper boundary x 3 = 1 , the linearized non-dimensional boundary conditions have the form

(34) v 3 ( 1 ) = ∂ v 3 ( 1 ) ∂ x 3 = 0 ,

(35) θ ( 1 ) = 0 ,

(36) ∂ ψ ( 1 ) ∂ x k + Φ 0 ϕ ¯ N A ∂ θ ( 1 ) ∂ x k + η T d ψ ( 1 ) d T d x 3 = 0 .

As is customary in this type of problem we now apply the curl curl operator to the momentum equation (24), then take the third component of the resulting equation to obtain

(37) 1 Pr f 1 ∇ 2 ∂ v 3 ∂ t + d f 1 d x 3 ∂ ∂ x 3 ∂ v 3 ∂ t = f 2 ∇ 4 v 3 + 2 d f 2 d x 3 ∂ ∂ x 3 ( ∇ 2 v 3 ) + d 2 f 2 d x 3 2 ∂ 2 v 3 ∂ x 3 2 − ∇ H 2 v 3 + ∇ H 2 Rn ψ − ∇ H 2 ( 1 − Φ ) Ra θ .

4 Normal mode analysis

The stability of perturbations to the steady-state solution is investigated using a normal mode analysis. Solutions of equations (25), (26), and (37) satisfying the boundary conditions (31)–(36) are sought in the form

(38) ( v 3 , θ , ψ ) = ( w ( x 3 ) , θ ( x 3 ) , ψ ( x 3 ) ) e σ t + i n x 1 + i m x 2 ,

where n , m are the wave numbers along the e 1 and e 2 directions, respectively, and σ is a growth rate. Substitute (38) into equations (25), (26), and (37) to obtain the following eigenvalue problem in σ

(39) σ Pr f 1 d 2 w d x 3 2 − a 2 w + d f 1 d x 3 d w d x 3 = f 2 d 2 w d x 3 2 − a 2 w 2 + 2 d f 2 d x 3 d d x 3 d 2 w d x 3 2 − a 2 w + d 2 f 2 d x 3 2 d 2 w d x 3 2 + a 2 w + a 2 Rn ψ − a 2 ( 1 − Φ ) Ra θ ,

(40) Le σ ψ = − Le ∣ Φ d − Φ 0 ∣ d Φ d x 3 w + T T 0 d 2 ψ d x 3 2 − a 2 ψ + 1 T 0 + η T d T d x 3 d ψ d x 3 + 2 N A η θ ϕ ¯ d 2 Φ d x 3 2 + N A ϕ ¯ 2 η + T 0 T d Φ d x 3 d θ d x 3 + N A T 0 T Φ ϕ ¯ d 2 θ d x 3 2 − a 2 θ − N A T 0 T 2 Φ ϕ ¯ d T d x 3 d θ d x 3 + d d x 3 η T d T d x 3 ψ ,

(41) σ f 3 θ = − f 3 ∣ T d − T 0 ∣ d T d x 3 w + f 4 d 2 θ d x 3 2 − a 2 θ + d f 4 d x 3 d θ d x 3 + d d x 3 1 ∣ T d − T 0 ∣ d T d x 3 f 5 ψ + 1 ∣ T d − T 0 ∣ N B Le d T d x 3 T T 0 d ψ d x 3 + N A Φ ϕ ¯ T 0 T d θ d x 3 + 2 N A η d d x 3 Φ ϕ ¯ θ + η T d T d x 3 ψ .

The eigenvalue problem together with the boundary conditions are solved numerically using the Chebyshev spectral Tau method when the fluid layer is heated from below. This method has high accuracy and allows stationary and overstable modes to be treated simultaneously, which is important whenever the critical eigenvalue moves between stationary and overstable modes in response to changing the values of the parameters. We begin by using the transformation z = 2 x 3 − 1 to map the interval x 3 ∈ [ 0 , 1 ] into the interval z ∈ [ − 1 , 1 ] and then introduce the Chebyshev spectral expansions

(42) y r ( z ) = ∑ k = 0 N α k r T k ( z ) , 1 ≤ r ≤ 8 ,

where T k ( z ) are the Chebyshev polynomials of the first kind, α k r are its coefficients, N is a user-specified number of Chebyshev polynomials, and the variables y 1 , … , y 8 have the definitions

(43) y 1 = w , y 2 = D w , y 3 = D 2 w − a 2 w , y 4 = D ( D 2 w − a 2 w ) , y 5 = ψ , y 6 = D ψ , y 7 = θ , y 8 = D θ ,

where the operator D denotes differentiation with respect to z . Applying the definitions (43) to the governing equations (39)–(41), and boundary conditions (31)–(36) we obtain a system of equations which leads to the generalized eigenvalue problem A Y = σ B Y . The eigenvalues, σ , and the corresponding eigenvectors are calculated using a specialized routine. The complexity in this problem stems from the presence of the non-constant coefficients, f 1 , … , f 5 , multiplying the variables of the eigenvalue problem.

5 Results

To study the stability of this problem, the critical thermal Rayleigh numbers are obtained numerically for different values of the non-dimensional parameters. Figure 1 shows the relation between the average volume fraction of nanoparticles, ϕ ¯ , and the critical thermal Rayleigh number, Ra, for a DW/alumina nanofluid (solid line) and DW/cupric oxide nanofluid (dashed line) based on the model of Kanafer and Vafai. In this figure, we can see that the layer of DW/alumina (solid line) nanofluid is more stable than the layer of DW/cupric oxide nanofluid (dashed line) if ϕ ¯ greater than 0.4%. A similar relation is displayed in Figure 2 for the same nanofluid using the model of Brinkman and Hamilton-Crosser. In this figure, we can see that the layer of DW/cupric oxide nanofluid is more stable than the layer of DW/alumina nanofluid, for all values of the nanoparticle volume fraction.

Figure 1 The relation between the volume fraction of nanoparticles and the critical thermal Rayleigh number for a DW/alumina nanofluid and DW/cupric oxide nanofluid based on the model of Kanafer and Vafai.
Figure 1

The relation between the volume fraction of nanoparticles and the critical thermal Rayleigh number for a DW/alumina nanofluid and DW/cupric oxide nanofluid based on the model of Kanafer and Vafai.

Figure 2 The relation between the volume fraction of nanoparticles and the critical thermal Rayleigh number for a DW/alumina nanofluid and DW/cupric oxide nanofluid based on the model of Brinkman and Hamilton-Crosser.
Figure 2

The relation between the volume fraction of nanoparticles and the critical thermal Rayleigh number for a DW/alumina nanofluid and DW/cupric oxide nanofluid based on the model of Brinkman and Hamilton-Crosser.

Figures 3 and 4 compare the model of Khanafer and Vafai (solid line) and the model of Brinkman and Hamilton-Crosser (dashed line) for the DW/alumina nanofluid and the DW/cupric oxide nanofluid, respectively. Clearly, the nanofluid layer for Brinkman and Hamilton-Crosser model is more stable than the nanofluid layer for Khanafer and Vafai model.

Figure 3 The relation between the volume fraction of nanoparticles and the critical thermal Rayleigh number for a DW/alumina nanofluid based on the model of Khanafer and Vafai and the model of Brinkman and Hamilton-Crosser.
Figure 3

The relation between the volume fraction of nanoparticles and the critical thermal Rayleigh number for a DW/alumina nanofluid based on the model of Khanafer and Vafai and the model of Brinkman and Hamilton-Crosser.

Figure 4 The relation between the volume fraction of nanoparticles and the critical thermal Rayleigh number for DW/cupric oxide nanofluid based on the model of Khanafer and Vafai and the model of Brinkman and Hamilton-Crosser.
Figure 4

The relation between the volume fraction of nanoparticles and the critical thermal Rayleigh number for DW/cupric oxide nanofluid based on the model of Khanafer and Vafai and the model of Brinkman and Hamilton-Crosser.

The relation between the concentration Rayleigh number, Rn , and the critical thermal Rayleigh number, Ra, is displayed in Figure 5 for both nanofluids and both models ϕ ¯ = 0.2 % . The figure shows that as Rn increases, the critical thermal Rayleigh number decreases for both nanofluids and for both models, which indicates that the concentration Rayleigh number has a destabilizing effect. Moreover, it is clear that the layer of DW/alumina nanofluid is more stable than the layer of DW/cupric oxide nanofluid for Khanafer and Vafai model. By contrast the layer of DW/cupric oxide nanofluid is more stable than the layer of DW/alumina oxide nanofluid for Brinkman and Hamilton-Crosser. Also, it is clear that the Brinkman and Hamilton-Crosser model is more stable than the Khanafer and Vafai model for both nanofluids. The numerical results for this figure are listed in Table 1.

Figure 5 The relation between the concentration Rayleigh number and the critical thermal Rayleigh number for a DW/alumina nanofluid and a DW/cupric oxide nanofluid based on the model of Brinkman and Hamilton-Crosser and the model of Khanafer and Vafai.
Figure 5

The relation between the concentration Rayleigh number and the critical thermal Rayleigh number for a DW/alumina nanofluid and a DW/cupric oxide nanofluid based on the model of Brinkman and Hamilton-Crosser and the model of Khanafer and Vafai.

Table 1

Numerical results of the effect of the concentration Rayleigh number, Rn , on the critical thermal Rayleigh number, Ra

Present Nield and Kuznetsov [15]
Rn DW/alumina DW/cupric oxide Critical Ra
Critical Ra Critical Ra
KV BHC KV BHC
0.000 12.98769 15.93058 5.53184 16.07164 1707.1707
0.050 12.91351 15.84176 5.46652 15.99588 1629.1629
0.100 12.83913 15.75286 5.40119 15.92007 1549.1549
0.150 12.76478 15.66389 5.33578 15.84423 1468.1468
0.200 12.69042 15.57484 5.27036 15.76835 1386.1386
0.250 12.61593 15.48570 5.20487 15.69245 1302.1302
0.300 12.54144 15.39641 5.13938 15.61646 1216.1216
0.350 12.46688 15.30708 5.07375 15.54046 1128.1128
0.400 12.39228 15.21764 5.00813 15.46444 1038.1038
0.450 12.31759 15.12812 4.94246 15.38838 944.944
0.500 12.24287 15.03852 4.87674 15.31225 846.846
0.550 12.16811 14.94877 4.81095 15.23611 742.742
0.600 12.09326 14.85895 4.74519 15.15992 630.630
0.650 12.01841 14.76904 4.67929 15.08370 503.503
0.700 11.94345 14.67907 4.61337 15.00744 337.337

BHC: Brinkman and Hamilton-Crosser; KV: Khanafer and Vafai.

The numerical results of this model are compared with the numerical results of the same problem discussed by Nield and Kuznetsov [15]. Nield and Kuznetsov [15] discussed the same problem assuming that Brownian and thermophoresis diffusions are constants in the expression of the diffusive mass flux of the nanoparticles and that the material properties of nanofluid such as effective thermal conductivity and effective viscosity behave as parameters across the nanofluid layer. In their work, they did not produce any numerical results, so we had to solve their problem numerically and compare the results with the results obtained in our revised model. These results are listed in Table 1 where we compare the critical Rayleigh numbers obtained from Nield and Kuznetsov’s problem with the results obtained in this problem for the two types of nanofluids, DW/alumina oxide and DW/cupric oxide, and for the two models, Brinkman and Hamilton-Crosser and Khanafer and Vafai. The results indicate that using the revised model, which is more realistic, produce critical Rayleigh numbers which are less by one to two order of magnitude compared to the results obtained from the work of Nield and Kuznetsov [15]. In other words, our revised model is less stable. Finally, the numerical results obtained show that all critical eigenvalues were real valued, indicating that the mechanism of instability is by a non-oscillatory mode.

6 Conclusion

This study has proposed a revised model for the nanoparticle mass flux and used it to study the thermal instability of the Rayleigh-Benard convection for layers of DW/cupric oxide and DW/alumina nanofluids heated from below. The material properties of the nanofluids are allowed to depend on the local volume fraction of nanoparticles and are modelled by non-constant constitutive expressions developed by Kanafer and Vafai and Brinkman and Hamilton-Crosser. The analysis shows that the profile of the nanoparticle volume fraction of the steady-state solution is of exponential type. Stability results obtained show that

  1. Based on the model of Khanafer and Vafai, the layer of DW/alumina nanofluid is more stable than the layer of DW/cupric oxide nanofluid if ϕ ¯ is greater than 0.4%.

  2. Based on the model of Brinkman and Hamilton-Crosser, the layer of DW/cupric oxide nanofluid is more stable than the layer of DW/alumina nanofluid for all values of the nanoparticle volume fraction.

  3. In general, the Brinkman and Hamilton-Crosser model is more stable than the Khanafer and Vafai model for both nanofluids.

  4. The critical thermal Rayleigh number, Ra, decreases as the concentration Rayleigh number, Rn , increases, which indicates that Rn has a destabilizing effect for both nanofluids and for both models of constitutive properties.

  5. The critical Rayleigh numbers obtained for both nanofluids and for both models are compared with the results of Nield and Kuznetsov [15]. The results indicate that using the revised model, which is more realistic, produces critical Rayleigh numbers which are less by one to two order of magnitude compared to the results obtained from the work of Nield and Kuznetsov [15]. In other words, our revised model is less stable.

  6. All the critical eigenvalues were real valued, indicating that the mechanism of instability is via a non-oscillatory mode.

  1. Funding information: Authors state no funding involved.

  2. Author contributions: Authors contributed equally to this work and approved its submission.

  3. Conflict of interest: Authors state no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2021-07-20
Revised: 2021-10-30
Accepted: 2021-11-16
Published Online: 2021-12-31

© 2021 Ozwah S. Alharbi and Abdullah A. Abdullah, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2021-0045
Keywords for this article
linear stability; Rayleigh-Benard convection; nanofluid; thermophoresis; Brownian; Chebyshev tau method
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Articles in the same Issue

  1. Regular Articles
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