# Effect of Variable Viscosity on Vortex Instability of Non-Darcy Mixed Convection Boundary Layer Flow Adjacent to a Nonisothermal Horizontal Surface in a Porous Medium

Authors: A. M. Elaiw; A. A. Bakr; M. A. Alghamdi; F. S. Ibrahim
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

We study the effect of variable viscosity on the flow and vortex instability for non-Darcy mixed convection boundary layer flow on a nonisothermal horizontal plat surface in a saturated porous medium. The variation of viscosity is expressed as an exponential function of temperature. The analysis of the disturbance flow is based on linear stability theory. The base flow equations and the resulting eigenvalue problem are solved using finite difference schemes. It is found that the variable viscosity effect enhances the heat transfer rate and destabilizes the flow for liquid heating, while the opposite trend is true for gas heating.


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## Body

## 1. Introduction

The study of convective heat transfer from surfaces embedded in porous media has received considerable attention in the literature. The interest for such studies is motivated by several thermal engineering applications, such as storage of radioactive nuclear waste materials transfer, separation processes in chemical industries, filtration, transpiration cooling, transport processes in aquifers, ground water pollution, and thermal insulation. The presence of a buoyancy force component in the direction normal to the surface leads to vortex instability of the flow. The problem of the vortex mode of instability in mixed or natural convection flow over a horizontal or an inclined heated plate in a saturated porous medium has recently received considerable attention (see [1–10]). These studies used either Darcy model [1–6], or non-Darcy model [7–10]. A comprehensive literature survey on this subject can be found in the recent book by Nield and Bejan [11].All the above studies dealt with constant viscosity. The fundamental analysis of convection through porous media with temperature-dependent viscosity is driven by several contemporary engineering applications from cooling of electronic devices to porous journal bearings and is important for studying the variations in constitutive property. The effect of variable viscosity for convective heat transfer through porous media is studied by several investigators [12–20]. The effect of variation of viscosity to study the instability of flow and temperature fields is discussed by Kassoy and Zebib [12] and Gray et al. [13]. Ling and Dybbs [14] presented a very interesting theoretical investigation of temperature-dependent fluid viscosity influence on the forced convection flow through a semi-infinite porous medium bounded by an isothermal flat plate. Lai and Kulacki [15] considered the variable viscosity effect for mixed convection flow along a vertical plate embedded in saturated porous medium. The effect of variable viscosity on combined heat and mass transfer in mixed convection about a wedge embedded in a saturated porous media for the case of uniform heat mass flux (UHF/UMF) is analyzed by Hassanien et al. [17]. The effect of variable viscosity on non-Darcy, free or mixed convection flow on a horizontal surface in a saturated porous medium, is studied by Kumari [16]. The effect of the temperature-dependent viscosity on mixed convection boundary layer assisting and opposing flows over a vertical surface embedded in a porous medium is investigated by Chin et al. [18]. Seddeek [19] studied the effects of magnetic field and variable viscosity on forced non-Darcy flow about a flat plate with variable wall temperature in porous media in the presence of suction and blowing. In [17], the linear variation of fluid viscosity is assumed while, in [15] and [16], the viscosity of the fluid is assumed to vary as an inverse linear function of temperature. In [20], the variation of viscosity with temperature is represented by an exponential function.The effect of variable viscosity on vortex instability of a horizontal free convection boundary layer flow in a saturated porous medium for an isothermal surface was studied by Jang and Leu [20]. However, the variable viscosity behavior on the flow and vortex instability of non-Darcy mixed convection boundary layer flow over a nonisothermal horizontal plate dose not seem to have been investigated. This motivated the present investigation.The present study examines in details the effects of temperature-dependent viscosity on the flow and vortex instability of non-Darcy mixed convection boundary layer flow adjacent to a heated horizontal surface embedded in a porous medium with variable wall temperature. The variation of viscosity with temperature is represented by an exponential function, which is more accurate than a linear function for large temperature differences. The transformed boundary layer equations, which are given by using nonsimilar solution approach, are solved by means of a finite difference method. The analysis of the disturbance flow is based on linear stability theory. The disturbance quantities are assumed to be in the form of a stationary vortex roll that is periodic in the spanwise direction, with its amplitude function depending primarily on the normal coordinate and weakly on the streamwise coordinate. The resulting eigenvalue problems are solved by using a finite difference scheme.


## 2. Analysis

### 2.1. The Main Flow

We considered a semi-infinite nonisothermal horizontal surface (Tw) embedded in a porous medium (T∞), where x represents the distance along the plate from its leading edge, and y the distance normal to the surface. The wall temperature is assumed to be a power function of x, that is, Tw=T∞+Axλ, where A is a constant, and λ is the parameter representing the variation of the wall temperature. In order to study transport through high porosity media, the original Darcy model is improved by including inertia. For the mathematical analysis of the problem, we assume that (i) local thermal equilibrium exists between the fluid and the solid phase, (ii) the physical properties are considered to be constant, except for the viscosity μ and the density ρ in the buoyancy force, (iii) we consider the non-Darcy model given by Ergun [21], and (iv) the Boussinesq approximation is valid. With these assumptions, the governing equations are given by(2.1)∂u∂x+∂v∂y=0,(2.2)u+K*νu2=-Kμ∂P∂x,(2.3)v+K*νv2=-Kμ(∂P∂y+ρg),(2.4)u∂T∂x+v∂T∂y=α∂2T∂y2,
where ρ=ρ∞[1-β(T-T∞)] is the fluid density, u and v are the velocities in the x and y directions, respectively, P is the pressure, T is the temperature, μ is the dynamic viscosity, K is the permeability of the porous medium, g is the gravitational acceleration, β is the thermal expansion coefficient of the fluid, and α represents the equivalent thermal diffusivity. Note that the second term on the left-hand side of (2.2) and (2.3) represents the inertia force, where K* is the inertia coefficient in Ergun model. As K*=0, (2.2) and (2.3) reduce to Darcy model.The viscosityμ of the fluid is assumed to vary with temperature according to an exponential function(2.5)μ=μ∞eA1((T-T∞)/(Tw-T∞)),
where μ∞ is the absolute viscosity at ambient temperature, and A1 is constant adopted from the least square fitting for a particular fluid. Formula (2.5) is a generalization of that used in [20], where the wall temperature is taken to be constant.The pressure terms appearing in (2.2) and (2.3) can be eliminated through cross-differentiation. The boundary layer assumption yields ∂/∂x≪∂/∂y and v≪u. With ψ being a stream function such that u=∂ψ/∂y and v=-∂ψ/∂x, the equations (2.1)–(2.4) become(2.6)(μ+2ρ∞K*∂ψ∂y)∂2ψ∂y2+u∂μ∂y=-Kρ∞gβ∂T∂x,(2.7)∂ψ∂y∂T∂x-∂ψ∂x∂T∂y=α∂2T∂y2.
The boundary conditions are defined as follows:(2.8)v(x,0)=-∂ψ∂x=0,T(x,0)=Tw=T∞+Axλ,u(x,∞)=BxmT(x,∞)=T∞.
Here m=0 for assisting flow over a horizontal flat plate at zero incident and m=1 for stagnation point flow about a horizontal surface.On applying the following transformations:(2.9)η(x,y)=yxPex1/2,f(ξ,η)=ψ(x,y)αPex1/2,θ(ξ,η)=T-T∞Tw-T∞,ξ=(xd)(2λ-3m-1)/2,
into (2.5)–(2.7) lead to the following:(2.10)μμ∞=eA1((T-T∞)/(Tw-T∞))=(μ*)θ,(2.11)(1+2ErPedξ2m/(2λ-3m-1)(μ*)-θf′)f′′=-(ln⁡μ*)f′θ′-Mξ(μ*)-θ(λθ+m-12ηθ′+2λ-3m-12ξ∂θ∂ξ),(2.12)θ′′=(λθ+2λ-3m-12ξ∂θ∂ξ)f′-(m+12f+2λ-3m-12ξ∂f∂ξ)θ′,
where μ*=μw/μ∞=eA1 is the wall to ambient viscosity ratio parameter, Pex=u∞x/α is the local Peclet number, M=Rad/Ped3/2 is the mixed convection parameter, Ped=u∞d/α is the Peclet number based on the pore diameter, and Er=K*α/dν∞ is the Ergun number, in which the dynamic viscosity is evaluated at T∞. The transformed boundary conditions are(2.13)f(ξ,0)=0,θ(ξ,0)=1,f′(ξ,∞)=1,θ(ξ,∞)=0.
The physical quantities of major interest are the velocity components u and v, and the local nusselt number(2.14)u(x,y)=u∞f′(ξ,η),v(x,y)=-αPex1/22x((m+1)f(ξ,η)+(m-1)ηf′+(2λ-3m-1)ξ∂f(ξ,η)∂ξ),NuxPex1/2=-θ′(ξ,0).


### 2.2. The Disturbance Flow

In the usual manner for linear stability analysis, the velocities, pressure, temperature, and viscosity are assumed to be the sum of mean and fluctuating components, here denoted as subscripts 0 and 1 quantities, respectively,(2.15)u(x,y,z,t)=u0(x,y)+u1(x,y,z,t),v(x,y,z,t)=v0(x,y)+v1(x,y,z,t),w(x,y,z,t)=w1(x,y,z,t),P(x,y,z,t)=P0(x,y)+P1(x,y,z,t),T(x,y,z,t)=T0(x,y)+T1(x,y,z,t),μ(T)=μ0(T0)+μ1(T1).
After substituting (2.15) into the governing equations for the three dimensional convective flow in a porous medium, the base flow quantities are subtracted, with the terms higher than first order in disturbance quantities being neglected. Then we get the following disturbance equations:(2.16)∂u1∂x+∂v1∂y+∂w1∂z=0,(2.17)μ0u1+μ1u0+2K*ρ∞u0u1=-K∂P1∂x,(2.18)μ0v1+μ1v0+2K*ρ∞v0v1=-K(∂P1∂y-ρ∞gβT1),(2.19)μ0w1=-K∂P1∂z,(2.20)u0∂T1∂x+v0∂T1∂y+u1∂T0∂x+v1∂T0∂y=α[∂2T1∂x2+∂2T1∂y2+∂2T1∂z2].
Following the method of order of magnitude analysis described in detail by Hsu and Cheng [1], the terms ∂u1/∂x and ∂2T1/∂x2 in (2.16) and (2.20) can be neglected. The omission of ∂u1/∂x in (2.16) implies the existence of a disturbance stream function Ψ1, such as(2.21)w1=∂Ψ1∂y,v1=-∂Ψ1∂z.
Eliminating P1 from (2.17)–(2.19) with the aid of (2.21) leads to(2.22)u0∂μ1∂z+(μ0+2K*ρ∞u0)∂u1∂z=μ0∂2Ψ1∂x∂y+∂Ψ1∂y∂μ0∂x,-(μ0+2K*ρ∞v0)∂2Ψ1∂z2+v0∂μ1∂z=μ0∂2Ψ1∂y2+∂μ0∂y∂Ψ1∂y+Kρ∞gβ∂T1∂z,u0∂T1∂x+v0∂T1∂y+u1∂T0∂x-∂Ψ1∂z∂T0∂y=α(∂2T1∂y2+∂2T1∂z2).
As in Hsu and Cheng [1], we assume that the three dimensionless disturbances for neutral stability are of the form(2.23)(Ψ1,u1,T1)=[Ψ¯(x,y),u¯(x,y),T¯(x,y)]eiaz,
where a is the spanwise periodic wave number. Substituting (2.23) into (2.22) yields(2.24)u¯=(11+(2K*/ν∞)(μ*)-θu0)[1ia(∂2Ψ¯∂x∂y+(ln⁡μ*)∂Ψ¯∂y∂θ∂x)-(ln⁡μ*)u0T¯Tw-T∞],∂2Ψ¯∂y2-a2(1+2K*ν∞(μ*)-θv0)Ψ¯-iav0(ln⁡μ*)T¯Tw-T∞+(ln⁡μ*)∂θ∂y∂Ψ¯∂y=-iaKρ∞gβμ∞(μ*)-θT¯,u0∂T¯∂x+v0∂T¯∂y+u¯∂T0∂x-iaΨ¯∂T0∂y=α(∂2T¯∂y2-a2T¯).
Equation (2.24) is solved based on the local similarity approximation (see [22]), wherein the disturbances are assumed to have weak dependence in the streamwise direction (i.e., ∂/∂x≪∂/∂η).To facilitate the analysis, the following transformations are introduced to nondimensionalize the preceding equations(2.25)k=axPex1/2,F(η)=Ψ¯iαPex1/2,Θ(η)=T¯Tw-T∞.
Introducing (2.25) into (2.24) gives(2.26)F′′+(ln⁡μ*)θ′F′-k2[1-2ErPedξ2m/(2λ-3m-1)(μ*)-θPex1/2H1]F+k(ln⁡μ*)Pex1/2H1Θ=-MkPex1/2ξ(μ*)-θΘ,(2.27)Θ′′+H3Θ′-[k2+λf′-(ln⁡μ*)f′H21+2ErPedξ2m/(2λ-3m-1)(μ*)-θf′]Θ-H2[((m-1)/2)ηF′′+(m+ln⁡μ*H4)F′]kPex1/2(1+2ErPedξ2m/(2λ-3m-1)(μ*)-θf′)=kPex1/2θ′F,
subject to the boundary conditions(2.28)F(0)=F(∞)=Θ(0)=Θ(∞)=0,
where the coefficients H1-H4 are given by(2.29)H1=m+12f+m-12ηf′+2λ-3m-12ξ∂f∂ξ,H2=λθ+m-12ηθ′+2λ-3m-12ξ∂θ∂ξ,H3=m+12f+2λ-3m-12ξ∂f∂ξ,H4=m-12ηθ′+2λ-3m-12ξ∂θ∂ξ,
Equations (2.26) and (2.27) constitute a second-order system of linear ordinary differential equations for the disturbance amplitude distributions F(η) and Θ(η). For fixed values of ξ,λ,m,M,Er,Ped, k, and μ*, the solution F and Θ is an eigenfunction for the eigenvalue Pex. We note that (2.26) and (2.27) under boundary conditions (2.28) for μ*=1, Er=0, ξ=1, and ∂/∂ξ=0 are reduced to those given in Hsu and Cheng [1], where the Darcy model with constant viscosity is considered.


## 2.1. The Main Flow

We considered a semi-infinite nonisothermal horizontal surface (Tw) embedded in a porous medium (T∞), where x represents the distance along the plate from its leading edge, and y the distance normal to the surface. The wall temperature is assumed to be a power function of x, that is, Tw=T∞+Axλ, where A is a constant, and λ is the parameter representing the variation of the wall temperature. In order to study transport through high porosity media, the original Darcy model is improved by including inertia. For the mathematical analysis of the problem, we assume that (i) local thermal equilibrium exists between the fluid and the solid phase, (ii) the physical properties are considered to be constant, except for the viscosity μ and the density ρ in the buoyancy force, (iii) we consider the non-Darcy model given by Ergun [21], and (iv) the Boussinesq approximation is valid. With these assumptions, the governing equations are given by(2.1)∂u∂x+∂v∂y=0,(2.2)u+K*νu2=-Kμ∂P∂x,(2.3)v+K*νv2=-Kμ(∂P∂y+ρg),(2.4)u∂T∂x+v∂T∂y=α∂2T∂y2,
where ρ=ρ∞[1-β(T-T∞)] is the fluid density, u and v are the velocities in the x and y directions, respectively, P is the pressure, T is the temperature, μ is the dynamic viscosity, K is the permeability of the porous medium, g is the gravitational acceleration, β is the thermal expansion coefficient of the fluid, and α represents the equivalent thermal diffusivity. Note that the second term on the left-hand side of (2.2) and (2.3) represents the inertia force, where K* is the inertia coefficient in Ergun model. As K*=0, (2.2) and (2.3) reduce to Darcy model.The viscosityμ of the fluid is assumed to vary with temperature according to an exponential function(2.5)μ=μ∞eA1((T-T∞)/(Tw-T∞)),
where μ∞ is the absolute viscosity at ambient temperature, and A1 is constant adopted from the least square fitting for a particular fluid. Formula (2.5) is a generalization of that used in [20], where the wall temperature is taken to be constant.The pressure terms appearing in (2.2) and (2.3) can be eliminated through cross-differentiation. The boundary layer assumption yields ∂/∂x≪∂/∂y and v≪u. With ψ being a stream function such that u=∂ψ/∂y and v=-∂ψ/∂x, the equations (2.1)–(2.4) become(2.6)(μ+2ρ∞K*∂ψ∂y)∂2ψ∂y2+u∂μ∂y=-Kρ∞gβ∂T∂x,(2.7)∂ψ∂y∂T∂x-∂ψ∂x∂T∂y=α∂2T∂y2.
The boundary conditions are defined as follows:(2.8)v(x,0)=-∂ψ∂x=0,T(x,0)=Tw=T∞+Axλ,u(x,∞)=BxmT(x,∞)=T∞.
Here m=0 for assisting flow over a horizontal flat plate at zero incident and m=1 for stagnation point flow about a horizontal surface.On applying the following transformations:(2.9)η(x,y)=yxPex1/2,f(ξ,η)=ψ(x,y)αPex1/2,θ(ξ,η)=T-T∞Tw-T∞,ξ=(xd)(2λ-3m-1)/2,
into (2.5)–(2.7) lead to the following:(2.10)μμ∞=eA1((T-T∞)/(Tw-T∞))=(μ*)θ,(2.11)(1+2ErPedξ2m/(2λ-3m-1)(μ*)-θf′)f′′=-(ln⁡μ*)f′θ′-Mξ(μ*)-θ(λθ+m-12ηθ′+2λ-3m-12ξ∂θ∂ξ),(2.12)θ′′=(λθ+2λ-3m-12ξ∂θ∂ξ)f′-(m+12f+2λ-3m-12ξ∂f∂ξ)θ′,
where μ*=μw/μ∞=eA1 is the wall to ambient viscosity ratio parameter, Pex=u∞x/α is the local Peclet number, M=Rad/Ped3/2 is the mixed convection parameter, Ped=u∞d/α is the Peclet number based on the pore diameter, and Er=K*α/dν∞ is the Ergun number, in which the dynamic viscosity is evaluated at T∞. The transformed boundary conditions are(2.13)f(ξ,0)=0,θ(ξ,0)=1,f′(ξ,∞)=1,θ(ξ,∞)=0.
The physical quantities of major interest are the velocity components u and v, and the local nusselt number(2.14)u(x,y)=u∞f′(ξ,η),v(x,y)=-αPex1/22x((m+1)f(ξ,η)+(m-1)ηf′+(2λ-3m-1)ξ∂f(ξ,η)∂ξ),NuxPex1/2=-θ′(ξ,0).


## 2.2. The Disturbance Flow

In the usual manner for linear stability analysis, the velocities, pressure, temperature, and viscosity are assumed to be the sum of mean and fluctuating components, here denoted as subscripts 0 and 1 quantities, respectively,(2.15)u(x,y,z,t)=u0(x,y)+u1(x,y,z,t),v(x,y,z,t)=v0(x,y)+v1(x,y,z,t),w(x,y,z,t)=w1(x,y,z,t),P(x,y,z,t)=P0(x,y)+P1(x,y,z,t),T(x,y,z,t)=T0(x,y)+T1(x,y,z,t),μ(T)=μ0(T0)+μ1(T1).
After substituting (2.15) into the governing equations for the three dimensional convective flow in a porous medium, the base flow quantities are subtracted, with the terms higher than first order in disturbance quantities being neglected. Then we get the following disturbance equations:(2.16)∂u1∂x+∂v1∂y+∂w1∂z=0,(2.17)μ0u1+μ1u0+2K*ρ∞u0u1=-K∂P1∂x,(2.18)μ0v1+μ1v0+2K*ρ∞v0v1=-K(∂P1∂y-ρ∞gβT1),(2.19)μ0w1=-K∂P1∂z,(2.20)u0∂T1∂x+v0∂T1∂y+u1∂T0∂x+v1∂T0∂y=α[∂2T1∂x2+∂2T1∂y2+∂2T1∂z2].
Following the method of order of magnitude analysis described in detail by Hsu and Cheng [1], the terms ∂u1/∂x and ∂2T1/∂x2 in (2.16) and (2.20) can be neglected. The omission of ∂u1/∂x in (2.16) implies the existence of a disturbance stream function Ψ1, such as(2.21)w1=∂Ψ1∂y,v1=-∂Ψ1∂z.
Eliminating P1 from (2.17)–(2.19) with the aid of (2.21) leads to(2.22)u0∂μ1∂z+(μ0+2K*ρ∞u0)∂u1∂z=μ0∂2Ψ1∂x∂y+∂Ψ1∂y∂μ0∂x,-(μ0+2K*ρ∞v0)∂2Ψ1∂z2+v0∂μ1∂z=μ0∂2Ψ1∂y2+∂μ0∂y∂Ψ1∂y+Kρ∞gβ∂T1∂z,u0∂T1∂x+v0∂T1∂y+u1∂T0∂x-∂Ψ1∂z∂T0∂y=α(∂2T1∂y2+∂2T1∂z2).
As in Hsu and Cheng [1], we assume that the three dimensionless disturbances for neutral stability are of the form(2.23)(Ψ1,u1,T1)=[Ψ¯(x,y),u¯(x,y),T¯(x,y)]eiaz,
where a is the spanwise periodic wave number. Substituting (2.23) into (2.22) yields(2.24)u¯=(11+(2K*/ν∞)(μ*)-θu0)[1ia(∂2Ψ¯∂x∂y+(ln⁡μ*)∂Ψ¯∂y∂θ∂x)-(ln⁡μ*)u0T¯Tw-T∞],∂2Ψ¯∂y2-a2(1+2K*ν∞(μ*)-θv0)Ψ¯-iav0(ln⁡μ*)T¯Tw-T∞+(ln⁡μ*)∂θ∂y∂Ψ¯∂y=-iaKρ∞gβμ∞(μ*)-θT¯,u0∂T¯∂x+v0∂T¯∂y+u¯∂T0∂x-iaΨ¯∂T0∂y=α(∂2T¯∂y2-a2T¯).
Equation (2.24) is solved based on the local similarity approximation (see [22]), wherein the disturbances are assumed to have weak dependence in the streamwise direction (i.e., ∂/∂x≪∂/∂η).To facilitate the analysis, the following transformations are introduced to nondimensionalize the preceding equations(2.25)k=axPex1/2,F(η)=Ψ¯iαPex1/2,Θ(η)=T¯Tw-T∞.
Introducing (2.25) into (2.24) gives(2.26)F′′+(ln⁡μ*)θ′F′-k2[1-2ErPedξ2m/(2λ-3m-1)(μ*)-θPex1/2H1]F+k(ln⁡μ*)Pex1/2H1Θ=-MkPex1/2ξ(μ*)-θΘ,(2.27)Θ′′+H3Θ′-[k2+λf′-(ln⁡μ*)f′H21+2ErPedξ2m/(2λ-3m-1)(μ*)-θf′]Θ-H2[((m-1)/2)ηF′′+(m+ln⁡μ*H4)F′]kPex1/2(1+2ErPedξ2m/(2λ-3m-1)(μ*)-θf′)=kPex1/2θ′F,
subject to the boundary conditions(2.28)F(0)=F(∞)=Θ(0)=Θ(∞)=0,
where the coefficients H1-H4 are given by(2.29)H1=m+12f+m-12ηf′+2λ-3m-12ξ∂f∂ξ,H2=λθ+m-12ηθ′+2λ-3m-12ξ∂θ∂ξ,H3=m+12f+2λ-3m-12ξ∂f∂ξ,H4=m-12ηθ′+2λ-3m-12ξ∂θ∂ξ,
Equations (2.26) and (2.27) constitute a second-order system of linear ordinary differential equations for the disturbance amplitude distributions F(η) and Θ(η). For fixed values of ξ,λ,m,M,Er,Ped, k, and μ*, the solution F and Θ is an eigenfunction for the eigenvalue Pex. We note that (2.26) and (2.27) under boundary conditions (2.28) for μ*=1, Er=0, ξ=1, and ∂/∂ξ=0 are reduced to those given in Hsu and Cheng [1], where the Darcy model with constant viscosity is considered.


## 3. Numerical Scheme

In this section, we compute the approximate value ofPex for (2.26) and (2.27) with the boundary conditions (2.28). An implicit finite difference method is used to solve first the base flow (2.11) and (2.12) with the boundary conditions (2.13), and the results are stored for a fixed step size h, which is small enough to predict accurate linear interpolation between mesh point. The domain is 0≤η≤η∞, where η∞ is the edge of the boundary layer of the basic flow. For a positive integer N, let h=η∞/N and ηi=ih, i=0,1,…,N, the problem is discretized with standard centered finite differences of order 2, following Usmani [23]. Solving eigenvalue problem is achieved by using the subroutine GVLRG of the IMSL library Inc., see [24].


## 4. Results and Discussion

Numerical results for the local Nusselt number, neutral stability curves, the critical Peclet, and associate wave numbers at the onset of vortex instability are presented for a range of wall to ambient viscosity ratio parameterμ* from 0.1 to 10. As the temperature is increased, the gas viscosity increases, while the liquid viscosity decreases [20]. Therefore, for a heated wall, values of μ*>1 correspond to the case of gas heating, and values of μ*<1 corresponds to the case of liquid heating. The effect of the nonuniform temperature profile on the wall is also studied, and corresponds to variations of the parameter λ. Because of lack of space, we shall only outline the numerical results of the case m=0 (assisting flow over a horizontal flat plate at zero incident) and results of the case m=1 (stagnation point flow about a horizontal surface) can be omitted.The local Nusselt number as functions ofM and ξ for various values of μ*, n, and Er are shown, respectively, in Figures 1 and 2. It is seen that higher Nusselt number occurs at higher values of M, n, ξ, and lower values of Er, μ*.Figure 1
The local Nusselt number as a function ofM for selected values of μ* and n for the case m=0.Figure 2
The local Nusselt number as a function ofξ for various values of Er for the case m=0.Figures3, 4, and 5, respectively, show the neutral stability curves, in terms of the Peclet number Pex and the dimensionless wave number k for selected values of μ*, M, and ξ. It is observed that, as μ* increases, the neutral stability curves shift to higher Peclet number, while as ξ and M increase, the neutral stability curves shift to lower Peclet number Pex.Figure 3
Neutral stability curves for selected values ofμ* for the case m=0.Figure 4
Neutral stability curves for selected values ofM for the case m=0.Figure 5
Neutral stability curves for selected values ofξ for the case m=0.The critical Peclet number and wave number are plotted as a function ofM for selected values of μ* and Ped in Figures 6 and 7. It can be seen that as the variable viscosity parameter μ* increases, the critical Peclet number Pex* increases, while as Pedor M increases, the critical Peclet number Pex* decreases. Further, higher critical wave number occurs at higher values of M, and lower values of μ* or Ped.Figure 6
Critical Peclet numbers as a function ofM for selected values of Ped and μ* for the case m=0.Figure 7
Critical wave numbers as a function ofM for the selected values of Ped and μ* for the case m=0.Finally, we conclude that the variable viscosity effect enhances the heat transfer rate and destabilizes the flow for liquid heating while the opposite tend is true for gas heating.


## 5. Conclusions

The non-Darcy mixed convection flow on a semi-infinite, nonisothermal horizontal plate embedded in a porous medium with variable viscosity, is investigated. The non-Darcy model, which includes the Ergun extension, is employed to describe the base and disturbed flows in the porous medium. The variation of viscosity is expressed as an exponential function of temperature. The effects of variable viscosity characterized by the parameterμ* on the flow and vortex instability are examined. The surface temperature is assumed to vary as a power function of the distance from the origin. The governing partial differential equations are transformed to a nonsimilar form by introducing appropriate transformations and are solved numerically using an implicit finite difference scheme. The resulting eigenvalue problem is solved by using a finite difference scheme. The effects of all involved parameters on the local Nusselt number, critical Peclet and associated wave number are presented. It is shown that, for liquid heating, the variable viscosity effect enhances the heat transfer rate and destabilizes flow, while, for gas heating, the opposite trend is true.



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*Source: 691802-2012-03-05.xml*