We consider a semi-infinite non-isothermal horizontal surface (T_w) embedded in a porous medium (T_∞) as shown in Figure 1, 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, i.e., T_w(x) = T_∞ + Ax^λ, 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 constant, except for the viscosity μ and the density ρ in the bouncy force, (iii) Ergun non-Darcy model 21 is considered, and (iv) the Boussinesq approximation is valid. With these assumptions, the governing equations are given by:

∂ u ∂ x + ∂ v ∂ y = 0 ,

u + K * v u 2 = - K μ ∂ P ∂ x ,

v + K * v v 2 = - K μ ∂ P ∂ y + ρ g ,

u ∂ T ∂ x + v ∂ T ∂ y = α ∂ 2 T ∂ y 2 ,

where ρ = ρ_∞[1 - β(T - T_∞)] is the fluid density. Note that the second term on the left-hand side of Equations (2) and (3) represents the inertia force, where K* is the inertia coefficient in Ergun model. As K* = 0, Equations (2) and (3) reduce to Darcy model.

The viscosity of the fluid μ is assumed to vary with temperature according to the following exponential form

μ = μ ∞ e A 1 T - T ∞ T w - T ∞ ,

where μ_∞ is the absolute viscosity at ambient temperature and A₁ is a constant adopted from the least square fitting for a particular fluid. Formula (5) is a generalization of that used in 19, where the wall temperature is taken to be constant.

The pressure terms appearing in Equations (2) and (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 (1)-(4) become

μ + 2 ρ ∞ K * ∂ ψ ∂ y ∂ 2 ψ ∂ y 2 + u ∂ μ ∂ y = - K ρ ∞ g β ∂ T ∂ x ,

∂ ψ ∂ y ∂ T ∂ x - ∂ ψ ∂ x ∂ T ∂ y = α ∂ 2 T ∂ y 2 .

The boundary conditions are defined as follows

v ( x , 0 ) = - ∂ ψ ∂ x = 0 , T ( x , 0 ) = T w = T ∞ + A x λ , u ( x , ∞ ) = 0 , T ( x , ∞ ) = T ∞ .

By introducing the following similarity variables:

η ( x , y ) = y x R a x 1 / 3 , f ( ξ , η ) = ψ ( x , y ) α R a x 1 / 3 , θ ( ξ , η ) = T - T ∞ T w - T ∞ , ξ = x d ( 2 λ - 1 ) / 3 ,

Equations (5)-(7) become

μ μ ∞ = e A 1 T - T ∞ T w - T ∞ = ( μ * ) θ ,

( 1 + 2 E r R a d 2 / 3 ξ ( μ * ) - θ f ′ ) f ′ ′ = - ( ln μ * ) f ′ θ ′ - ( μ * ) - θ λ θ + λ - 2 3 η θ ′ + 2 λ - 1 3 ξ ∂ θ ∂ ξ ,

θ ′ ′ = λ θ + 2 λ - 1 3 ξ ∂ θ ∂ ξ f ′ - λ + 1 3 f + 2 λ - 1 3 ξ ∂ f ∂ ξ θ ′ ,

where μ * = μ w μ ∞ = e A 1 is the wall to ambient viscosity ratio parameter, R a x = ρ ∞ g β K ( T w - T ∞ ) x α μ ∞ is the local Rayleigh number, R a d = ρ ∞ g β K A d λ + 1 α μ ∞ is the Rayleigh number based on the pore diameter and E r = K * α d ν ∞ is the Ergun number.

The boundary conditions are transformed as follows:

f ( ξ , 0 ) = 0 , θ ( ξ , 0 ) = 1 , f ′ ( ξ , ∞ ) = 0 , θ ( ξ , ∞ ) = 0 ,

where the primes denote the derivatives with respect to η. The velocity components and the heat transfer coefficient in term of the local Nusselt number are given by:

u ( x , y ) = α R a x 2 / 3 x f ′ ( ξ , η ) ,

v ( x , y ) = - α R a x 1 / 3 3 x ( λ + 1 ) f ( ξ , η ) + λ - 2 η f ′ ( ξ , η ) + 2 λ - 1 ξ ∂ f ( ξ , η ) ∂ ξ ,

N u x R a x 1 / 3 = - θ ′ ( ξ , 0 ) .

The standard linear stability method yields the following:

∂ u 1 ∂ x + ∂ v 1 ∂ y + ∂ w 1 ∂ z = 0 ,

μ 0 u 1 + μ 1 u 0 + 2 K * ρ ∞ u 0 u 1 = - K ∂ P 1 ∂ x ,

μ 0 v 1 + μ 1 v 0 + 2 K * ρ ∞ v 0 v 1 = - K ∂ P 1 ∂ y - ρ ∞ g β T 1 ,

μ 0 w 1 = - K ∂ P 1 ∂ z ,

u 0 ∂ T 1 ∂ x + v 0 ∂ T 1 ∂ y + u 1 ∂ T 0 ∂ x + v 1 ∂ T 0 ∂ y = α ∂ 2 T 1 ∂ x 2 + ∂ 2 T 1 ∂ y 2 + ∂ 2 T 1 ∂ z 2 .

where the subscripts 0 and 1 signify the mean flow and disturbance components, respectively.

Following the order of magnitude analysis method described in detail by Hsu et al. 1, the terms ∂u₁/∂x and ∂²T₁/∂x² in Equations (17) and (21) can be neglected. The omission of ∂u₁/∂x in Equation (17) implies the existence of a disturbance stream function Ψ₁, such as

w 1 = ∂ Ψ 1 ∂ y , v 1 = - ∂ Ψ 1 ∂ z .

Eliminating P₁ from Equations (18)-(20) with the aid of (22) leads to

u 0 ∂ μ 1 ∂ z + ( μ 0 + 2 K * ρ ∞ u 0 ) ∂ u 1 ∂ z = μ 0 ∂ 2 Ψ 1 ∂ x ∂ y + ∂ Ψ 1 ∂ y ∂ μ 0 ∂ x ,

- ( μ 0 + 2 K * ρ ∞ v 0 ) ∂ 2 Ψ 1 ∂ z 2 + v 0 ∂ μ 1 ∂ z = μ 0 ∂ 2 Ψ 1 ∂ y 2 + ∂ μ 0 ∂ y ∂ Ψ 1 ∂ y + K ρ ∞ g β ∂ T 1 ∂ z ,

u 0 ∂ T 1 ∂ x + v 0 ∂ T 1 ∂ y + u 1 ∂ T 0 ∂ x - ∂ Ψ 1 ∂ z ∂ T 0 ∂ y = α ∂ 2 T 1 ∂ y 2 + ∂ 2 T 1 ∂ z 2 .

As in Hsu et al. 1, we assume that the three-dimensionless disturbances for neutral stability are of the form

( Ψ 1 , u 1 , T 1 ) = Ψ ̄ ( x , y ) , ū ( x , y ) , T ̄ ( x , y ) e i a z ,

where a is the spanwise periodic wave number. Substituting Equation (26) into Equations (23)-(25) yields

ū = 1 1 + 2 K * ν ∞ ( μ * ) - θ u 0 1 i a ∂ 2 Ψ ̄ ∂ x ∂ y + ( ln μ * ) ∂ Ψ ̄ ∂ y ∂ θ ∂ x - ( ln μ * ) u 0 T ̄ T w - T ∞ ,

∂ 2 Ψ ̄ ∂ y 2 - a 2 1 + 2 K * ν ∞ ( μ * ) - θ v 0 Ψ ̄ - i a v 0 ( ln μ * ) T ̄ T w - T ∞ + ( ln μ * ) ∂ θ ∂ y ∂ Ψ ̄ ∂ y = - i a K ρ ∞ g β μ ∞ ( μ * ) - θ T ̄ ,

u 0 ∂ T ̄ ∂ x + v 0 ∂ T ̄ ∂ y + ū ∂ T 0 ∂ x - i a Ψ ̄ ∂ T 0 ∂ y = α ∂ 2 T ̄ ∂ y 2 - a 2 T ̄ .

Equations (27)-(29) are solved based on the local similarity approximation 22, wherein the disturbances are assumed to have weak dependence in the streamwise direction (i.e., ∂/∂x ≪ ∂/∂η).

On applying the following transformations

k = a x R a x 1 / 3 , F ( η ) = Ψ ̄ i α R a x 1 / 3 , Θ ( η ) = T ̄ T w - T ∞ ,

into Equations (27)-(29) leads to the following:

F ′ ′ + ( ln μ * ) θ ′ F ′ - k 2 1 - 2 E r R a d 2 / 3 R a x 1 / 3 ξ ( μ * ) - θ G 1 F + k ( ln μ * ) R a x 1 / 3 G 1 Θ = - k R a x 1 / 3 ( μ * ) - θ Θ ,

Θ ′ ′ + G 3 Θ ′ - k 2 + λ f ′ - ( ln μ * ) f ′ G 2 1 + 2 E r R a d 2 / 3 ξ ( μ * ) - θ f ′ Θ - G 2 λ - 2 3 η F ′ ′ + 2 λ - 1 3 + ( ln μ * ) G 4 F ′ k R a x 1 / 3 1 + 2 E r R a d 2 / 3 ξ ( μ * ) - θ f ′ = k R a x 1 / 3 θ ′ F ,

with the boundary conditions

F ( 0 ) = F ( ∞ ) = Θ ( 0 ) = Θ ( ∞ ) = 0 ,

where G₁ - G₄ are given by

G 1 = λ + 1 3 f + λ - 2 3 η f ′ + 2 λ - 1 3 ξ ∂ f ∂ ξ , G 2 = λ θ + λ - 2 3 η θ ′ + 2 λ - 1 3 ξ ∂ θ ∂ ξ , G 3 = λ + 1 3 f + 2 λ - 1 3 ξ ∂ f ∂ ξ , G 4 = λ - 2 3 η θ ′ + 2 λ - 1 3 ξ ∂ θ ∂ ξ ,

For fixed values of ξ, λ, k, Er, Ra_d, and μ*, the solution F and Θ is an eigenfunction for the eigenvalue Ra_x. When μ* = 1, Er = Ra_d = ξ = 0, Equations (31) and (32) reduce to those presented in 1 and when μ* = 1, ξ = 1, ∂ ∂ ξ = 0 , λ = 0.5 they reduce to those given in 8. It may also remarked that for λ = Er = Ra_d = ξ = 0, Equations (31) and (32) reduce to those presented in Jang and Leu 19.