Mixed Convection Boundary Layer Flow over a Permeable Vertical Flat Plate Embedded in an Anisotropic Porous Medium

An analysis is performed to study the heat transfer characteristics of steady mixed convection flow over a permeable vertical flat plate embedded in an anisotropic fluid-saturated porous medium. The effects of uniform suction and injection on the flow field and heat transfer characteristics are numerically studied by employing an implicit finite difference Keller-box method. It is found that dual solutions exist for both assisting and opposing flows. The results indicate that suction delays the boundary layer separation, while injection accelerates it.


Introduction
Transport processes through porous media play important roles in diverse applications, such as in geothermal operations, petroleum industries, and many others. Excellent reviews on this topic can be found in the books by Ingham and Pop 1 , Vafai 2 , Nield and Bejan 3 , Vadasz 4 , and in the review paper by Magyari et al. 5 . The study of convective heat transfer and fluid flow in porous media has received great attention in recent years. Most of the earlier studies Minkowycz and Cheng 6 , Cheng and Minkowycz 7 , and Badr and Pop 8 were based on Darcy's law which states that the volume-averaged velocity is proportion to the pressure gradient. Kaviany 9 used the line integral method to study the heat transfer from a semi-infinite flat plate embedded in a fluid-saturated porous medium. Jang and Shiang 10 studied the mixed convection along a vertical adiabatic surface embedded in a porous medium. Few studies of convective boundary-layer flows in porous media using All of the works mentioned above are conducted to flows over an impermeable surface embedded in a Darcian porous medium. The free convections with injection or suction over permeable vertical and horizontal plates in a porous medium were studied by Cheng 17 ,Merkin 18 and Minkowycz et al. 19 . Lai and Kulacki 20, 21 investigated the effects of injection and suction on mixed convection over horizontal and inclined surfaces embedded in fluid-saturated porous media. Elbashbeshy and Bazid 22, 23 analyzed the heat transfer over a continuously moving plate and mixed convection along a vertical plate embedded in a non-Darcian porous medium. Further, Elbashbeshy 24,25 investigated the effects of suction and injection on mixed convection boundary layer flow over horizontal flat plate and mixed convection boundary layer flow along a vertical plate embedded in a non-Darcian porous medium.
The aim of this paper is to study the effects of suction and injection on the mixed convection boundary layer flow over a permeable vertical plate embedded in an anisotropic porous medium. Injection or withdrawal of fluid through a porous bounding heated or cooled wall is of general interest in practical problems involving film cooling, control of boundary layers, and so forth. This can lead to enhance heating or cooling of the system and can help to delay the transition from laminar flow see Chaudhary and Merkin 26 . We mention to this end that such a study has also been done by Massoudi 27 ,Weidman et al. 28,29 ,and Ishak et al. 30 for the classical problems of the boundary layers over a permeable wedge, moving flat plates, and permeable vertical flat plates. To the best of our knowledge, this problem has not been studied before and the results are new and original.

Problem Formulation
Consider the steady mixed convection boundary layer flow over a semi-infinite vertical permeable surface with a uniform surface temperature T w x embedded in an anisotropic fluid-saturated porous medium, as shown in Figure 1. The uniform temperature of the ambient fluid is T ∞ , where T w x > T ∞ for a heated plate and T w x < T ∞ for a cooled plate. The corresponding velocity components in the x and y directions are u and v, respectively, and the surface mass flux V w is assumed to be constant with V w > 0 for injection and V w < 0 for suction. The permeabilities along the two principal axes of the porous matrix are denoted by K 1 and K 2 . The anisotropy of the porous medium is characterized by the anisotropy ratio K * K 1 /K 2 and the orientation angle φ, defined as the angle between the horizontal direction and the principal axis with permeability K 2 . Under the Boussinesq approximation, the basic equations of continuity, the generalized Brinkman-extended Darcy's law, and energy are given by see Vasseur

2.5
Here u e x is the free stream velocity, p is the fluid pressure, g is the acceleration due to gravity, α m is the effective thermal diffusivity, β is the coefficient of volumetric thermal expansion, μ is the effective dynamic viscosity, μ is the dynamic viscosity, and υ is the kinematic viscosity. We now introduce the following nondimensional boundary-layer variables: where U ∞ is the characteristic velocity, L is the characteristic length, ΔT is the characteristic temperature difference, and Pe U ∞ L/α m is the Péclet number. Substituting the nondimensional variables 2.6 into 2.1 -2.3 , eliminating the pressure gradients from 2.2 , and imposing the usual boundary layer approximations, we obtain the following boundarylayer equations for the present problem: Here Da is the Darcy-Brinkman parameter, λ is the mixed convection parameter, and ε is the modified Péclet number, which are defined as where Ra gK 1 βΔTL/υα m is the Rayleigh number for the anisotropic porous medium. It should be noted that λ > 0 is for the assisting flow, λ < 0 is for the opposing flow, and λ 0 corresponds to forced convection flow. Integrating 2.8 with the boundary conditions 2.10b and introducing the stream function ψ, which is defined as u ∂ψ/∂y and v −∂ψ/∂x, we obtain It should be noted that this similarity solution corresponds to the mixed convection flow in a porous medium near the stagnation point on a vertical surface with a linear variation in the wall temperature. The corresponding situation for using the Darcy-Brinkman formulation of the governing equations and the slip condition on the surface was studied by Harris et al. 16 . Substituting 2.18 into 2.12 , we obtain the following system of ordinary differential equations: where A a/ εDa is the anisotropy parameter and Λ λ/ εDa is the modified mixed convection parameter with εDa / 0, and primes denote differentiation with respect to y. The transformed boundary conditions are

2.21
where f 0 − V w is the suction or injection parameter with f 0 > 0 for suction and f 0 < 0 is for injection. When εDa 0 inertial effect is neglected , 2.19 can be reduced to where λ * λ/a, which is identical to that derived by Merkin 33 and subjected to the associated boundary conditions 2.21 with f 0 0 in his paper. Thus, this case will not be considered here.
Expressions 2.16 for the skin friction coefficient C f and the Nusselt number Nu become

Results and Discussion
Equations 2.19 and 2.20 subject to the boundary conditions 2.21 have been solved numerically for some values of the governing parameters f 0 and Λ using a very efficient finite-difference scheme known as the Keller-box method, which is described in the book by Cebeci and Bradshaw 34 , and in the review paper by Keller 35 . This method has been successfully used by the present authors to study various boundary value problems cf. 36-41 . The variations of the skin friction coefficient f 0 with Λ together with their velocity profiles are shown in Figures 2-4 for A 1, f 0 0.2, and f 0 −0.2, respectively, while the respective Nusselt number −θ 0 together with their temperature profiles is shown in Figures 5-7, to support the validity of the numerical results obtained. It is worth mentioning that all the velocity and temperature profiles satisfy the far field boundary conditions 2.21 asymptotically. In these figures the solid lines and the dash lines are for the first solution and second solution, respectively. The results for the skin friction coefficient f 0 and the Nusselt number −θ 0 as a function of Λ show that it is possible to get dual solutions of the similarity equations 2.19 and 2.20 subject to the boundary conditions 2.21 for the assisting flow Λ > 0 as well, beside that usually reported in the literature for the opposing flow Λ < 0 . Also for Λ > 0, there is a favorable pressure gradient due to the buoyancy effects, which results in the flow being accelerated in a larger skin friction coefficient than in the nonbuoyant case Λ considered in this study, whereas for the opposing flow, the solutions exist up to certain values of Λ, that is, Λ c . Beyond these critical values, the boundary layer separates from the surface; thus no solution is obtained using the boundary layer approximations. Moreover, from Figures 2 and 5, we found that the values of |Λ| for which the solution exists increase as f 0 increases. Hence, suction delays the boundary layer separation. Numerical results for the local Nusselt number as presented in Figure 5 show that −θ 0 approaches ∞ as Λ → 0 , and −∞ as Λ → 0 − . In Figure 2  which is in agreement with Ridha 44 . However, as discussed by Ridha 44 and Ishak et al. 30 , the second solutions have no physical sense. Although such solutions are deprived of physical significance, they are nevertheless of mathematical interest as well as of physical terms so far as the differential equations are concerned. Besides, similar equations may arise in other situations where the corresponding solutions could have more realistic meaning Ridha 45 .

Conclusions
We have theoretically studied the existence of dual similarity solutions in mixed convection boundary layer flow over a permeable vertical plate embedded in an anisotropic porous medium with suction and injection. The governing boundary layer equations have been solved numerically for both assisting and opposing flow regimes using the Keller-box method. Discussions for the effects of suction or injection parameter f 0 and the modified mixed convection parameter Λ on the skin friction coefficient f 0 and the Nusselt number −θ 0 for A 1 have been done. It is found that dual solutions exist for both assisting and opposing flows. It is shown that introducing suction effect increases the range of Λ for which the solution exists and in consequence delays the boundary layer separation, while it is found that injection acts in the opposite manner.