Heat Transfer to MHD Oscillatory Viscoelastic Flow in a Channel Filled with Porous Medium

The combined effect of a transverse magnetic field and radiative heat transfer on unsteady flow of a conducting optically thin viscoelastic fluid through a channel filled with saturated porous medium and nonuniform walls temperature has been discussed. It is assumed that the fluid has small electrical conductivity and the electromagnetic force produced is very small. Closed-form analytical solutions are constructed for the problem. The effects of the radiation and the magnetic field parameters on velocity profile and shear stress for different values of the viscoelastic parameter with the combination of the other flow parameters are illustrated graphically, and physical aspects of the problem are discussed.


Introduction
The flow of an electrically conducting fluid has important applications in many branches of engineering science such as magnetohydrodynamics (MHD) generators, plasma studies, nuclear reactor, geothermal energy extraction, electromagnetic propulsion, and the boundary layer control in the field of aerodynamics. In the light of these applications, MHD flow in a channel has been studied by many authors; some of them are Nigam and Singh [1], Soundalgekar and Bhat [2], Vajravelu [3], and Attia and Kotb [4]. A survey of MHD studies in the technological fields can be found in Moreau [5]. The flow of fluids through porous media is an important topic because of the recovery of crude oil from the pores of the reservoir rocks; in this case, Darcy's law represents the gross effect. Raptis et al. [6] have analysed the hydromagnetic free convection flow through a porous medium between two parallel plates. Aldoss et al. [7] have studied mixed convection flow from a vertical plate embedded in a porous medium in the presence of a magnetic field. Makinde and Mhone [8] have considered heat transfer to MHD oscillatory flow in a channel filled with porous medium.
In this study, an attempt has been made to extend the problem studied by Makinde and Mhone [8] to the case of viscoelastic fluid characterised by second-order fluid.
The constitutive equation for the incompressible secondorder fluid is of the form where σ is the stress tensor, p is the hydrostatic pressure, I is the unit tensor, A n (n = 1, 2) are the kinematic Rivlin-Ericksen tensors, μ 1 , μ 2 , and μ 3 are the material coefficients describing viscosity, elasticity, and cross-viscosity, respectively. The material coefficients μ 1 , μ 2 , and μ 3 have taken constants with μ 1 and μ 3 as positive and μ 2 as negative (Markovitz and Coleman [9]). Equation (1) was derived by Coleman and Noll [10] from that of the simple fluids by assuming that stress is more sensitive to the recent deformation than to the deformation that occurred in the distant past.

Mathematical Formulation of the Problem
Consider the flow of a conducting optically thin fluid in a channel filled with saturated porous medium under the influence of an externally applied homogeneous magnetic field and radiative heat transfer as shown in Figure 1. It is assumed that the fluid has small electrical conductivity and the electromagnetic force produced is very small. The x-axis 2 Physics Research International is taken along the centre of the channel, and the y-axis is taken normal to it. Then, assuming a Boussinesq incompressible fluid model, the equations governing the motion are given ∂T ∂t subject to boundary conditions where u is the axial velocity, t is the time, T is the fluid temperature, P is the pressure, g is the gravitational force, q is the radiative heat flux, β is the co-efficient of volume expansion due to temperature, C p is the specific heat at constant pressure, k is the thermal conductivity, K is the porous medium permeability co-efficient, B 0 (= μ e H 0 ) is the electromagnetic induction, μ e is the magnetic permeability, H 0 is the intensity of the magnetic field, σ e is the conductivity of the fluid, ρ is the fluid density, and υ i = μ i /ρ, (i = 1, 2). It is assumed that both walls of temperature T 0 , T w are high enough to induce radiative heat transfer. Following Cogley et al. [11], it is assumed that the fluid is optically thin with a relatively low density and the radiative heat flux is given by where α is the mean radiation absorption co-efficient.
The following nondimensional quantities are introduced: where U is the flow mean velocity. The dimensionless governing equations together with the appropriate boundary conditions (neglecting the bars for clarity) can be written as Pe ∂θ ∂t with where Gr, H, N, Pe, Re, Da, S(= 1/Da), and γ = (υ 2 Re)/a 2 are Grashoff number, Hartmann number, Radiation parameter, Péclet number, Reynolds number, Darcy number, porous medium shape factor parameter, and viscoelastic parameter, respectively.

Method of Solution
In order to solve (7) and (8) for purely oscillatory flow, let where λ is a constant and ω is the frequency of oscillation. Substituting the above expressions into (7) and (8) and using (9), we get The nondimensional shear stress σ at the wall y = 0 is given by The rate of heat transfer across the channel's wall is given as

Discussions and Conclusion
The purpose of this study is to bring out the effects of the viscoelastic parameter γ on the governing flow with the combination of the other flow parameters. The corresponding results for Newtonian fluid can be deduced from the above results by setting γ = 0, and it is worth mentioning here that these results coincide with that of Makinde and Mhone [8].
We have considered the real parts of the results throughout for numerical validation. The velocity profile u against y is plotted in Figures 2-4  It has also been observed that the temperature field is not significantly affected by the viscoelastic parameter.