Effects of thermal radiation and space porosity on MHD mixed convection flow in a vertical channel using homotopy analysis method

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Abstract

A study of MHD mixed convection flow through porous space in the presence of a temperature dependent heat source in a vertical channel with radiation has been analyzed. The Rosseland approximation is considered in the modeling of the conduction radiation heat transfer and temperatures of the walls are assumed constants. The governing equations are expressed in non-dimensional form and the series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). The effects of various involved parameters on the velocity and temperature field are shown and discussed. The coefficient of skin friction, and the rate of heat transfer coefficient are obtained and illustrated graphically.

Introduction

The study of combined free and forced convection flow in vertical channel has received considerable attention because of its wide range of applications from cooling of electronic devices to that of solar energy collectors. A comprehensive review of the work on mixed convection can be found in [1], [2]. One of the earliest study on laminar, fully developed mixed convection in a vertical channel with uniform wall temperatures was by Tao [3]. Recently Aung and Worku [4], Cheng et al. [5] and Hamadah and Wirtz [6] have studied the mixed convection in a vertical channel with symmetric and asymmetric heating of the walls. Barletta [7], [8] has studied the fully developed combined free and forced convection flow in a vertical channel with viscous dissipation. Chamkha et al. [9] have presented both analytical and numerical results of the problem of fully developed free convection flow of a micropolar fluid between a parallel-plate vertical channel with asymmetric wall temperature distribution. Umavathi et al. [10] have numerically investigated the problem of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipation. Prathap kumar et al. [11] have analyzed the problem of fully developed combined free and forced convective flow in a fluid saturated porous medium channel bounded by two vertical parallel plates. Pop et al. [12] have investigated the steady fully developed mixed convection flow in a vertical channel with constant temperature walls when there is a heat generated by an exothermic reaction inside the channel.

The analysis of magnetohydrodynamic flow through ducts has received considerable attention. This class of flow has many applications in the design of MHD generators, cross-field accelerators, shock tubes, pumps and flow meters. In many cases the flow in these devices will be accompanied by heat either that dissipated internally through viscous or Joule heating or that produced by electric currents in the walls. Recently, Umavathi and Malashetty [13] studied the problem of combined free and forced magnetoconvection flow in a vertical channel with symmetric and asymmetric boundary heating in the presence of viscous and Joulean dissipations. More recently Srinivas and Kothandapani [14] investigated the effects of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls. Zueco et al. [15] analyzed the problem of unsteady MHD free convection of a micropolar fluid between two parallel porous vertical walls with convection from the ambient.

In spite of complexity of solution for convection and radiation equation, there are few studies about simultaneous effect of convection and radiation for internal flows ([16], [17] and several references therein). Radiation heat transfer in porous media has also been studied by some researchers ([18], [19], [20] and several references therein). There has been renewed interest in studying magnetohydrodynamic (MHD) flow and heat transfer in porous media due to the effect of magnetic fields on the flow control and on the performance of many systems using electrically conducting fluids. Raptis et al. [21] have considered MHD free convection flow through a porous medium between two parallel plates. Aldoss et al.[22] have studied mixed convection from a vertical plate embedded in a porous medium in the presence of a magnetic field. Chamka and Ben-Nakhi [23] have studied the numerical modeling of steady, laminar, heat and mass transfer by MHD mixed convection in the presence of thermal radiation and Dufour and Soret effects.

To the best of our knowledge the influence of thermal radiation on MHD mixed convection flow through a vertical porous space has not been studied before. Therefore the main goal of the present investigation is to analyze the thermal radiation effects on the steady fully developed mixed convection MHD flow in a vertical channel with porous medium. The walls of the channels are subjected to uniform but different wall temperatures. The homotopy analysis method (HAM) proposed by Liao [24], [25] has been employed for the analytic solution. The HAM is recently developed powerful technique and has been used successfully in solving different types of non-linear problems arising in heat transfer, fluid flow, oscillatory systems etc. [26], [27], [28], [29], [30], [31]. The organization of the paper is as follows. The problem is formulated in Section 2. Sections 3 Basic idea of homotopy analysis method (HAM), 4 Solution of the problem comprise the series solutions for flow and heat transfer analysis, respectively. The convergence of the solution is discussed in Section 5. The graphical results are presented and discussed in Section 6. Section 7 contains the concluding remarks.

Section snippets

Formulation of the problem

Consider the steady, laminar, hydromagnetic fully developed flow in a parallel plate vertical channel filled with a porous material. The x-axis is chosen parallel to gravitational field, but with opposite direction and y-axis is transverse to the channel walls. The origin is such that the channel walls are at positions y = d and y = d respectively as shown in Fig. 1. The wall y = d has the given uniform temperature T1, while the wall at y = d is subjected to a uniform temperatureT2, where T2>T1. A

Basic idea of homotopy analysis method (HAM)

Let us assume the following non-linear differential equation in the form of:N[u(τ)]=0where N is a non-linear operator, τ is an independent variable and u(τ) is the solution of equation. We define the function, ϕ(τ,p) as follows:limϕ(τ,p)p0=u0(τ)where p[0,1] and u0(τ) is the initial guess which satisfies initial or boundary conditions andlimϕ(τ,p)p1=u(τ)And by using the generalized homotopy method, Liao’s so-called zero-order deformation equation (17) is(1-p)L[ϕ(τ,p)-u0(τ)]=phH(τ)N[ϕ(τ,p)],

Solution of the problem

For HAM solutions, we choose the initial guesses and auxiliary linear operators in the following form:u0(η)=0θ0(η)=ηL1(u)=uL2(θ)=θL1(c1η+c2)=0L2(c3η+c4)=0and ci(i=1-4) are constants. Let p[0,1] denotes the embedding parameter and h indicates non-zero auxiliary parameters. We then construct the following equations.

Zeroth-order deformation equations:(1-p)L1[u(η,p)-u0(η)]=phN1[u(η,p),θ(η,p)],u(-1,p)=0;u(1,p)=0(1-p)L2[θ(η,p)-θ0(η)]=phN2[θ(η,p)],θ(-1,p)=-1;θ(1,p)=1whereN1[u(η,p),θ(η,p)]=d2u(η,p)dη

Convergence of the solution

The expression for velocity and temperature field contains the auxiliary parameter h. As pointed out by Liao [24], the convergence region and rate of approximations given by the homotopy analysis method are strongly dependent upon the auxiliary parameter. In Fig. 2, Fig. 3 the h-curves are plotted to see the range of admissible values for -1h0. Our calculations clearly indicate that the series (47), (48) converge for whole region of η when h = −0.2, h = −0.5.

Results and discussion

In order to get a clear insight of the physical problem, extensive calculations have been performed to obtain the flow and temperature field with magnetic parameter (H), radiation parameter (Rd), mixed convection parameter (Gr/Re), thermal parameter (θR), temperature dependent heat source parameter (α) and porosity parameter (Da) and are shown graphically in Fig. 4, Fig. 5.

The variation of velocity profiles for various values of H, Da,Rd and Gr/Re are depicted in Fig. 4. It can be noticed from

Concluding remarks

The thermal radiation effects on the steady fully developed MHD mixed convection flow through porous space in a vertical channel has been examined in this treatise. The radiative heat flux term in the energy equation is simplified by using Rosseland approximation. The governing equations are expressed in the non-dimensional form and are solved by using HAM. This method provides highly accurate analytic solutions of nonlinear problems compared with previous analytic and numerical methods [24],

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