Influence of temperature-dependent viscosity and thermal radiation on MHD forced convection over a non-isothermal wedge

doi:10.1016/j.amc.2009.02.013

Applied Mathematics and Computation

Volume 212, Issue 1, 1 June 2009, Pages 194-208

https://doi.org/10.1016/j.amc.2009.02.013 Get rights and content

Abstract

An analysis has been carried out to study heat transfer characteristics of an incompressible Newtonian electrically conducting and heat generating/absorbing fluid having temperature-dependent viscosity over a non-isothermal wedge in the presence of thermal radiation. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The wedge surface is assumed to be permeable so as to allow for possible wall suction or injection. The effects of viscous dissipation, Joule heating, stress work and thermal radiation are included in the model. The governing differential equations are derived and transformed using a non-similarity transformation. The transformed equations are solved numerically by applying a fifth-order Runge–Kutta–Fehlberg scheme with shooting technique. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results for the velocity and temperature profiles for a prescribed magnetic field parameter as well as the development of the local skin-friction coefficient and local Nusselt number with the magnetic field and radiation parameters are presented graphically and in tabulated form to elucidate the influence of the various physical parameters.

Introduction

The problems of heat transfer in the boundary layers on continuous stretching surfaces with a given temperature or heatflux distribution, moving in an otherwise quiescent fluid medium, have attracted considerable attention during the last few decades due to their numerous applications in several industrial manufacturing processes. Few examples of such technological processes are the extrusion of plastic sheet, hot rolling, wire drawing, glass-fibre and paper production, drawing of plastic films, metal spinning and the cooling of a metallic plate in a cooling bath. Steady two-dimensional laminar forced convection heat transfer of incompressible Falkner–Skan flow from a wedge has been reviewed extensively by Lin and Lin [1]. They proposed a similarity solution method for either an isothermal surface or a uniform flux boundary to fluids of any Prandtl number. Within the field of aerodynamics, the analysis of boundary-layer problems for two-dimensional steady and incompressible laminar flow passing a wedge is a common area of interest. Falkner and Skan [2] considered two-dimensional wedge flows and developed a similarity transformation method in which the partial differential boundary-layer equation was reduced to a non-linear third-order ordinary differential equation and then solved it numerically. Rajagopal et al. [3] studied the Falkner–Skan boundary-layer flow of a homogeneous incompressible second grade fluid past a wedge placed symmetrically with respect to the flow direction. Asaithambi [4] presented a finite-difference method for solving the Falkner–Skan equation.

When the free convective flows occur at high temperature, radiation effects on the flow can not be neglected. Many processes in new engineering areas occur at high temperature and knowledge of radiative heat transfer becomes very important for the design of the pertinent equipment. Nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering areas. In fact, very little is known about the effects of radiation on the boundary layer flow of a radiating fluid past a body because of the difficulties arising when attempting to study such problems. Actually, three major difficulties arise in the study of fluid radiation. First, when radiative heat transfer takes place in a system, the radiation is absorbed and emitted not only at the system boundaries but also in the interior of the system, which makes the prediction of fluid absorption a very difficult task. Second, the absorption coefficients of the absorbing/emitting fluids are in general strongly dependent on wavelength. Third, the inclusion of a radiation term in the energy equation leads to a highly non-linear partial differential equation and this is a computational difficulty. However, many researchers studied the effects of radiation on convective flows. Hossain and Takhar [5] studied the effects of radiation on mixed convection flow of an optically dense viscous incompressible fluid past a heated vertical plate with uniform free stream velocity and surface temperature. The effects of radiation on the free convection of an optically dense viscous incompressible fluid along a heated inclined flat surface maintained at uniform temperature placed in a saturated porous medium was studied by Hossain and Pop [6] by considering only the suction boundary condition in the absence of viscous dissipation. Grief et al. [7] first studied the mixed free-forced flow of a radiating gas between two vertical plates and used the small optical thickness approximation to neglect the fluid self-absorption. Using the same approximation, Takhar et al. [8] studied radiation effect on the free convection flow of a gas past a semi-infinite flat plate. Recently, under the same approximation, Abo-Eldahab [9] studied the radiation effect on heat transfer in an electrically conducting fluid at a stretching surface.

The study of magnetohydrodynamic(MHD) viscous flows is important to industrial, technological, and geothermal applications, such as high-temperature plasmas, cooling of nuclear reactors, liquid metal fluids, Magnetohydrodynamic generators and accelerators. As a result, a significant amount of interest has been carried out to study the effects of electrically conducting fluids such as liquid metals, water mixed with a little acid and others in the presence of a magnetic field on the flow and heat transfer aspects in various geometries. Watanabe and Pop [10] presented numerical results of MHD free convection flow over a wedge in the presence of a magnetic field,while Kafoussias and Nanousis [11] analyzed the MHD laminar boundary layer flow of a micropolar fluid over a permeable wedge. Yih [12] extended the work of Watanabe and Pop [10], by considering the MHD forced convection flow adjacent to a non-isothermal wedge in the presence of viscous and magnetic dissipations and stress work.

Moreover, when radiative heat transfer takes place, the fluid involved can be electrically conducting in the sense that it is ionized due to the high operating temperature. Accordingly, it is of interest to examine the effect of the magnetic field on such flow due to its great importance in the application fields where thermal radiation and MHD are correlative. The process of fusing of metals in the electrical furnace by applying a magnetic field and the process of cooling of the first wall inside the nuclear reactor containment vessel where the hot plasma is isolated from the wall by applying a magnetic field are examples of such field. Very little is known about the effects of radiation on the boundary layer of a radiate-MHD fluid past a body. The inclusion of radiation effects in the energy equation leads to a highly non-linear partial differential equation. More recently, Duwairi and Damseh [13], [14] studied the radiation–conduction interaction in free and mixed convection fluid flow for a vertical flat plate with the presence of a magnetic field effect.

All the above studies were combined to the fluid with uniform viscosity. However, it is known that their physical property may change significantly with temperature.For instance, the viscosity of water decreases by about 240 when the temperature increases from 10 °C $(μ = 0.0131 g {cm}^{- 1} s^{- 1})$ to 50 °C $(μ = 0.00548 g {cm}^{- 1} s^{- 1})$ . Thus it is necessary to take into account their variation of viscosity in order to predict the flow behavior accurately since real fluid viscosity are functions of temperature and pressure. For many liquids, such as water and oils, the viscosity variation according to temperature is the most dominant effect. In fact, in many thermal transport processes, the temperature distribution within the flow field is never uniform. In other words, the fluid viscosity may change noticeably if a large temperature difference exists between the solid surface and the surroundings. Therefore, it is highly necessary to ponder the effect of temperature-dependent viscosity on the momentum and thermal transport predictions. Takhar et al. [15] considered the influence of temperature-dependent properties for monatomic gases, diatomic gases, air and water vapour on laminar boundary-layer flow over a continuously moving flat surface. Pop et al. [16] and Elbashbeshy and Bazid [17] have shown that when this effect is included, the flow characteristics changed substantially compared to the constant viscosity case. Kafoussias and Williams [18] have investigated the effect of temperature-dependent viscosity on the mixed convection flow from a vertical flat plate in the region near the leading edge using the local non-similarity method. Abo-Eldahab and El Gendy [19] studied radiation effect on convective heat transfer in an electrically conducting fluid at a stretching surface with variable viscosity and uniform free stream. Recently, Abo-Eldahab and Salem [20] studied radiation effect on MHD free convective flow of a gas past a semi-infinite vertical plate with variable viscosity. They showed that the flow characteristics are markedly affected by the variation of viscosity with temperature.

Therefore, the objective of this paper is to study the combined effects of thermal radiation and temperature-dependent viscosity on the momentum and heat transfer in the presence of magnetic field. The Rosseland approximation is considered to describe the radiative heat flux in the energy equation. Hence the radiation contribution comes only through the energy equation. The transformed dimensionless governing equations are solved numerically. These results are compared with those which employ the constant viscosity assumption. We also discuss the effect of variable viscosity on the behaviour of the skin-friction coefficient and the Nusselt number over an adequate range of influential parameters.

Section snippets

Mathematical formulations

Consider the steady boundary-layer flow past a wedge in an electrically conducting viscous incompressible fluid in the presence of a magnetic field $B_{0}$ applied in the normal direction to the walls of the wedge as shown in Fig. 1. The induced magnetic field is assumed to be small. The radiation heat flux on x-direction is negligible compared to the flux in the y-direction. The x-axis is chosen along the plate and the y-axis is taken normal to it. The viscous dissipation and velocity of the fluid

Local non-similarity method

We now discuss the local non-similarity method to solve Eqs. (19), (20). Consideration of equation up to the second level of truncation gives almost accurate results comparable with the solutions from other methods such as finite-difference method. To do this, we introduce the following new functions [6]: $g = \frac{\partial f}{\partial ξ}, ϕ = \frac{\partial θ}{\partial ξ} .$ Introducing these functions into Eqs. (19), (20) we get $f^{‴} - \frac{1}{θ - θ_{r}} θ^{'} f^{″} - \frac{1 + m}{2} (\frac{θ}{θ_{r}} - 1) {ff}^{″} = (1 - \frac{θ}{θ_{r}}) [ξ (1 - m) (f^{'} g^{'} - f^{″} g) + ξ (f^{'} - 1) + m (f^{' 2} - 1)],$ $\frac{1 + N_{r}}{\Pr} θ^{″} + \frac{1 + m}{2} f θ^{'} - 2 {mf}^{'} θ + λ ξ θ + Ec [f^{″ 2} - (m + ξ) f^{'} + ξ f^{' 2}] = ξ (1 - m) (f^{'} ϕ - θ^{'} g) .$

Numerical method

The system of ordinary differential Eqs. (26), (27), (30), (31) with the boundary condition (32) are solved numerically using the Runge–Kutta–Fehlberg scheme with shooting method. The details of the solution method are presented in Pal and Shivakumara [25]. The accuracy of the numerical method used was checked by performing comparisons with the previously published work (when $θ_{r} \to \infty$ in present case) as shown in Table 1, Table 2, Table 3, Table 4. Results for $f^{'} (0, 0)$ for various values of m in the

Discussion of the results

The similarity Eqs. (26), (27), (30), (31) are non-linear, coupled ordinary differential equations, which posses no closed-form solution. Thus, these equations are solved numerically subject to the boundary conditions given by Eq. (32). The solution convergence criterion employed in the present work is based on the difference between the values of the dependent variables of the current and the previous iterations. When the value of this difference reaches $10^{- 5}$ which showed that the solution was

Conclusion

The problem of steady, laminar, hydromagnetic forced convection boundary layer flow of a Newtonian, heat generating/absorbing fluid considering time-dependent viscosity over a non-isothermal wedge with permeable surface in the presence of thermal radiation was investigated. Power-law variation of the wall temperature was assumed. A transformed set of non-similar equations was obtained and were solved numerically by the Runge–Kutta–Fehlberg scheme with shooting technique. Comparisons with

Acknowledgement

University Grants Commission (UGC), New Delhi, India financially supported the work of DP under Special Assistance Programme (DRS Phase-I) [Grant No. F.510/8/DRS/2004(SAP-I)]. Authors acknowledge with gratitude the constructive comments and suggestions of the reviewers which improved the quality of the paper greatly.

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Influence of temperature-dependent viscosity and thermal radiation on MHD forced convection over a non-isothermal wedge

Abstract

Introduction

Section snippets

Mathematical formulations

Local non-similarity method

Numerical method

Discussion of the results

Conclusion

Acknowledgement

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Int. J. Heat Mass Transfer

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Some approximate solutions of the boundary-layer equations

Philos. Mag.

Radiation effect on mixed convection along a vertical plate with uniform surface temperature

Heat Mass Transfer

Radiation effect on Darcy free convection in boundary layer flow along an inclined surface placed in porous media

Heat Mass Transfer

Laminar convection of a radiating gas in a vertical channel

J. Fluid Mech.

Radiation effects on MHD free convection flow of a radiation gas past a semi-infinite vertical plate

Int. J. Numer. Meth. Heat Fluid Flow

Convective heat transfer in an electrically conducting fluid at a stretching surface by the presence of radiation

Can. J. Phys.

Thermal boundary layers in magnetohydrodynamic flow over a flat plate in the presence of a transverse magnetic field

Acta Mech.

Magnetohydrodynamic laminar boundary layer flow over a wedge with suction or injection

Can. J. Phys.