Journal of Fluids

Volume 2015, Article ID 358053, 9 pages

http://dx.doi.org/10.1155/2015/358053

## Unsteady/Steady Hydromagnetic Convective Flow between Two Vertical Walls in the Presence of Variable Thermal Conductivity

^{1}Department of Mathematics, Usmanu Danfodiyo University, PMB 2346, Sokoto, Nigeria^{2}Department of Mathematics, Zamfara State College of Education, PMB 1002, Maru, Nigeria

Received 16 September 2014; Revised 25 March 2015; Accepted 1 April 2015

Academic Editor: Miguel Onorato

Copyright © 2015 M. M. Hamza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Unsteady as well as steady natural convection flow in a vertical channel in the presence of uniform magnetic field applied normal to the flow region and temperature dependent variable thermal conductivity is studied. The nonlinear partial differential equations governing the flow have been solved numerically using unconditionally stable and convergent semi-implicit finite difference scheme. For steady case, approximate solutions have been derived for velocity, temperature, skin friction, and the rate of heat transfer using perturbation series method. Results of the computations for velocity, temperature, skin friction, and the rate of heat transfer are presented graphically and discussed quantitatively for various parameters embedded in the problem. An excellent agreement was found during the numerical computations between the steady-state approximate solutions and unsteady numerical solutions at steady-state time. In addition, comparison with previously published work is performed and the results agree well.

#### 1. Introduction

In recent years, the interest in the study of hydromagnetic flow in a channel region has been growing rapidly because of its extensive engineering applications. The experimental investigation of modern MHD flow in a laboratory was first carried out by [1]. This study provided the basic knowledge for the development of many MHD devices, such as MHD pumps, MHD generators, brakes, flow meters, plasma studies, and geothermal energy extraction. Unsteady free convection heat transfer with MHD effects in a channel region can be found in [2]. Unsteady hydromagnetic flows in rotating systems have been studied by [3–9]. An exact solution for unsteady hydromagnetic free convection flow with constant heat flux is to be found in [10]. All the above mentioned studies assumed the thermal conductivity of the fluid to be constant. However, it is known that the fluid physical properties may change significantly with temperature changes. To accurately predict the flow behavior and heat transfer rate, it is necessary to take into account the variation of thermal conductivity with temperature (see [11]). Thermal properties, particularly thermal conductivity and diffusivity, are essential materials parameters of bedrock controlling the heat transfer and temperature increases in the vicinity of repository. There has been considerable published work dealing with steady flow with variable thermal conductivity (see [12–18]). Recently (see [19]) studied steady MHD flow with variable thermal conductivity over an inclined radiative isothermal permeable surface.

To the best of our knowledge, the problem of unsteady/steady hydromagnetic convective flow between two vertical walls heated symmetrically/asymmetrically in the presence of variable thermal conductivity has not been studied. The present paper is committed to study unsteady as well as steady natural convection flow of a viscous, incompressible fluid between two parallel vertical walls in the presence of transverse magnetic field and temperature dependent variable thermal conductivity when convection between the vertical parallel walls is set up by a change in the temperature of the walls compared to the fluid temperature.

#### 2. Governing Equations

Consider the unsteady natural convection flow of viscous, incompressible, and electrically conducting fluid between two vertical walls in the presence of a transversely imposed magnetic field of strength . Initially, it is assumed that both the fluid and the walls are at rest and at the same temperature . At time , the temperature of the walls and is instantaneously raised or lowered to and , respectively, such that which is thereafter maintained constant. We chose a Cartesian coordinate system with axis along the upward direction and the axis normal to it as shown in Figure 1. Thermal conductivity () of the fluid is assumed to vary as a linear function of temperature in the form (see [20]), , where is the fluid free stream thermal conductivity and is a constant depending on the nature of the fluid, where for fluids such as water and air, while for fluids such as lubrication oils (see [20]). The governing equations under Boussinesq’s approximation can be written asThe initial and boundary conditions for the present problem arewhere is the coefficient of the thermal expansion, is the kinematic viscosity, is the gravitational force, is the conductivity of the fluid, is the electromagnetic induction, is the density of the fluid, and is the specific heat at constant pressure.