Journal of Thermodynamics

Volume 2016 (2016), Article ID 3487182, 14 pages

http://dx.doi.org/10.1155/2016/3487182

## Effect of Magnetic Field on Mixed Convection Heat Transfer in a Lid-Driven Square Cavity

^{1}Centre for Research in Computational Mathematics, Faculty of Science, Technology and Human Development, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia^{2}Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

Received 8 October 2015; Revised 1 February 2016; Accepted 14 February 2016

Academic Editor: Pedro Jorge Martins Coelho

Copyright © 2016 N. A. Bakar 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

The effect of magnetic field on fluid flow and heat transfer in two-dimensional square cavity is analyzed numerically. The vertical walls are insulated; the top wall is maintained at cold temperature, while the bottom wall is maintained at hot temperature, where . The dimensionless governing equations are solved using finite volume method and SIMPLE algorithm. The streamlines and isotherm plots and the variation of Nusselt numbers on hot and cold walls are presented.

#### 1. Introduction

Due to the effect on many engineering applications and nature phenomena, fundamental problem of combined free and forced convection has received considerable attention from researchers. Problem of heat transfer in low speed lid-driven cavity is treated as a mixed convection problem. A forced convection conditions is created by the movement by one or two walls of the cavity while the temperature difference across the cavity caused a buoyancy driven flow. Hence, complicated heat and mass transfer flow occur inside the cavity. There have been many investigations in the past on mixed convection flow in lid-driven cavity. Many different configurations and combinations of thermal boundary conditions have been considered by various investigators. One of the earliest investigations on mixed convection in lid-driven cavity was conducted by [1]. Later, [2] studied numerically the effect of the Prandtl () number, on the laminar mixed convection heat transfer in a lid-driven cavity. The numerical simulations showed that, for higher values of , the effect of thermal buoyancy force on the flow and heat transfer inside the cavity is more pronounced. Reference [3] investigated numerically mixed convection heat transfer in a driven cavity with a stable vertical temperature gradient. It was found that, for high values of the Richardson number, much of the fluid in both the middle and bottom portions of the cavity interior is stagnant. Also, results in [3] results showed that the flow features are similar to those of a conventional driven cavity of a nonstratified fluid for small values of the Richardson number. Meanwhile, [4] carried out experimental study on mixed convection heat transfer and fluid flow in a cavity where the upper lid was cooled and heated from bottom. Reference [5] examined numerically mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium and reported on the effects of the Darcy and Richardson numbers on the flow and heat transfer characteristics. Reference [5] observed that the Darcy number is most important parameter in convective flows. Moreover, the presence of internal heat generation provides an additional dynamic in overall convective flow system, which has significant influence on the features of isotherms and streamlines for small values of the Richardson number.

Subsequently, various investigations on mixed convection heat transfer in a lid-driven cavity are conducted. For instance, [6] investigated the mixed convection fluid flow and heat transfer in a square cavity with moving and differentially heated side walls. They found that the fluid flow and heat transfer in the cavity are affected by both, Richardson number and and the direction of moving walls. Reference [7] investigated mixed convection fluid flow in a square cavity with linearly heated left wall, uniformly cooled bottom wall, insulated on the top while the right wall is either heated linearly or cooled uniformly. Reference [7] showed that by increase in Grashof number the strength of convection increased. Later, [8] investigated a uniformly heated bottom wall with linearly heated left side wall. Reference [9] analyzed four cases of mixed convection in a square cavity. For the first and second cases, the top and bottom walls are either heated or cooled while both side walls are insulated. In the third and fourth cases, the top and bottom walls are adiabatic while the side walls are either heated or cooled. Reference [9] reported that both and direction of temperature gradient affect the flow pattern and heat transfer rate in the cavity. Mixed convection heat transfer and fluid flow in lid-driven cavities with different lengths of the heating portion and different location of it are studied by [10]. Reference [10] concluded that the heat transfer rate is enhanced on reducing the heating portion and when the portion is at the middle or top of the hot wall. Investigation of mixed convection fluid flow and heat transfer that combined linear and sinusoidal heating of different walls of a two-dimensional square cavity with a moving lid had been done by [11]. The effects of inclination angle, , and various aspects ratios on the flow and heat transfer in an air-filled cavity had been discussed by [12]. Reference [12] reported that, among the three aspects ratio investigated, the increase of inclination angle does not affect the flow structures and heat transfer when the flow is in a forced convection dominated regime.

Investigations on the significance of magnetic field on the fluid flow and performance of different systems have been growing through these past decades. Reference [13] studied the effects of magnetic field on free convection in a rectangular cavity. Then, [14] investigated the significance of transverse magnetic field in tilted enclosure numerically. Different thermal boundary conditions with the existence of magnetic field on natural convection are discussed by [15, 16]. Recently, [17–19] investigated natural convection heat transfer and fluid flow in a cavity filled with nanofluid using the Lattice Boltzmann method. Reference [17] studied the effects of magnetic field to the natural convection flow in a rectangular cavity. Reference [18] investigated the MHD natural convection with sinusoidal temperature distribution on the right wall. Later, [19] extended the work by [18] when sinusoidal temperature distribution is applied on both side walls. Reference [20] examined the natural convection in a square enclosure that is filled with nanofluid and influenced by a magnetic field. Reference [20] reported that the values of Hartmann number and Rayleigh number influence the heat transfer performance for the increment of solid volume fraction. Very recently, [21] investigated the effects of magnetic field on natural convection of nanofluid in a three-dimensional cubic cavity that is heated from below. It is reported that the applied magnetic field results in opposite force to the flow direction which leads to dragging the flow and then reduces the convection by reducing the velocities.

Reference [22] investigated combined convection flow in a two-dimensional square cavity filled with an electrically conducting fluid. Reference [22] also studied the effects of magnetic field and internal heat generation or absorption to the opposing and aiding flow. It was found that the presence of magnetic field strongly affected the heat transfer and flow behaviors. Very recently, the significance of both internal heating or absorption and magnetic field to the fluid flow and heat transfer rate in a lid-driven cavity is studied by [23]. Reference [23] showed that, by increasing the strength of magnetic field, the average Nusselt number dropped significantly. In addition, the effect of heat generation or absorption became insignificant for forced convection region. Then, [24] reported on the conjugate effect of Joule heating and magnetic field in an obstructed lid-driven cavity to the flow and heat transfer characteristics. It is found that the obstacle strongly affected the thermal field at forced convection regime and the flow field at mixed convection regime.

References [25, 26] studied the effects of different heating locations with the presence of magnetic field in a lid-driven cavity to the flow and heat transfer rate. They reported that heating location and existence of magnetic field have significant role in the heat transfer rate and fluid flow behavior. Hydromagnetic effects on mixed convection in a lid-driven cavity are studied by [27, 28]. Reference [27] investigated the effects with sinusoidal temperature on both side walls while [28] investigated them with sinusoidal corrugated bottom surface cavity. Both studies reported that magnetic field plays important role in the convection heat transfer rate and fluid flow behaviors. Later, [29] extended the work of [27] and studied the effects of both magnetic field and inclination angle on the mixed convection heat transfer and flow behaviors. The results showed that heat transfer rate is enhanced by increasing either the Hartmann number or inclination angle. References [30, 31] studied magnetohydrodynamic mixed convection in a lid-driven cavity with linearly heated wall. Very recently, [32] investigated the effects of magnetohydrodynamic mixed convection over an isothermal circular cylinder in the presence of an aligned magnetic field. The problems of mixed convection of a nanofluid in a differentially heated lid-driven cavity with the presence of magnetic field is investigated by [33, 34]. Using different type of nanofluid, both researches reported that the heat transfer rate increased with an increase of the Reynolds number but it decreased with an increase of the Hartmann number. Then, [35] executed a numerical investigation of mixed convection in a nanofluid-filled cavity under the influence of magnetic field. Reference [35] compared the results obtained from the Maxwell method and modified Maxwell method. The results showed that heat transfer is higher based on the modified Maxwell method and the heat transfer mechanism is strongly dependent on the strength of magnetic field. Recently, [36] reported on mixed convection heat transfer of nanofluid flow in a vertical channel with sinusoidal walls under magnetic field effect. It is found that the average Nusselt number and Poiseuille number increased with an increase of the Hartman number. Numerical investigation of magnetic field on mixed convection and entropy generation of nanofluid in a trapezoidal enclosure was done by [37]. The results showed that, with imposing the magnetic field and enhancing it, the nanofluid convection and the strength of flow decrease and the flow tend toward natural convection and finally toward pure conduction. Thus, for all of the considered Reynolds numbers and volume fractions, the average Nusselt number decreased with an increase of the Hartmann number.

Based on the literature review, no work has been dealt on mixed convection flow and heat transfer in a lid-driven square cavity with the presence of magnetic field. Thus, the main objective of present work is to examine the influence of Hartmann number and the Richardson number on the characteristics of mixed convection flow inside a square cavity having the top lid moving with uniform speed.

#### 2. Mathematical Modelling

Two-dimensional square cavity is illustrated in Figure 1. The width and height of the cavity are denoted as . Both of the vertical walls are insulated while the bottom and top walls are maintained at constant and different temperature, and , respectively, such that . Furthermore, the top lid is assumed to have a constant speed that slides in the positive direction of -axis. A uniform magnetic field of strength is applied in the horizontal direction normal to the adiabatic wall. The induced magnetic field due to the motion of the electrically conducting fluid is neglected in comparison to the external magnetic field. Moreover, the imposed and induced electrical fields are assumed to be negligible. The Joule heating of the fluid and the effect of viscous dissipation are also negligible. The working fluid is assumed to be Newtonian, incompressible, unsteady, and laminar flow. The gravitational force acts vertically downward. It is assumed that the thermophysical properties of the fluid are constant except the density in the body force term of the momentum equations which varies according to the Boussinesq approximation. Hence, based on the aforementioned assumptions, the dimensional forms of the continuity, momentum, and energy equations are as follows [25]:where and are the coordinate directions and is the time. The variables , , , and are the velocity components of the fluid in the - and -directions, the pressure, and the temperature, respectively. The parameters , , , , and are the fluid thermal expansion coefficient, the gravity, the electrical conductivity, the kinematic viscosity, and the density, respectively. Parameter is the thermal diffusivity, is the thermal conductivity, is the specific heat capacity, and is the magnetic induction coefficient. The initial and boundary conditions of the problem are as follows: