Effect of thermal radiation on conjugate natural convection flow of a micropolar fluid along a vertical surface

Abstract

This article investigates the behavior of conjugate natural convection over a finite vertical surface immersed in a micropolar fluid in the presence of intense thermal radiation. The governing boundary layer equations are made dimensionless and then transformed into suitable form by introducing the non-similarity transformations. The reduced system of parabolic partial differential equations is integrated numerically along the vertical plate by using an implicit finite difference Keller-box method. The features of fluid flow and heat transfer characteristics for various values of micropolar or material parameter, $K$, conjugate parameter, $B$, and thermal radiation parameter, $R_d$, are analyzed and presented graphically. Results are presented for the local skin friction coefficient, heat transfer rate and couple stress coefficient for high Prandtl number. It is found that skin friction coefficient and couple stress coefficient reduces whereas heat transfer rate enhances when the micro-inertia parameter increases. All the physical quantities get augmented with thermal radiation.

Introduction

Conjugate heat transfer problems involve the interaction of convective fluid and conduction through the bounding wall. Such type of problems are quite different from traditional problems of boundary-layer flow (for example see [1]) since the heat transport rate or thermal boundary conditions are a part of problem’s solution rather than being defined as in conventional articles. Conjugate heat transfer problems are important in a situation when analysis of heat transfer is done for extended geometries and the thermal boundary conditions are imposed only at the edge of the surface. The interaction among convection and conduction in adjacent boundary-layer flows and solid surface, respectively, shows interesting characteristics of heat transfer rate. Such problems were initially discussed by Martynenko and Sokovishin [2]. Later, Pozzi and Lupo [3] investigated the coupling of conduction with laminar natural convection along a flat plate and obtained a series solution. They also examined the behavior of forced convection regime [4]. The analytical solutions for a stationary conjugate heat transport problem focusing on conduction in heat-generating slab and the natural convection in participating liquid were reported by Kelleher and Yang [5]. Similarly, conjugate heat transfer analysis has also been done by various authors, for instance see [6], [7], [8], [9], [10], [11], in which various convection mechanisms were considered, induced by the objects such as a cylinder or a flat plate.

Keeping in view the industrial needs, the dynamic attributes of fluid are needed. Among these, micropolar fluids are used extensively which are based on the concept of microcontinuum. Besides the standard velocity field, micropolar fluid theory introduces the microrotation vector and the gyration parameter $j$ in order to explore the kinematics of microrotations which results in a comprehensive mathematical model for describing the non-Newtonian behavior observed in numerous natural fluids such as animal blood and artificial liquids like polymeric fluids. Initially, Eringen [12] proposed the micropolar fluids theory to study the important class of fluids, for instance liquid crystals, polymeric fluids, suspension and colloidal solutions, animal blood and lubricants. Comprehensive study of ‘simple microfluent’ media was presented by [12] in which the author discussed the special case when gyrations were small and micro-deformation rates were linear. The theory of microfluid, discussed by Eringen [12], was also further extended by Eringen [13] which includes the heat conduction and heat dissipation effects. Since then, theory of micropolar fluids has generated a lot of interest and several different physical situations have been studied as reported in [14], [15], [16], [17], [18], [19], [20], [21].

Calculation of combined heat transfer of thermal radiation and natural convection becomes indispensable for many manufacturing and engineering processes taking place in high temperature environments, for example in furnace designing, casting and levitation, glass production, gas turbines, steel rolling, fins designing, nuclear power plants, space vehicles, electronic equipments, and satellites [22]. In this regard, several studies have been reported in which radiative heat transfer was observed with the help of Rosseland diffusion approximation [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. In view of industrial applications, the influence of thermal radiation for the conjugate heat transfer problem is proposed in the present study. Therefore, consideration has been given to the conjugate natural convection boundary-layer flow over a finite vertical surface immersed in the micropolar fluid in the presence of radiant heat flux. The governing equations are made dimensionless and then further converted into a non-conserved form via non-similarity transformations. These equations are examined in the entire regime and the non-similar transport equations for momentum, angular momentum and energy are solved for the whole length of the finite plate by using implicit finite difference Keller-box method. Computational results are expressed in terms of rate of heat transfer, skin friction coefficient, velocity profile and temperature profile.