MHD boundary-layer flow of a micropolar fluid past a wedge with constant wall heat flux

Abstract

The steady laminar magnetohydrodynamic (MHD) boundary-layer flow past a wedge with constant surface heat flux immersed in an incompressible micropolar fluid in the presence of a variable magnetic field is investigated in this paper. The governing partial differential equations are transformed into a system of ordinary differential equations using similarity variables, and then they are solved numerically by means of an implicit finite-difference scheme known as the Keller-box method. Numerical results show that micropolar fluids display drag reduction and consequently reduce the heat transfer rate at the surface, compared to the Newtonian fluids. The opposite trends are observed for the effects of the magnetic field on the fluid flow and heat transfer characteristics.

Introduction

The boundary-layer theory has been successfully applied to non-Newtonian fluids models and has received substantial attention during the last few decades. One of the important non-Newtonian fluids is the micropolar fluid in which the theory was first introduced by Eringen [1], [2]. This theory takes into account the microscopic effects arising from the local structure and micromotions of the fluid elements. The theory is expected to provide a mathematical model for non-Newtonian fluid behaviour, which can be used to analyze the behaviour of exotic lubricants, polymeric fluids, liquid crystals, animal blood, colloidal fluids, ferro-liquid, real fluids with suspensions, etc., for which the classical Navier–Stokes theory is inadequate. Extensive reviews of the theory and applications of micropolar fluids can be found in the review articles by Ariman et al. [3], [4] and the recent books by Łukaszewicz [5] and Eringen [6]. The micropolar fluid theory requires that one must solve an additional transport equation representing the principle of conservation of local angular momentum, as well as the usual transport equations for the conservation of mass and momentum.

Recently, Kim [7] and Kim and Kim [8] have considered the steady boundary-layer flow of a micropolar fluid past a fixed wedge with constant surface temperature and constant surface heat flux, respectively. The similarity variables found by Falkner and Skan [9] were employed to reduce the governing partial differential equations to ordinary differential equations. Unfortunately, the angular momentum equation was not correctly derived in the papers by Kim [7] and Kim and Kim [8] so that the results of these papers are inaccurate. Therefore, the objective of this paper is to improve and extend the work of Kim and Kim [8] by considering the effect of variable magnetic field on the fluid flow and heat transfer characteristics for a fixed wedge with constant surface heat flux. The effect of a transverse magnetic field on a permeable wedge placed symmetrically with respect to the flow direction in a non-Newtonian fluid has been considered by Hady and Hassanien [10]. Watanabe and Pop [11] presented the numerical results for MHD free convection flow over a wedge in the presence of a magnetic field, while Kafoussias and Nanousis [12] investigated the MHD laminar boundary-layer flow over a permeable wedge. Both of these papers considered a wedge immersed in a Newtonian fluid. Later, Yih [13] extended the work of Watanabe and Pop [14], by considering the MHD forced convection flow adjacent to a non-isothermal wedge. The former considered MHD boundary-layer flow over a flat plate in the presence of a transverse magnetic field. The effects of variable magnetic field on the fluid flow and heat transfer characteristics were considered by Cobble [15], [16], Helmy [17], [18], Chiam [19], Anjali Devi and Thiyagarajan [20] and very recently by Zhang and Wang [21], [22], Amkadni et al. [23] and Hoernel [24]. On the other hand, the existence of the similarity solutions for the case of variable magnetic field has been established by an experiment reported by Papailiou and Lykoudis [25], when they re-examined the theoretical work done by Lykoudis [26]. They found that similarity solutions exist when the intensity of the magnetic field changes with $x^{-1/4}$, where x is the coordinate measured in the direction of the flow.