Convective heat transfer in an electrically conducting micropolar fluid at a stretching surface with uniform free stream

Abstract

The problem of convective heat transfer for a micropolar fluid in the presence of uniform magnetic field is investigated near an isothermal-stretching sheet. This has been done under the effect of buoyancy force. The sheet is linearly stretched in the presence of a uniform free stream of constant velocity and temperature. Numerical solutions for the governing boundary layer equations for a range of values of magnetic, vortex viscosity and microrotation parameters, also, Grashoff number obtained by applying an efficient numerical technique based on shooting method. The effects of these parameters examined on the velocity of fluid, temperature distribution and angular velocity of microstructures as well as the coefficient of heat flux and shearing stress at the plate.

Introduction

In recent years, the dynamics of micropolar fluids, originated from the theory of Eringen [1] has been a popular area of search. This theory takes into account the effect of local rotary inertia and couple stresses arising from practical microrotation. The theory is expected to provide a mathematical model for the non-Newtonian fluid behavior observed in certain man-made liquids such as polymers, colloidal suspensions, fluids with additives, suspension solutions, and animal blood, etc. This theory also capable of explaining the experimentally observed phenomena of drag reduction near a rigid body in fluids containing small amount of additives when compared with the skin friction in the same fluids without additives. Peddieson and McNitt [2] and Willson [3] have introduced the boundary layer concept in such fluid. Ahmadi [4] has studied the boundary layer flow of micropolar fluids past a semi-infinite plate. He obtained self-similar solutions when the microinertia is not constant. The theory of micropolar [1] has been extended by Eringen [5] to take into account thermal effects, i.e. heat conduction and heat dissipation and this extension has been termed as the theory of thermomicropolar fluids. Convective heat transfer through a vertical channel has been studied by Balarm and Sastary [6]. Hassanien et al. [7] studied the mixed convection boundary layer flow of a micropolar fluid along vertical slender cylinders. Heat transfer in the stagnation point of a micropolar fluid has been studied by Ramachandran and Mathur [8].

Two-dimensional boundary layer flow caused by a moving plate or a stretching sheet is of interest in manufacture of sheeting material through an extrusion process.

Rajagopal et al. [9] studied a boundary layer flow of a non-Newtonian fluid over a stretching sheet with a uniform free stream. Hady [10] studied the solution of heat transfer to micropolar fluid from a non-isothermal stretching sheet with injection.

Na and Pop [11], investigated the boundary layer flow of a micropolar fluid due to a stretching wall. Hassanien et al. studied the numerical solution for heat transfer in a micropolar fluid over a stretching sheet [12]. Desseaux and Kelson studied the flow of a micropolar fluid bounded by a stretching sheet [13]. Hassanien and Gorla [14] studied the heat transfer to a micropolar fluid from a non-isothermal stretching sheet with suction and blowing.

In all the above studies, the authors have taken the stretching sheet to be oriented in horizontal direction. However, of late, the effects of magnetic field to the micropolar fluid problem are very important. Mansour and Gorla studied the Joule-heating effects on unsteady natural convection from a heated vertical plate in a micropolar fluid [15]. Siddeshwar and Pranech investigated the magneto-convection in a micropolar fluid [16].

The physical situation discussed by Rajagopal et al. [9] is one of the possible cases. An another physical phenomenon is the case in which the difference between the surface temperature T_w and the free stream temperature T_∞, namely T_w−T_∞, is appreciably large causing the free convection currents to flow in the boundary layer. The finding of such physical phenomena would have definite bearing on the fabric, plastic and polymer industries. Hence, it is interesting to study the effects of the free convection in the presence magnetic field.

The Grashoff number represents the effects of free convection currents. Due to the presence of free convection currents, the problem is governed by the coupled non-linear partial differential equations. The mathematical formulation and solution of the problem is considered, the results for velocity, angular velocity, temperature and the rate of heat transfer are discussed. Since the free convection currents are in existence, T_w−T_∞ may be positive, zero or negative. Hence, the free convection parameter (Grashoff number) $\text{Gr}\text{[Gr}\text{α(T}{}_{\mathrm{<mtext>w</mtext>}}\text{−T}{}_{\mathrm{\infty }}\text{)]}$ assumes positive, zero or negative values. Physically, Gr>0 corresponds to heating of the fluid (or cooling of the surface), Gr<0 corresponds to cooling of the fluid (or heating of the surface), and Gr=0 corresponds to the absence of the free convection currents.

In the present work, we consider the problem of convective heat transfer for a micropolar fluid in the presence of uniform magnetic field is investigated near an isothermal-stretching sheet. This has been done under the effect of buoyancy force. The sheet is linearly stretched in the presence of a uniform free stream of constant velocity and temperature. Numerical solutions for the governing boundary layer equations for a range of values of magnetic, vortex viscosity and microrotation parameters, also, Grashoff number obtained by applying an efficient numerical technique based on shooting method. The effects of these parameters examined on the velocity of fluid, temperature distribution and angular velocity of microstructures as well as the coefficient of heat flux and shearing stress at the plate.