Effects of Magnetic Field and Nonlinear Temperature Profile on Marangoni Convection in Micropolar Fluid

M. N. Mahmud,R. Idris,I. Hashim

doi:10.1155/2009/748794

Abstract

The combined effects of a uniform vertical magnetic field and a nonuniform basic temperature profile on the onset of steady Marangoni convection in a horizontal layer of micropolar fluid are studied. The closed-form expression for the Marangoni numberMfor the onset of convection, valid for polynomial-type basic temperature profiles upto a third order, is obtained by the use of the single-term Galerkin technique. The critical conditions for the onset of convection have been presented graphically.

Highlights

Convective flow in a thin layer of fluid, free at the upper surface and heated from below, is of fundamental importance and a prototype to a more complex configuration in experiments and industrial processes
The convective flows in a liquid layer can be driven by buoyancy forces due to temperature gradients and/or thermocapillary Marangoni forces caused by surface tension gradients
Most of the previous studies were concerned with convection in Newtonian fluids

Summary

Introduction

Convective flow in a thin layer of fluid, free at the upper surface and heated from below, is of fundamental importance and a prototype to a more complex configuration in experiments and industrial processes. The combined effects of nonuniform temperature gradients and a magnetic field on Marangoni convection were investigated by Rudraiah et al. The work of Friedrich and Rudraiah 27 was further extended to include the effect of buoyancy by Rudraiah and Ramachandramurthy. The effects of nonuniform temperature gradients on the onset of oscillatory Marangoni and Benard-Marangoni convection in a magnetic field were analyzed in 31, , respectively. Rudraiah and Siddheshwar analyzed the effects of nonuniform temperature gradients of parabolicand stepwise-types on the onset of Marangoni convection in a micropolar fluid. Idris et al studied the effect of Dupont et al 30 cubic temperature profile on the onset of Benard-Marangoni convection in a micropolar fluid. We shall investigate the combined effects of Dupont et al 30 cubic temperature profile and a magnetic field on the onset of Marangoni convection in a micropolar fluid. Comparisons with the other polynomial-type temperature profiles normally used by previous investigators shall be undertaken

Mathematical Formulation

Solution of the Linearized Problem

N3 DΩ1 2

Discussion

Conclusion