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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 number

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. Thermal convective problems have long been studied extensively since the pioneering experimental and theoretical works of Bénard [

Most of the previous studies were concerned with convection in Newtonian fluids. However, much less work has been done on convection in non-Newtonian fluids such as the micropolar fluids. The theory of micropolar fluids, as developed by Eringen [

There has also been much less work focused on the effect of nonuniform temperature gradient on convection. Friedrich and Rudraiah [

Thermal convection in micropolar fluids has also been studied. Rudraiah and Siddheshwar [

In this paper, we shall investigate the combined effects of Dupont et al. [

We wish to examine the stability of a horizontal layer of quiescent micropolar fluid of thickness

The governing equations for the problem are the continuity equation, conservation of momentum, conservation of angular momentum, conservation of energy, and magnetic induction, compare [

The surface tension

The infinitesimal perturbations

Substituting (

Eliminating

Equations (

Following [

^{-}) denotes dimensional quantities,

Reference steady-state temperature gradients.

Model | Reference steady-state | ||||

temperature gradient | |||||

1 | Linear | 1 | 1 | 0 | 0 |

2 | Inverted parabolic | 0 | 0 | ||

3 | Cubic 1 | 0 | 0 | 1 | |

4 | Cubic 2 | 0.6 | 0 | 0.34 |

Equations (

First we multiply (

The critical Marangoni number

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The problem of Marangoni convection in a micropolar fluid in the presence of a cubic basic state temperature profile and a vertical magnetic field has been studied theoretically. The results indicate that it is possible to delay the onset of convection by the application of a cubic basic state temperature profile. In addition, the presence of a magnetic field is to suppress Magnetomarangoni convection and hence leads to a more stable system. As expected, the presence of the micron-sized suspended particles adds to the stabilizing effect of the magnetic field.

The authors acknowledge the financial support received under the Grant UKM-GUP-BTT-07-25-173 and from Universiti Kuala Lumpur (UniKL MICET).