Effect of Non-Uniform Temperature Gradient and Magnetic Field on Marangoni and Benard-Marangoni Convection

The problem of thermal convection in a fluid layer driven by either buoyancy (Bénard) or thermocapillary (Marangoni) effects has recently been assumed importance in material processing. In this study, the problems of Marangoni and Bénard-Marangoni convection in a horizontal fluid layer are theoretically considered. The fluid layer is bounded from below by a rigid boundary and above by a non-deformable free surface. A linear stability analysis is applied to the problem, and the effect of non-uniform temperature profiles and magnetic field are examined. The critical Marangoni numbers are obtained for free-slip and isothermal, and no-slip and adiabatic lower boundary with adiabatic temperature on upper free surface. Six non-uniform basic temperature profiles which are linear, inverted parabola, parabola, step function (superposed two-fluid layer), piecewise linear (heated from below) and piecewise linear (cooled from above) are considered. The eigenvalues are obtained and solved using single-term Galerkin expansion procedure. The influence of various parameters such as Chandrasekhar number and thermal depth on the convection has been analysed. Finally, we showed that the inverted parabola is most stabilizing basic temperature distribution, and the step function is the most destabilizing basic temperature distribution. We have also proved that magnetic field suppressed Marangoni and Bénard-Marangoni convection.

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