Effect of magnetic field on the buoyancy and thermocapillary driven convection of an electrically conducting fluid in an annular enclosure

We investigate the long-wave Marangoni instability in a binary-liquid layer in the limit of a small Biot number B. The surface deformation and the Soret effect are both taken into account. It is shown that the problem is characterized by two distinct asymptotic limits for the disturbance wave number k, k∼B1∕4 and k∼B1∕2, which are caused by the action of two instability mechanisms, namely, the thermocapillary and solutocapillary effects. The asymptotic limit of k∼B1∕2 is novel and is not known for pure liquids. A diversity of instability modes is revealed. Specifically, a new long-wave oscillatory mode is found for sufficiently small values of the Galileo number.

Long-wave Marangoni instability in a binary-liquid layer with deformable interface in the presence of Soret effect: Linear theory

The study of convection in bounded geometries is often facilitated by consideration of model problems which require little numerical computation. Such problems typically result in a sixth-order ordinary differential system which is non-selfadjoint in the inner product associated with the space of square integrable functions in the sense of Lebesgue. Using theorems of Karlin (1971) and Naimark (1967), we prove the simplicity of the eigenvalues and completeness of the eigenspace. We illustrate some of the simplicity results with numerical calculations. The use of completeness in future problems is explained. We also consider the extended Graetz problem with homogeneous reaction and heterogeneous reaction at the wall. Similar results are shown here. Some of the above-mentioned results can be obtained by other means, but we provide this analysis as an interesting alternative.

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Chu Xiao-Lin

Papers

Free surface convection in a bounded cylindrical geometry: note on the role of surface adsorption and solute accumulation at the air-liquid interface