Thermocapillary convection in a rectangular cavity with a deformable interface

A finite-volume method and a boundary element technique are used to compute two-dimensional (2D) thermocapillary convection in a rectangular cavity. The free surface is deformable and the deviations from a flat interface, h, are assumed small in the finite-volume calculations as appropriate for small Capillary (Ca). Two asymptotic approaches are employed; the first is an expansion valid as Ca tends towards 0 and the second assumes h tends towards 0, retaining Ca explicitly as a parameter. On the other hand, the boundary elements approach can be used with O(1) surface deformations. These three different formulations are used to calculate thermocapillary motions in fluids with small Prandtl numbers of about 0.01, Ca less than 0.05, aspect ratios (width/height) 1, 2 and 4, and various values of the Maragoni number (Ma). The same solutions are calculated with these different approaches and are found in good agreement for values of Ca up to 0.05. All solutions calculated are steady, which is both in agreement and disagreement with recently published results by different authors employing different numerical techniques.