High Marangoni number convection in a square cavity

We consider the steady thermocapillary motion in a square cavity with a top free surface in the absence of gravitational forces. The cavity is heated from the side with the vertical boundaries isothermal while the horizontal boundaries are adiabatic. The relative change in the surface tension is very small, i.e., an appropriate capillary number tends to zero, so that the free surface is assumed to remain flat at leading order. A finite difference method is employed to compute the flow field. Numerically accurate solutions are obtained for a range of Prandtl numbers and for Reynolds numbers as high as 5 x 1000. Surface deflections are computed as a domain perturbation for small capillary number. In addition, asymptotic methods are used to infer the bounary layer structure in the cavity, in the limit of large values of the Reynolds number Re and the Marangoni number Ma.