High Marangoni number convection in a square cavity

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Abstract
The steady thermocapillary motion in a square cavity with a top free surface in the absence of gravitational forces is considered. 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 Re as high as 5×104. Surface deflections are computed as a domain perturbation for small capillary number. In addition, asymptotic methods are used to infer the boundary layer structure in the cavity, in the limit of large values of the Reynolds and Marangoni numbers. For a fixed Prandtl number Pr, it is shown that the Nusselt number, liquid circulation, and maximum vorticity are asymptotic to Re1/3, Re−1/3, and Re2/3, respectively. These results are in agreement with the computed solutions. The leading‐order solution for the free‐surface deformation is sensitive to the value of Pr. With Pr>1, the depression near the hot corner may exceed the elevation near the cold corner, while a secondary elevation may be induced near the hot corner when Pr<1.

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