Thermocapillary instabilities in the liquid layer with two deformable surfaces

A liquid layer with two deformable free surfaces subject to thermocapillary instabilities is studied. The critical Marangoni number Mac is determined as a function of the surface tension number S, the Prandtl number Pr and the smallest wave number k0 limited by the length of the layer. The results show that Mac always increases with both S and k0. When S is relatively small, the preferred mode is either the streamwise wave or the spanwise stationary mode. However, as S is increases, it changes to the hydrothermal waves while Mac tends to a constant. The streamwise wave at small Pr is a type of surface wave instability (SWI), without any rolls inside the layer. Other preferred modes belong to the convective instability (CI), which consists of counter-rotating rolls within the layer and small fluctuations on the surface. Energy analysis shows that the CI obtains most of its power from Marangoni forces, while the main energy source of the SWI is the work done by the shear stress caused by surface deformation. Additionally, Mac often increases with the Biot number Bi, while the opposite case appears in the SWI at small Pr.