Convection in a two-layer liquid system in a finite cylinder

This paper describes a cylindrical container of finite dimensions, filled with two resting immiscible heat-conducting liquids with a common flat interface. The side walls and base of the vessel are solid, there are no external forces, and the contact angle of the interface with the side wall of the container is $\pi/2$. The interface has a surface tension whose strength linearly depends on temperature. When one of the container bases is heated to a critical temperature, there is movement inside the vessel. When modeling takes into account the energy spent on the interface deformation. The emerging spectral problem is solved by the modified Galerkin method. For various liquids, in the case of monotonous vibrations, the dependence of the critical Marangoni number on the container size and the temperature ratio, specified on the cylinder bases, is obtained, and a perturbed motion velocity field is constructed.