Study of stability and secondary regimes of thermocapillary flow in a liquid layer under localized heating

The problem of thermocapillary convection in a thin horizontal layer of viscous incompressible fluid with a deformable free boundary under the action of space-inhomogeneous localized temperature field is considered. The system of nonlinear differential equations for temperature, surface deformation and vorticity amplitudes in the long-wave approximation is solved numerically. The Galerkin method with polynomial basic functions is used to investigate steady states and their stability to infinitesimal two-dimensional disturbances in the cases of planar and axisymmetrical heat fluxes. The dependences of a disturbance decrement on the wave number are obtained for different steady states and parameter values. The forms of eigenfunctions for the most dangerous disturbances are presented. The nonlinear behavior of the disturbances of localized one-dimensional equilibrium states is investigated by the pseudo-spectral method in the two-dimensional formulation. It is shown that there is a domain of parameters in which these states are steady. Beyond its limits, diverse variants of the nonlinear development of disturbances can be realized, which, depending on the parameter values and disturbance magnitude, could lead to formation of the localized structures of another symmetry, global cellular structures of different symmetry and oscillatory regimes of thermocapillary convection.