Marangoni effect on the instability of non-uniformly heated falling liquid films

The linear and nonlinear instability of a liquid film flowing down an inclined, nonuniformly heated plate is studied. A constant temperature gradient is imposed along the plate. Based on the solution of a long-wave evolution equation of Benney type, the existence of supercritically equilibrated finite-amplitude waves is predicted. The calculated parameters of small- and finite-amplitude waves axe strongly influenced by thermocapillarity. In the case of linear decrease of plate temperature, the thermocapillary force has a stabilizing effect, while for linear increase in plate temperature, thermocapillarity destabilizes the film.