Marangoni instability of a liquid film flow with viscous dissipation

A linear stability analysis of a thin liquid film flowing over a plate is performed. The plate is considered as impermeable and adiabatic. The upper surface of the film is assumed to be a free boundary with a non-negligible surface tension, characterized by a Robin thermal boundary condition. The thermoconvective instability is generated by the interplay between the heating due to viscous dissipation and the temperature-dependent surface tension at the free boundary. A basic parallel flow, arbitrarily oriented, is assumed and the basic temperature profile is determined analytically. In order to investigate the linear stability of the system, the normal mode method is employed. A system of ordinary differential equations defining an eigenvalue problem is thus obtained. The case of longitudinal rolls, where the base flow velocity is parallel to the axis rolls, is solved both analytically and numerically. Other possible inclinations of the base flow are investigated by means of a numerical procedure based on combining the Runge-Kutta and the shooting methods.