Marangoni instability of a thin liquid film heated from below by a local heat source

We consider the motion of a liquid film falling down a heated planar substrate. Using the integral-boundary-layer approximation of the Navier-Stokes/energy equations and free-surface boundary conditions, it is shown that the problem is governed by two coupled nonlinear partial differential equations for the evolution of the local film height and temperature distribution in time and space. Two-dimensional steady-state solutions of these equations are reported for different values of the governing dimensionless groups. Our computations demonstrate that the free surface develops a bump in the region where the wall temperature gradient is positive. We analyse the linear stability of this bump with respect to disturbances in the spanwise direction. We show that the operator of the linearized system has both a discrete and an essential spectrum. The discrete spectrum bifurcates from resonance poles at certain values of the wavenumber for the disturbances in the transverse direction. The essential spectrum is always stable while part of the discrete spectrum becomes unstable for values of the Marangoni number larger than a critical value. Above this critical Marangoni number the growth rate curve as a function of wavenumber has a finite band of unstable modes which increases as the Marangoni number increases.