A Theoretical Study of Instabilities at the Advancing Front of Thermally Driven Coating Films

A thin liquid coating can spread vertically beyond the equilibrium meniscus position by the application of a temperature gradient to the adjacent substrate. So called super-meniscus films experience a surface shear stress which drives flow toward regions of higher surface tension located at the cooler end of the substrate. The Marangoni stresses responsible for this spreading process can also be used to coat horizontal surfaces rapidly and efficiently. Experiments in the literature have shown that in either geometry, the advancing front can develop a pronounced ridge with lateral undulations that develop into long slender rivulets. These rivulets, which prevent complete surface coverage, display a remarkable regularity in height, width, and spacing which suggests the presence of a hydrodynamic instability. We have performed a linear stability analysis of such thermally driven films to determine the most dangerous wavenumber. Our numerical solutions indicate the presence of an instability at the advancing front of films which develop a sufficiently thick capillary ridge. Our results for the film thickness profiles and spreading velocities, as well as the wavenumber corresponding to the most unstable mode, compare favorably with recent experimental measurements. An energy analysis of the perturbed flow reveals that the increased mobility in the thickened portions of the films strongly promotes unstable flow, in analogy with other coating processes using gravitational or centrifugal forces.