Stability of free-surface thin-film flows over topography

Abstract

We consider the stability of the steady free-surface thin-film flows over topography examined in detail by Kalliadasis et al. (2000). For flow over a step-down, their computations revealed that the free surface develops a ridge just before the entrance to the step. Such capillary ridges have also been observed in the contact line motion over a planar substrate, and are a key element of the instability of the driven contact line. In this paper we analyse the linear stability of the ridge with respect to disturbances in the spanwise direction. It is shown that the operator of the linearized system has a continuous spectrum for disturbances with wavenumber less than a critical value above which the spectrum is discrete. Unlike the driven contact line problem where an instability grows into well-defined rivulets, our analysis demonstrates that the ridge is surprisingly stable for a wide range of the pertinent parameters. An energy analysis indicates that the strong stability of the capillary ridge is governed by rearrangement of fluid in the flow direction flowing to the net pressure gradient induced by the topography at small wavenumbers and by surface tension at high wavenumbers.