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.