Front and Back Instability of a Liquid Film on a Slightly Inclined Plate

We study the transverse instability of a liquid ridge on horizontal and inclined substrates using a film evolution equation based on a long wavelength approximation. The equation incorporates an additional pressure term--the disjoining pressure--accounting for the efefctive interaction of the film with the substrate. On a horizontal substrate the dominant instability mode is varicose, but may turn into a zigzag mode on a slightly inclined substrate depending on the inclination angle and ridge volume. For larger angles or volumes the instabilities at the front and back decouple. The linear stability properties of a one-dimensional ridge-like state are studied in detail, and an energy analysis is used to demonstrate that the disjoining pressure provides the dominant instability mechanism at both the front and the back, while the body force is responsible for the main differences between these two instabilities. An amplitude equation for the time evolution of perturbations with small transverse wavenumbers is derived that predicts correctly the linear crossing of the most dangerous eigenvalues at zero wavenumber in the inclined case, in contrast to the situation on a horizontal substrate