Contact line instability of a liquid rivulet partially wetting an inclined plane*

We analyze the stability of a liquid rivulet of cross section, $A$, positioned across a plane with inclination angle, $\alpha$. The liquid partially wets the substrate with a static contact angle, $\theta_0$, when the substrate is horizontal. The contact line stability is studied using the lubrication approximation and with a slip model. Both normal and parallel components of gravity are included A static solution exists for small $\alpha$'s and its linear stability is considered. We use an pseudo-spectral Chebyshev method with a combination of basis functions that automatically satisfies the conditions at the contact lines. We analyze the effects of $A$, $\theta_0$ and $\alpha$ on the predictions of the model, such as stability regions, the maximum growth rate and the behavior of most unstable perturbation. Experiments with silicone oils spreading on a coated glass substrate are considered for a number of different fluid volumes and $\alpha$'s. We find a good agreement between the wavelength of maximum growth predicted by the model and the experimental average distance between drops.