Instability of a transverse liquid rivulet on an inclined plane

This work concentrates on the stability of a viscous liquid rivulet positioned across an inclined plane under partial wetting conditions. The study is performed within the framework of lubrication approximation by employing a slip model. Both normal and parallel components of gravity are considered. We find the stability regions for given area of the cross section of the rivulet, A, plane inclination angle, α, and static contact angle, θ₀, characterizing the wettability of the substrate. For α's smaller than some critical angle, α*, a static solution exists. This solution is characterized by rear/front contact angles given by θ₀ ± δ. The linear stability analysis of this solution is performed using an efficient pseudo-spectral Chebyshev method. We analyze the effects of A, θ0, and α on the predictions of the model, such as the dominant wavelength, the maximum growth rate, and the behavior of the most unstable perturbation mode. To verify them, we also carry out experiments with silicone oils spreading on a coated glass substrate for several different fluid volumes and inclination angles. We find very good agreement between the wavelength of maximum growth rate given by the theory and the average distance between the drops after rivulet breakup. An analysis of finite size effects shows that the inclusion of normal gravity effects leads to a better agreement between theoretical and experimental results.