Stability study of a constant-volume thin film flow

We study the stability of a constant volume of fluid spreading down an incline. In contrast to the commonly considered flow characterized by constant fluid flux, in the present problem the base flow is time dependent. We present a method to carry out consistently linear stability analysis, based on simultaneously solving the time evolution of the base flow and of the perturbations. The analysis is performed numerically by using a finite-difference method supplemented with an integral method developed here. The computations show that, after a short transient stage, imposed perturbations travel with the same velocity as the leading contact line. The spectral analysis of the modes evolution shows that their growth rates are, in general, time dependent. The wavelength of maximum amplitude, λmax, decreases with time until it reaches an asymptotic value which is in good agreement with experimental results. We also explore the dependence of λmax on the cross sectional fluid area A, and on the inclination angle α of the substrate. For considered small A 's, corresponding to small Bond numbers, we find that the dependence of λmax on A is in good agreement with experimental data. This dependence differs significantly from the one observed for the films characterized by much larger A 's and Bond numbers. We also predict the dependence of λmax on the inclination angle α.