Flow of an infinite fluid strip down an inclined plane: Contact line stability

We present a computational study on the flow of a fluid strip down an inclined plane. Unlike the commonly considered case of constant film thickness far behind the contact line, this flow involves a finite cross section and therefore it does not admit a traveling wave solution; instead, the base flow is time- dependent. Consequently, the equations that governs the evolution of both the base state and of the perturbation must be solved simultaneously. The computations show that, 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 time-dependent. The wavelength of the mode with maximum amplitude decreases with time until it reaches an asymptotic value, $\lambda_{max}$. We explore the dependence of $\lambda_{max}$ on the cross sectional fluid area, $A$, and on the inclination angle of the plane, $\alpha$. For the considered small $A$, corresponding to small Bond numbers, we find that the dependence of $\lambda_{max}$ on $A$ is in good agreement with experimental data (Gomba et al, Phys. Rev. E 71, 016304, 2005). This dependence differs from the one observed for films characterized by much larger cross sectional areas and Bond numbers.