A falling film on a porous medium

A gravity-driven falling film on a saturated porous inclined plane is studied via a continuum approach, where the liquid and porous layers are considered as a single composite layer. Using a weighted residual technique, a two-equation model is derived in terms of the local flow rate q(x, t) and the entire layer thickness H(x, t). Its linear stability analysis has been satisfactorily compared to the results of the Orr-Sommerfeld problem. The principal effect of the porous substrate on the film flow is to displace the liquid-porous interface to an effective liquid-solid interface located at the lower boundary of the upper momentum boundary layer in the porous medium. The stability and dynamics of the film is thus only weakly affected by the presence of a permeable substrate. In both the linear and the nonlinear regimes, the spatial response of a falling film on a porous medium is not very different from that observed on an impermeable inclined wall. However, the wavy motion of the film triggers a significant exchange of mass at the liquid-porous interface.