Dynamics of liquid films falling down a porous substrate in the presence of an ambient gas flow

We study the dynamics of a gravity-driven liquid film flowing over a porous substrate where the liquid-gas interface is sheared by an ambient gas flow. Using the long-wave approximation, we first derive a nonlinear evolution equation for the film thickness. Based on the evolution model, the impact of both wall permeability and gas shear stress on the linear stability, weakly nonlinear response, and fully nonlinear evolution of the liquid film are comprehensively examined. Linear stability analysis shows that wall permeability promotes instability, while the impact of external shear depends on the direction, namely, the co-current shear enhances the disturbance growth, and the countercurrent shear suppresses the instability. The weakly nonlinear analysis further reveals that the coupling between wall permeability and gas shear governs the transition between supercritical and subcritical regimes. Furthermore, fully nonlinear simulations uncover that the liquid-gas interface remains close to the initially imposed disturbance for small wavenumbers, however, disturbances with wavenumbers approaching the critical value develop finite-amplitude oscillations and may even diverge. In particular, increasing the permeability of porous wall strengthens and prolongs these oscillations; In addition, oscillations are further amplified by a co-current shear, however, suppressed by the countercurrent shear. These nonlinear behaviors are consistent with the predictions from the linear and weakly nonlinear analyses. Our findings provide new insight into how the intricate interplay between wall permeability and external gas flow shapes the evolution of thin liquid films flowing down porous substrates.