Stability of viscous film flow coating the interior of a vertical tube with a porous wall

The stability of the gravity-driven flow of a viscous film coating the inside of a tube with a porous wall is studied theoretically. We used Darcy's law to describe the motion of fluids in a porous medium. The Beaver-Joseph condition is used to describe the discontinuity of velocity at the porous-fluid interface. We derived an evolution equation for the film thickness using a long-wave approximation. The effect of velocity slip at the porous wall is identified by a parameter beta. We examine the effect of beta on the temporal stability, the absolute-convective instability (AI-CI), and the nonlinear evolution of the interface deformation. The results of the temporal stability reveal that the effect of velocity slip at the porous wall is destabilizing. The parameter beta plays an important role in determining the AI-CI behavior and the nonlinear evolution of the interface. The presence of the porous wall promotes the absolute instability and the formation of the plug in the tube.