Elastic stability of viscoelastic liquid films flowing on a porous substrate

This paper investigates the linear stability of viscoelastic liquid films flowing down an inclined porous substrate analytically and numerically. It focuses on the Stokes flow of viscoelastic films and uncovers two unstable modes triggered by elasticity. The elastic surface mode with a long wave number is solved analytically and numerically. Our results also indicate elasticity can trigger an elasto-porous mode at small incline angle and ratio of film thickness to substrate thickness. The Oldroyd-B model is used for the constitutive relation between the strain and polymer stress. The classical Beavers–Joseph condition is applied to describe the boundary conditions at the fluid-porous interface (Beavers and Joseph, 1967). This condition represents the linear relationship between velocity gradient of fluid layer and velocity difference between two layers, ${\partial }_{z}u=\frac{\alpha }{\sqrt{\kappa }}(u-u_m)$, where $\alpha$ is the Beavers–Joseph coefficient, representing slip flow at the interface; $\kappa$ is the permeability of the porous medium. Effects of porous medium properties, including permeability and depth ratio, as well as the impact of slip flow at the interface on the unstable modes are examined.