Steady flow of a viscoelastic film over an inclined plane featuring periodic slits

We consider the steady flow of a viscoelastic film over an inclined plane featuring a periodic arrangement of slits, which are oriented normal to the main direction of flow. The film creates a second gas-liquid interface connecting the two hydrophilic sidewalls of a slit. This interface forms two three-phase contact lines and supports a widely varying amount of liquid under different physical and geometrical conditions. We develop a computational model and carry out detailed numerical simulations, based on the finite element method to investigate this flow. To this end, we solve the two-dimensional momentum and mass conservation equations, while employing the Phan-Thien-Tanner (PTT) constitutive model to account for the rheology of the viscoelastic material. An elliptic grid generation scheme is used to follow the large deformation of the film shape. We perform a thorough parametric analysis to investigate the combined effects of elastic, inertia, capillary and viscous forces on the characteristics of the steady flow. The results of our simulations indicate that increasing fluid elasticity decreases the two wetting lengths along each side wall of a slit. On the contrary, the wetting of the slit is enhanced by the shear-thinning of the fluid. Multiple steady solutions connected by turning points forming a hysteresis loop and transcritical bifurcations as well as isolated solution branches are revealed by pseudo-arc-length continuation. In particular, it is predicted that under certain conditions, the transition from the capillary to the inertia regime is not smooth; instead a hysteresis loop arises. This is the signature of an abrupt decrease of the film penetration with increasing flow rate since higher deformations cannot be sustained. Additionally, we have performed calculations for a wide range of the geometrical characteristics of the substrate. We find that the elastic effects of the fluid become more pronounced when the slits are closer together in their periodic arrangement and of width decreasing to become almost comparable to the capillary length of the liquid. These can lead to almost no wetting of the slit, i.e. the Cassie-Baxter state, even in a hydrophilic surface.