Viscoelastic film flows over an inclined substrate with sinusoidal topography. I. Steady state

We consider the steady flow of a viscoelastic liquid film over an inclined wall with sinusoidal corrugations of arbitrary wavelength and depth. 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 deformations of the liquid film. We perform a thorough parametric analysis to investigate the combined effects of elasticity, inertia, and capillary and viscous forces on the characteristics of the steady flow. The results of our simulations indicate that fluid elasticity suppresses interfacial deformation at low flow rates, whereas at moderate values of Re it enhances the deformation considerably. In the latter case, elastic forces may even give rise to the formation of a static hump and a cusp at the free surface, the size of which increases with the relaxation time of the liquid. The resonance of the liquid film with the substrate undulations is also enhanced by shear thinning. Interestingly, it is predicted that under certain conditions the transition to the inertia regime is not smooth and a hysteresis loop arises, which is the signature of an abrupt change of the film shape, since its high deformations cannot be sustained. Additionally, we have performed calculations for a wide range of different geometrical characteristics of the substrate. We find that viscoelastic effects become more pronounced in the case of long-wavelength wall undulations, while for substrates with short wavelengths the effect of shear thinning is less significant due to the presence of vortices inside the corrugations.