Steady viscoelastic film flow over 2D Topography: II. The effect of capillarity, inertia and substrate geometry

We examine the two-dimensional, steady flow of a viscoelastic film under the action of gravity over a substrate with periodic topographical features. We account for the rheology of the viscoelastic material using the exponential Phan-Thien and Tanner (PTT) constitutive model. The conservation equations are solved via the mixed finite element method combined with a quasi-elliptic grid generation scheme, while the viscoelastic stresses are discretized using the EVSS-G/SUPG method. Our scheme allows the computation of accurate steady-state solutions up to high values of Deborah, Reynolds and capillary numbers. We perform a thorough parametric analysis to investigate the effect of the elastic, capillary and inertia forces on the flow characteristics. Our results indicate that surface tension and elasticity affect the film closer to the location with abrupt changes of the substrate topography; the sizes of the capillary ridge before a step down and of the depression before a step up are increased and move upstream as fluid elasticity or interfacial tension increase. It is shown that under creeping flow conditions the length scale of the capillary ridge increases with De following a power law of 1/4, which can also be predicted by simple scaling arguments. Inertia has a more global effect on the film affecting larger portions of it, while in its presence the length scale of the capillary features is not affected significantly by the material elasticity. Moreover, it is shown that similarly to the case of Newtonian liquids, high inertia causes the formation of a ridge just after the step up. We also explore the effect of the geometrical characteristics of the substrate as well as its inclination angle and it is shown that the interface shape becomes more deformed as the topography appears wider, deeper or it approaches the vertical plane. (C) 2016 Elsevier B.V. All rights reserved.