Tsouka S., Dimakopoulos Y., Tsamopoulos J.
We consider the two dimensional, steady flow of a dilute polymer solution over a solid substrate with periodic topography under the action of a body force. We examine how the distribution of polymer is affected by flow conditions, physical properties and substrate geometry and how it affects the dynamics of the free-surface. The Mavrantzas-Beris two-fluid Hamiltonian model is used in order to account for polymer migration due to stress gradients. The model is solved via the mixed finite element method combined with a quasi-elliptic grid generation technique. Results for polymer concentration, stress and velocity fields are presented as a function of the non-dimensional parameters of the problem. The basic phenomena that appear in the case of polymer migration in undulating channels (Tsouka et al., 2014), also appear in free surface flows. Increasing the elastic stresses, increases migration of macromolecules towards the free surface, developing a polymer-depleted layer especially over the substrate maxima, which finally gives rise to “apparent” slip. Increasing the cavity steepness also enhances migration and the thickness of the depletion layer and induces strong variation in the stresses away from the substrate wall, especially in low polymer concentrations. All these phenomena cannot be captured by the homogenous Oldroyd-B model. The evaluation of the Stokes number shows that flow resistance decreases as elasticity increases, increases monotonically up to an asymptote with the capillary number, and exhibits a non-monotonic dependence on the Reynolds number. Moderate inertial effects can give rise to large deformations of the free surface, developing waves with complex shapes. If we use a similarly modified ePTT model, we observe that its inherent shear-thinning is enhanced by the stress induced migration and tends to reduce the gradients on both polymeric stresses and polymer concentration along the film.
Film flow over topography, Finite element solution, Stress-induced migration, Viscoelastic fluids