Energy stable finite element strategy for simulating spreading, sliding and rolling flow dynamics of viscoelastic droplets

Abstract

Moving mesh finite element method (FEM) for simulating dynamics of viscoelastic droplets on inclined surfaces is proposed. Viscoelasticity is incorporated into the governing system employing the Oldroyd–B constitutive model. The supporting (inclined) surface is allowed to have non–homogeneous properties incorporated into the mathematical model through the generalized Navier boundary conditions (GNBC). The droplet motion (sliding and/or rolling) is handled by employing arbitrary Lagrangian Eulerian (ALE) framework. Energy balance is derived from the governing system and it is a starting point in the derivation of the numerical scheme. The overall numerical strategy is designed in such a way that a counterpart of the (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system which, in turn, guarantees the stability of the scheme in the energy norm. The framework proposed is very general and encapsulates both two and three dimensional scenarios. Numerical studies in this work focus primarily on the three dimensional scenarios since such are significantly more challenging and seem to be much scarcer in the literature. The newly proposed numerical strategy is validated on several examples to confirm the theoretical predictions. The role of viscoelasticity in the overall droplet dynamics is briefly investigated and the behaviors of Newtonian and non–Newtonian droplets are compared.