A variational approach to the contact angle dynamics of spreading droplets

The modeling and the numerical representation of the contact line between a two-phase interface and a solid surface are still open problems from the physical, mathematical and numerical point of view. This paper deals with the numerical simulation of the spreading of a single droplet impacting over horizontal dry surfaces. A new variational approach to study the droplet spreading is presented by coupling an interface front-tracking algorithm to the single-fluid finite element formulation of the incompressible Navier-Stokes equations which are solved on a fixed mesh. Standard no-slip boundary conditions near the contact line lead to a singular behavior that in the variational approach is removed by introducing a generalized boundary condition which is the sum of a dissipation term and the dynamical contact angle law. By chang- ing the intensity of the dissipation a large number of boundary conditions around the contact point are modeled, ranging from no-slip to free-slip. Since the impact is over horizontal surfaces, axisymmetric solu- tions are investigated with high mesh resolutions. A very precise implementation of the capillary force with a volumetric extension of the curvature has been adopted. We have considered a Lagrangian front-tracking method to advect the interface. The marker representing the contact point is simplyadvected by the computed velocity at the boundary without the need to extrapolate the vector field from the interior and to enforce locally mass-conservation. The model has been tested for the impact and the spreading of a droplet on solid substrates with a different wettability at low Reynolds numbers where the inertial, the viscous and the surface tension forces are all important. A number of droplet impacts with different outcomes, ranging from simple deposition to partial and complete rebound, have been reproduced. However, our simulations indicate that the formulas suggested in the literature for the dynamical contact angle should be modified to simulate a broad class of experiments