Finite element setting for fluid flow simulations with natural enforcement of capillary effects

TL;DR

This paper introduces a finite element framework for simulating capillary-driven fluid flows, naturally enforcing contact line conditions without explicit contact angle specification, and capable of handling complex 2D and 3D phenomena.

Contribution

It develops a variational formulation that naturally enforces mechanical equilibrium at interfaces and integrates Level-Set and mesh adaptation techniques for accurate interface tracking.

Findings

Successfully simulates droplet spreading, coalescence, and capillary rise.

Achieves equilibrium states with correct velocity, pressure, and contact angles.

Provides a versatile method for 2D and 3D capillary flow analysis.

Abstract

Capillary phenomena are involved in many industrial processes, especially those dealing with composite manufacturing. However, their modelling is still challenging. Therefore, a finite element setting is proposed to better investigate this complex issue. The variational formulation of a liquid-air Stokes system is established, while the solid substrate is described through boundary conditions. Expressing the weak form of Laplace's law over liquid-air, liquid-solid and air-solid interfaces, leads to a natural enforcement of the mechanical equilibrium over the wetting line, without imposing explicitly the contact angle itself. The mechanical problem is discretized by using finite elements, linear both in velocity and pressure, stabilized with a variational multiscale method, including the possibility of enrichment of the pressure space. The moving interface is captured by a Level-Set methodology, combined with a mesh adaptation technique with respect to both pressure and level-set fields. Our methodology can simulate capillary-driven flows in 2D and 3D with the desired precision: droplet spreading, droplet coalescence, capillary rise. In each case, the equilibrium state expected in terms of velocity, pressure and contact angle is reached.