Incompressible two-phase flow: Diffuse interface approach for large density ratios, grid resolution study, and 3D patterned substrate wetting problem

Abstract

Abstract We explore numerical simulations of incompressible and immiscible two-phase flows. The description of the fluid-fluid interface is captured via a diffuse interface approach. The two phase fluid system is represented by a coupled Cahn–Hilliard Navier–Stokes set of equations. We discuss challenges and solution approaches to solving this coupled set of equations using a SUPG stabilized finite element formulation, especially in the case of a large density ratio between the two fluids. Specific features that enabled efficient solution of the equations include: (i) a conservative form of the convective term in the Cahn–Hilliard equation which ensures conservation of mass of both fluid components; (ii) a continuous formula to compute the interfacial surface tension which results in lower requirement on the spatial resolution of the interface; and (iii) a four-step fractional scheme to decouple pressure from velocity in the Navier–Stokes equation, which provides an efficient time discretization. These are integrated with standard streamline-upwind Petrov–Galerkin (SUPG) stabilization to avoid spurious oscillations. We subsequently perform exhaustive numerical tests to determine the minimal resolution of spatial discretization required and showcase the robustness of our framework. We illustrate the accuracy of the framework using the analytical results of Prosperetti for a damped oscillating interface between two fluids with various density contrasts as well as a benchmark Rayleigh–Taylor instability problem. We also showcase the framework by modeling the crown ring effect during droplet impact, and the spread of a droplet on a wetting surface. Finally, we explore the affects of surface patterns on the droplet spreading process. Specifically, we investigate formations of wetting spots and air entrapment on grooved and checker-patterned surfaces. These results have implications for the design of tuned wettability surfaces for the manufacture of thin film devices.