A contact line treatment for second-order conservative phase field methods

In the field of two-phase flow modeling, a longstanding focus has been the treatment of the contact line, which is the intersection of a fluid-fluid interface with a solid wall boundary. Contact lines play an important role in the fluid mechanics of a variety of applications, including droplet impact, superhydrophobic surfaces, flow in porous media, microfluidics, and inkjet printing. Various contact line treatments have been developed for various two-phase flow methods. In this presentation, we describe a treatment for the Conservative Diffuse Interface (CDI) method, which is a second-order, conservative phase field (PF) method and which offers advantages over higher-order methods like Cahn-Hilliard in certain aspects, including better bound preservation and milder timestep restrictions. The key difficulty in modeling contact lines with CDI is that its second-order spatial operator admits a single PF boundary condition (BC) at wall boundaries, preventing simultaneous enforcement of the commonly used contact angle BC and a zero regularization flux BC. We propose a solution in which the PF BC only enforces zero regularization flux and has no effect on the contact angle, and contact line physics are then incorporated using a slip BC on the mixture velocity, which is motivated by the generalized Navier boundary condition. We also describe the near-wall discretizations for the capillary force and the regularization term, which are important for accurate simulations. We present numerical results demonstrating the ability of the treatment to accurately model both static and moving contact lines.