An alternative method to implement contact angle boundary condition and its application in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces

We propose an alternative method to implement the contact angle boundary condition on a solid wall and apply it in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces in which the flow equations are solved by the lattice-Boltzmann method and the interface equations are solved by the finite-difference method. Using the hyperbolic tangent profile of the order parameter across an interface in phase-field theory, we were able to obtain its unknown value at a ghost point from the information at only one point in the fluid domain. This is in contrast with existing approaches relying on interpolations involving several points. The special feature allows it to be more easily implemented on immersed surfaces cutting through the grid lines. It was properly incorporated into the framework of the hybrid lattice-Boltzmann finite-difference simulation, and applied successfully for several problems with different levels of complexity. First, the equilibrium shapes of a droplet on a sphere with different contact angles and radii were studied under cylindrical geometry and a good agreement with theoretical predictions was found. Preliminary studies on a three-dimensional droplet spreading on a sphere were also performed and the results agreed well with the corresponding axisymmetric results. Second, the spreading of a two-dimensional drop on an embedded inclined wall with a given contact angle was investigated and the results matched those on a flat wall at the domain boundary under the same condition. Third, capillary filling in a cylindrical tube was studied and the speed of the interface in the tube was found to follow Washburn’s law. Fourth, a droplet impacting on a sphere was investigated and several different outcomes were captured depending on the Reynolds number, the viscosity ratio, and the wettability and radius of the sphere. Finally, the proposed method was shown to be capable of studying even more complicated problems involving the interaction between a droplet and multiple objects of different sizes and contact angles