Meshfree phase-field modeling of three-phase flow using smoothed particle hydrodynamics with differential reproducing kernels and artificial compressibility

This paper presents a meshfree approach for phase-field (PF) modeling of three-phase flow using Smoothed Particle Hydrodynamics (SPH) with Differential Reproducing Kernels (DRK). In this method, the constructed approximating weight functions and their gradients satisfy the reproducibility condition. The computational domain is discretized into particles and a diffuse interface is assumed between three immiscible fluid phases. The propagation of the diffuse interface is handled by solving a Cahn-Hilliard (CH) equation based on the Helmholtz free energy functional minimization. We employ an artificial compressibility method based on the General Pressure Equation (GPE), exhibiting a stabilizing dissipative pressure term for solving the Navier–Stokes (NS) equations. As such, we depart from the density-based Equation of State (EOS) typically used in weakly-compressible SPH. The proposed PF-SPH method for solving the Cahn-Hilliard-Navier–Stokes (CHNS) system of equations was found to be particularly effective in handling the non-trivial surface tension effects at the fluid interfaces, where more than two immiscible fluid phases co-exist. Additionally, particle shifting was found to be critical in ensuring successful construction of the DRK weight functions. We present the mathematical formulation and discuss the results of numerical simulations conducted in both 2D and 3D, as well as in both Lagrangian and Eulerian frameworks, demonstrating the versatility and applicability of the proposed method.