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A Cahn-Hilliard-Bingham Model: Numerical Analysis and Simulation of Phase Separation in Viscoplastic Fluids

Abstract:

This paper focuses on the analysis and numerical solution of a coupled Cahn-Hilliard-Bingham model, designed to describe a two-phase incompressible flow system involving viscoplastic fluids. The Bingham component of the model is regularized using the Huber approach to address the nonsmoothness caused by the yield stress, while the Cahn-Hilliard equation remains unaltered. The resulting model is shown to satisfy the energy dissipation law, aligning with the physical principles governing the system. We extend the fully finite element scheme from [9] to solve the coupled system with matched densities, demonstrating its effectiveness in handling the nonlinearities arising from the viscoplastic stress and phase separation. Numerical experiments validate the efficiency and stability of the proposed scheme, accurately capturing the interplay between phase separation and yield-stress dynamic. The results highlight the robustness of the approach across various test cases.