Orateur
Description
Multiphase flows arise in a wide range of applications, including materials science, petrochemical engineering, and the oil industry. Despite their importance, the accurate numerical simulation of multiphase flows remains challenging due to the complex behaviour of the governing equations and the need for stable long-time computations. The key challenges in simulating multiphase flow include the complexity of the equations, computational resources, accuracy, and stability. To address these challenges, we propose a new Besse- type scheme for the Cahn-Hilliard equation (BSCH) and its coupling with the Navier-Stokes equation (BSCH-NS). The proposed scheme is second order in time and linearly implicit, requiring only the solution of a linear system at each time step. The scheme satisfies the energy dissipation law without imposing any condition on time or mesh size, providing unconditional stability. The key objective of the work is to develop a method that accurately preserves the energy law at the discrete level, and also, a method that shows improved long-time behaviour compared to the existing auxiliary variable method. We study the deformation of an initially rectangular bubble with two immiscible fluids using the proposed scheme, with the interface described by the level curve . The interface evolves under surface tension, driving the phase separation dynamics. The proposed method is compared with the existing auxiliary variable methods to investigate the long-time behaviour of the numerical solution. Numerical experiments are implemented in FreeFEM and MATLAB. The proposed method provides an improved numerical solution that is accurate and stable over a long-time computation, contributing to a better prediction of multiphase flow behaviour in industrial applications.
Keywords: Cahn–Hilliard equation, Navier–Stokes equations, Besse-type scheme, two-phase flow, energy dissipation law, unconditional stability, finite element scheme.