An inverse design framework of mesoscopic approaches for two-phase flows

This paper presents a unified inverse design framework for a mesoscopic model of two-phase flows based on the phase-field method. Traditional mesoscopic approaches often require case-specific formulations for different phase-field evolution equations (e.g., Cahn–Hilliard, Allen–Cahn, and their variants), leading to fragmented model development. To address this, we propose a generalized formulation for the order parameter equation that encapsulates both conservative and non-conservative forms. By predetermining the equilibrium distribution functions and inversely designing the source terms in the Boltzmann equations via Chapman–Enskog analysis, we derive constraints that ensure recovery of macroscopic governing equations for mass, momentum, and interface evolution. This framework accommodates diverse phase-field models within a single numerical structure, enhancing flexibility, consistency, and design efficiency. Numerical results demonstrate the validity of the proposed inverse design framework. The approach offers a robust foundation for future extensions to broader fluid dynamics applications.