  The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental
building blocks of hydrodynamic phase-field models for multiphase fluid flow
dynamics. Due to the coupling between the Navier-Stokes equation and the
Cahn-Hilliard equation, the CHNS system is non-trivial to solve numerically.
Traditionally, a numerical extrapolation for the coupling terms is used.
However, such brute-force extrapolation usually destroys the intrinsic
thermodynamic structures of this CHNS system. This paper proposes a new
strategy to reformulate the CHNS system into a constraint gradient flow
formation. Under the new formulation, the reversible and irreversible
structures are clearly revealed. This guides us to propose operator splitting
schemes. The operator splitting schemes have several advantageous properties.
First of all, the proposed schemes lead to several decoupled systems in smaller
sizes to be solved at each time marching step. This significantly reduces
computational costs. Secondly, the proposed schemes still guarantee the
thermodynamic laws of the CHNS system at the discrete level. It ensures the
thermodynamic laws, accuracy, and stability for the numerical solutions. In
addition, unlike the recently populated IEQ or SAV approach using auxiliary
variables, our resulting energy laws are formulated in the original variables.
Our proposed framework lays out a foundation to design decoupled and energy
stable numerical algorithms for hydrodynamic phase-field models. Furthermore,
given different splitting steps, various numerical algorithms can be obtained,
making this framework rather general. The proposed numerical algorithms are
implemented. Their second-order accuracy in time is verified numerically. Some
numerical examples and benchmark problems are calculated to verify the
effectiveness of the proposed schemes.
