In this work, we extend the discrete unified gas-kinetic scheme (DUGKS) [Guo et al., Phys. Rev. E 88, 033305 (2013)] to continue two-phase flows. In the framework of DUGKS, two kinetic model equations are used to solve the quasi-incompressible phase-field governing equations [Yang et al., Phys. Rev.E 93, 043303 (2016)].One is for the Chan-Hilliard (CH) equation and the other is for the Navier-Stokes equations. The DUGKS can correctly recover the quasi-incompressible phase-field governing equations through the Chapman-Enskog analysis. Unlike previous phase-field-based LB models, the Courant-Friedricks-Lewy condition in DUGKS is ajustable which can increase numerical stability. Furthermore, with the finite-volume formulation the model can be easily implemented on non-uniform meshes which can improve numerical precision. The proposed model is validated by simulating a stationary drop, layered Poiseuille flow, rising bubble and Rayleigh-Taylor instability and comparing with the quasi-incompressible lattice Boltzmann method (LBM). Numerical results show that the method can track the interface with high accuracy and stability. The model is also capable of dealing with a wider range of viscosity and density ratios than the quasi-incompressible lattice Boltzmann model. The present model is a promising tool for numerical simulation of two-phase flows. keywords: Multiphase flow; Finite-volume method; Discrete unified gas-kinetic scheme; Lattice Boltzmann method; Non-uniform gird IntroductionRecently, modeling multiphase flows based on kinetic descriptions has received particular attention. In kinetic schemes, intermolecular interactions that determine phase behaviors are incorporated at the mesoscopic level into a discretized Boltzmann equation such that the complex macroscopic fluid behaviour, such as phase separation or coalescence, is a result of intermolecular interactions. This feature brings some distinct advantages, such as free of interface tracking. The most popular kinetic method for two-phase flows may be the lattice Boltzmann Equation (LBE) method, which solves the discrete velocity Boltzmann equation *

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In this paper, a lattice Boltzmann model with the single-relaxation-time model for the Cahn-Hilliard equation (CHE) is proposed. The discrete source term is redesigned through a third-order Chapman-Enskog analysis. By coupling the Navier-Stokes equations, the time-derivative term in the source term is expressed as the relevant spatial derivatives. Furthermore, the source term on the diffusive time scale is also proposed though recovering the macroscopic CHE to third order.The model is tested by simulating diagonal motion of a circular interface, Zalesak's disk rotation, circular interface in a shear flow, a deformation field and the problem of Rayleigh-Taylor instability.It is shown that the proposed method can track the interface with high accuracy and stability. For the complex flow, the source term on the diffusive time scale should be considered for capturing the interface correctly.

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