A highly accurate bound-preserving phase field method for incompressible two-phase flows

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In this paper, we propose a phase-field-based spectral element method by solving the Navier–Stokes/Cahn–Hilliard equations for incompressible two-phase flows. With the use of the Newton–Raphson method for the Cahn–Hilliard equation and the time-stepping scheme for the Navier–Stokes equation, we construct three constant (time-independent) coefficient matrixes for the solutions of velocity, pressure, and phase variable. Moreover, we invoke the modified bulk free energy density to guarantee the boundness of the solution for the Cahn–Hilliard equation. The above strategies enhanced computation efficiency and accurate capture of the interfacial dynamics. For the canonical tests of diagonal motion of a circle and Zalesak's disk rotation, the lowest relative errors for the interface profile in contrast to the published solutions highlight the high accuracy of the proposed approach. In contrast to our previous work, the present method approximately produces only one tenth relative errors after one rotation cycle but saves 27.2% computation cost. Furthermore, we note that the mobility parameter adopted appears to produce convergent solutions for the phase field but the distribution of the chemical potential remains divergent, which thereby results in diverse coalescence processes in the two merging droplets example. Therefore, a criterion for the choice of the mobility parameter is proposed based on these observations, i.e., the mobility adopted should ensure the convergence solution for the chemical potential. Finally, the rising bubble is presented to verify the proposed method's versatility under large density (1000) and viscosity contrasts (100), and its advantage in efficiency over previous solver is manifested by 44.9% savings in computation cost.