Conservative phase-field lattice-Boltzmann model for ternary fluids

In this article, we propose a conservative phase-field method for tracking the interfaces among three immiscible (ternary) fluids. We formulate a new lattice Boltzmann model to numerically solve the interface-tracking and Navier-Stokes equations for incompressible flows at high density and viscosity contrasts. Two sets of numerical studies are considered to assess the accuracy and reliability of the proposed model. In the first set, the efficacy of the interface-tracking equation for ternary fluids is probed by considering several benchmark problems, including the diagonal translation of circular interfaces, Zalesak's disk rotation, and two circular interfaces in a shear flow. The numerical findings are compared with those based on the commonly used Cahn-Hilliard model, and noticeable improvement in the accuracy and consistency of the results is achieved. In the second set of the benchmark tests, the hydrodynamic interactions are included to account for the coupling between the LB-based interface-tracking and fluid flow solvers. Several dynamic problems in ternary-fluid systems, such as the classic example of two circular drops, spreading of a liquid lens, spinodal decomposition, and Rayleigh-Taylor instability, are studied and the numerical results are found to be in good agreement with available data. Among advantages of the proposed method is that it conserves mass and does not yield the appearance of an unphysical fluid at the interfaces of other two fluids, an artifact that is common to many existing interface-tracking models for ternary fluids.