We propose a hybrid lattice-Boltzmann finite-difference method to simulate axisymmetric multiphase flows. The hydrodynamics is simulated by the lattice-Boltzmann equations with the multiple-relaxation-time (MRT) collision model and suitable forcing terms that account for the interfacial tension and axisymmetric effects. The interface dynamics is captured by the finite-difference solution of the convective Cahn–Hilliard equation. This method is applied to simulate a quiescent drop, an oscillating drop, a drop spreading on a dry surface and a drop accelerated by a constant body force. It is validated through comparisons of the computed results for these problems with analytical solutions or numerical solutions by other different methods. It is shown that the MRT-based method is able to handle more challenging cases than that with the single-relaxation-time collision model for axisymmetric multiphase flows due to its improved stability.

Originally, the color-gradient model proposed by Rothman and Keller (RÀK) was unable to simulate immiscible two-phase°ows with di®erent densities. Later, a revised version of the RÀK model was proposed by Grunau et al. [D. Grunau, S. Chen and K. Eggert, Phys. Fluids A: Fluid Dyn. 5, 2557 (1993).] and claimed it was able to simulate two-phase°ows with high-density contrast. Some studies investigate high-density contrast two-phase°ows using this revised RÀK model but they are mainly focused on the stationary spherical droplet and bubble cases. Through theoretical analysis of the model, we found that in the recovered NavierÀStokes (NÀS) equations which are derived from the RÀK model, there are unwanted extra terms. These terms disappear for simulations of two-phase°ows with identical densities, so the correct NÀS equations are fully recovered. Hence, the RÀK model is able to give accurate results for°ows with identical densities. However, the unwanted terms may a®ect the accuracy of simulations signi¯cantly when the densities of the two°uids are di®erent. For the simulations of spherical bubbles and droplets immersed in another°uid (where the densities of the two°uids are di®erent), the extra terms may not be important and hence, in terms of surface tension, accurate results can be obtained. However, generally speaking, the unwanted term may be signi¯cant in many°ows and the RÀK model is unable to obtain the correct results due to the e®ect of the extra terms. Through numerical simulations of parallel two-phase°ows in a channel, we con¯rm that the RÀK model is not appropriate for general two-phase°ows with di®erent densities. A scheme to eliminate the unwanted terms is also proposed and the scheme works well for cases of density ratios less than 10.

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In this work, the motion of a two-dimensional drop on a surface with given wettability gradient is studied numerically by a hybrid lattice-Boltzmann finite-difference method using the multiple-relaxation-time collision model. We incorporate the geometric wetting boundary condition that allows accurate implementation of a contact angle hysteresis model. The method is first validated through three benchmark tests, including the layered Poiseuille flow with a viscosity contrast, the motion of a liquid column in a channel with specified wettability gradient and the force balance for a static drop attached to a surface with hysteresis subject to a body force. Then, simulations of a drop on a wall with given wettability gradient are performed under different conditions. The effects of the Reynolds number, the viscosity ratio, the wettability gradient, as well as the contact angle hysteresis on the drop motion are investigated in detail. It is found that the capillary number of the drop in steady state is significantly affected by the viscosity ratio, the magnitudes of the wettability gradient and the contact angle hysteresis, whereas it only shows very weak dependence on the Reynolds number.

SUMMARYBifurcations in capillarity-driven two-phase fluid systems, due to different mobilities in phase-field models for such systems, are studied by using a lattice Boltzmann method (LBM). Specifically, two-dimensional (2D) and three-dimensional (3D) droplets on a flat wall with given wettability variations are investigated. It is found that the mobility controls the rate of diffusive relaxation of the phase field from non-equilibrium toward equilibrium, and similar to previous findings on mechanically driven two-phase systems, the mobility is closely related to the contact line velocity. For the cases investigated, different mobilities across a critical value result in fundamentally different system evolution routes and final stable equilibrium states. These results may provide some implications for phase-field study of droplet manipulations by surface wettability adjustments in microfluidics.