In this work, the motion of a two-dimensional drop on a surface with stepwise wettability gradient (WG) is studied numerically by a hybrid lattice-Boltzmann finite-difference method. We incorporate the geometric wetting boundary condition that allows accurate implementation of a contact angle hysteresis (CAH) model. The method is first validated through a series of tests that check different constituents of the numerical model. Then, simulations of a drop on a wall with given stepwise WG are performed under different conditions. The effects of the Reynolds number, the viscosity ratio, the WG, as well as the CAH on the drop motion are investigated in detail. It was discovered that the shape of the drop in steady motion may be fitted by two arcs that give two apparent contact angles, which are related to the respective contact line velocities and the relevant contact angles (that specify the WG and CAH) through the relation derived by Cox [“The dynamics of the spreading of liquids on a solid surface. Part 1. viscous flow,” J. Fluid Mech. 168, 169–194 (1986)] if the slip length in simulation is defined according to Yue et al. [“Sharp-interface limit of the Cahn-Hilliard model for moving contact lines,” J. Fluid Mech. 645, 279–294 (2010)]. It was also found that the steady capillary number of the drop is significantly affected by the viscosity ratio, the magnitudes of the WG, and the CAH, whereas it almost shows no dependence on the Reynolds number.
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