  We propose an efficient numerical method for the simulation of multi-phase
flows with moving contact lines in three dimensions. The mathematical model
consists of the incompressible Navier-Stokes equations for the two immiscible
fluids with the standard interface conditions, the Navier slip condition along
the solid wall, and a contact angle condition which relates the dynamic contact
angle to the normal velocity of the contact line (Ren et al. (2010)
\cite{Ren10}). In the numerical method, the governing equations for the fluid
dynamics are coupled with an advection equation for a level-set function. The
latter models the dynamics of the fluid interface. Following the standard
practice, the interface conditions are taken into account by introducing a
singular force on the interface in the momentum equation. This results in a
single set of governing equations in the whole fluid domain. Similar to the
treatment of the interface conditions, the contact angle condition is imposed
by introducing a singular force, which acts in the normal direction of the
contact line, into the Navier slip condition. The new boundary condition, which
unifies the Navier slip condition and the contact angle condition, is imposed
along the solid wall. The model is solved using the finite difference method.
Numerical results, including a convergence study, are presented for the
spreading of a droplet on both homogeneous and inhomogeneous solid walls, as
well as the dynamics of a droplet on an inclined plate under gravity.
