  We present a fully Eulerian hybrid immersed-boundary/phase-field model to
simulate wetting and contact line motion over any arbitrary geometry. The solid
wall is described with a volume-penalisation ghost-cell immersed boundary
whereas the interface between the two fluids by a diffuse-interface method. The
contact line motion on the complex wall is prescribed via slip velocity in the
momentum equation and static/dynamic contact angle condition for the order
parameter of the Cahn-Hilliard model. This combination requires accurate
computations of the normal and tangential gradients of the scalar order
parameter and of the components of the velocity. However, the present algorithm
requires the computation of averaging weights and other geometrical variables
as a preprocessing step. Several validation tests are reported in the
manuscript, together with 2D simulations of a droplet spreading over a
sinusoidal wall with different contact angles and slip length and a spherical
droplet spreading over a sphere, showing that the proposed algorithm is capable
to deal with the three-phase contact line motion over any complex wall. The
Eulerian feature of the algorithm facilitates the implementation and
parallelisation. The procedure proposed here can be directly extended to impose
mixed boundary conditions for any field variables evolving over a complex
geometry, modelled with an immersed-boundary approach.
