Film flow down an inclined plane over a three-dimensional obstacle

The low-Reynolds-number flow of a liquid film down an inclined plane wall over a particle attached to the wall is considered. The effect of the particle is described on the assumption that the free surface suffers only a minor disturbance from its basic flat state, and the disturbance velocity over the free surface is small, even though the particle size may not be small compared to the unperturbed film thickness. The problem is formulated using the boundary integral method for Stokes flow to describe the surface velocity, the deformation of the film surface, and the distribution of the traction over the particle surface. Results are presented for small particles following the earlier asymptotic analysis of Pozrikidis and Thoroddsen, and for moderate-sized particles with semispherical, spherical, and semispheroidal shapes. The simulations reveal that, in all cases, the free surface causes an upstream hump and a horseshoe type of deformation downstream whose intensity depends on the Bond number and is largely insensitive to the specific particle shape.