Motion of a spherical particle in film flow

The motion of a spherical particle suspended in gravity-driven film flow down an inclined plane is considered in the limit of vanishing Reynolds and Bond numbers where the free-surface deformation is infinitesimal. Taking advantage of the axially symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and free surface and passes through the particle centre, the problem is formulated as a system of one-dimensional integral equations for the first Fourier coefficients of the unknown traction and velocity along the boundary contours in a meridional plane. It is found that the particle translational velocity scaled by the unperturbed velocity evaluated at the particle centre increases monotonically as the particle approaches the free-surface, whereas the corresponding angular velocity of rotation scaled by the unperturbed vorticity evaluated at the particle centre reaches a maximum at a certain intermediate position. The free-surface velocity vector field and deformation are displayed, the force and torque exerted on a spherical particle adhering to the wall are tabulated, and the associated flow pattern is discussed.