THE DEFORMATION OF A LIQUID-FILM FLOWING DOWN AN INCLINED PLANE WALL OVER A SMALL PARTICLE ARRESTED ON THE WALL

The shape of the free surface of a liquid film flowing down an inclined plane wall over a particle captured on the wall is studied in the asymptotic limit where the size of the particle is much smaller than the film thickness, and the Reynolds number is vanishingly small. Using the boundary integral method for Stokes flow, the problem is formulated in terms of a system of four linear integral and differential equations for the elevation of the free surface and the three components of the velocity at the free surface, and is solved using a semi-iterative method. The results demonstrate that the particle causes a marked deformation of the free surface upstream right before the particle, and a surface corrugation resembling a surface wake downstream behind the particle. The detailed features of the deformation and the disturbance flow are affected strongly by the geometry of the particle, surface tension, and the inclination of the wall.