The effect of surface tension on rimming flows in a partially filled rotating cylinder

We study the shape of the interface in a partially filled horizontal cylinder which is rotating about its axis. Two-dimensional steady solutions for the interface height are examined under the assumptions that the filling fraction is small, inertia may be neglected, and the fluid forms a continuous film covering the surface. Three different regimes of steady solutions have been reported in the literature, corresponding to limits in which the ratio of gravitational to viscous forces (as defined in the text) is small, moderate or large. In each case, solutions have only been described analytically in the limit that surface tension effects are negligible everywhere. We use analytical and numerical methods, include surface tension and study steady solutions in a regime when the ratio of gravitational to viscous forces is large. This solution comprises a fluid pool that sits near the bottom of the cylinder and a film that coats the sides and top of the cylinder, the thickness of which can be determined by Landau-Levich-Derjaguin type arguments. We also examine the effect of surface tension on the solutions in the limits of the ratio of gravity to viscous forces being moderate and small.