On rimming flows with shocks

Abstract

We examine rimming flows, i.e. flows of a liquid film on the inside of a horizontal rotating cylinder. So far this problem has mostly been explored using the so-called lubrication approximation (LA). It was shown that, if the volume of the liquid in the cylinder exceeds a certain threshold, then a shock similar to a tidal bore appears in the lower half of the cylinder on its rising side. The position of the shock can be characterized by the polar angle θ _s, with a value between θ _s = −90° (the bottom of the cylinder) and θ _s = 0° (the horizontal direction). In this study, we examine rimming flows without the LA, by solving numerically the exact Stokes equations. It is shown that a steady solution describing a (smoothed) shock exists only if ${-60^{\circ}\lesssim\theta_{\rm s} <0 ^{\circ}}$ . Shocks with lower locations overturn, so no steady solution exists. It is also shown that smoothed-shock solutions have an oscillating structure upstream from the shock. If, however, capillary effects are taken into account, the range of θ _s where solutions overturn contracts, and if surface tension is sufficiently strong, solutions exist for all values of θ _s.