Flow in a partly liquid-filled, rotating, horizontal cylinder is analysed by means of finite-element numerical simulation. Of alternative methods for locating the free surface, a boundary collocation scheme with Newton-Raphson iteration converges. This method forces the residual in the normal-stress boundary condition to zero at a finite set of points on the liquid meniscus. Solutions of the steady, two-dimensional, incompressible flow problem show circumferential variation of the liquid-film thickness and corresponding pressure and velocity fields, including recirculation zones. The complications of an unknown meniscus location and a nonlinear normal-stress condition when surface tension is significant are illustrated. The finite-element method proves an effective and convenient tool for such flows, in which inertial, gravitational, pressure, viscous and capillary forces are all important.