Numerical simulation of viscous flow with a free surface

A key feature of any approximate method for free surface flows is the way in which the free surface is represented. It is convenient to represent the surface as an expansion in locally defined polynomial trial functions. The expansion is then used in the finite element method to solve the nonlinear, elliptic Laplace-Young equation. Solutions are presented for the three-dimensional meniscus which forms between cylinders of circular and square cross section standing on an infinite square array in a pool of liquid. An iterative scheme is devised for the more difficult, and previously unsolved, class of free boundary meniscus problems which occur when the solid cross-sectional shape depends on elevation, and solutions for cones and spheres in a pool are given. The finite element method is then used to solve the Navier-Stokes equations for the flow of a liquid film inside a horizontal rotating cylinder.