Finite-element method for time-dependent incompressible free surface flow

We present a finite-element method for time-dependent incompressible free surface fluid flow problems described by the Navier-Stokes equations. The elements chosen have dimensions in both space and time, and the resulting system of equations is block-tridiagonal and lends itself to solution by standard techniques. In the present article we restrict our attention to two-dimensional problems although three-dimensional problems may be solved by a straightforward generalization. The method is essentially an implicit time stepping technique and therefore stable even for relatively large time steps. With this choice of elements, the method is completely adaptive to the changing nature of the solution. An iterative procedure is used to find the position of the free surface; this procedure is found to be rapidly convergent determining accurately the shape of the free surface within a few iterations. Numerical results are given for the problem of entrainment of fluid by a vertically moving plate, which has applications to the chemical engineering problems of the free coating of metals. We also consider the problem of circulation flow in a rectangular channel.