The Numerical Solution of Free -

This thesis describes a new approach for the solution of two-dimensional, timedependent, surface-tension-driven free-surface ows involving domains of arbitrary shape that may undergo large changes in shape during the course of a problem. Both Stokes and Navier-Stokes problems are considered, a mixed Lagrangian-Eulerian nite element formulation being employed for the latter. All meshes are generated automatically using a Delaunay mesh generator, the only user input required being the speci cation of the initial free-surface shape. Very few constraints are placed on the shape of the initial domain and arbitrarily large deformations of the domain are permitted. A key feature of the new method is its ability to dynamically re ne and de-re ne the free-surface discretisation as and when necessary to maintain an accurate representation of the free surface, as is essential for surface-tension-driven problems. Full implementation details are included. Semi-implicit time integration schemes are employed for both Stokes and NavierStokes problems, the resulting systems of linear equations being solved by the conjugate residual method preconditioned using high-quality, thresholded, incomplete LU factorisations. A novel scheme for the automatic selection of the maximum time step size that ensures free-surface stability is described. A number of challenging problems are considered. First a Stokesow problem with a known analytic solution is employed to con rm that the expected rates of convergence in the solution are obtained. Next the Stokesow evolution of a lm of viscous uid on a rotating cylinder is investigated, the time-dependent case being modelled for the rst time. Illustrations of the large free-surface deformations leading up to load shedding are presented. In addition, the unexpected existence of apparently stable oscillatory solutions is reported for certain con gurations. Finally the axisymmetric oscillations of droplets at low Reynolds numbers (Re 100) are considered. i