Boundary integral equation solution of viscous flows with free surfaces

Summary solutions of the biharmonic equation governing steady two-dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.An iterative modification of the classical BBIE is presented which is able to solve a large class of (nonlinear) viscous free surface flows for a wide range of surface tensions. The method requires a knowledge of the asymptotic behaviour of the free surface profile in the limiting case of infinite surface tension but this can usually be obtained from a perturbation analysis. Unlike space discretisation techniques such as finite difference or finite element, the BBIE evaluates only boundary information on each iteration. Once the solution is evaluated on the boundary the solution at interior points can easily be obtained.