Continuation in a parameter - Experience with viscous and free surface flows

The results of modifications in continuation methods applied to obtain solutions to the Navier-Stokes systems of equations for incompressible, two-dimensional, steady flows are reported. It is shown that parameter continuation permits prediction of accurate, initial estimates for iterative processing of nonlinear finite difference and finite element equations of motions. The new parameter steps are derived from values of the preceding parameter steps. The accuracy of the estimates is ensured with appropriate choices of the step size. The continuation predictor/iterative corrector is demonstrated to trace the branches of parameter space along which steady flow states are found, and techniques are available for tracing multiply branching paths. The techniques are applied to solving the Navier-Stokes equations for flow through a rotating square channel, the formation of a falling liquid curtain, and gyrostatic equilibria of rotating cylindrical drops.