Self-similar, time-dependent flows with a free surface

A simple derivation is given of the parabolic flow first described by John (1953) in semi-Lagrangian form. It is shown that the scale of the flow decreases liket−3, and the free surface contracts about a point which lies one-third of the way from the vertex of the parabola to the focus. The flow is an exact limiting form of either a Dirichlet ellipse or hyperbola, as the timettends to infinity. Two other self-similar flows, in three dimensions, are derived. In one, the free surface is a paraboloid of revolution, which contracts liket−2about a point lying one-quarter the distance from the vertex to the focus. In the other, the flow is non-axisymmetric, and the free surface contracts liket−5. The parabolic flow is shown to be one of a general class of self-similar flows in the plane, described by rational functions of degreen. The parabola corresponds ton= 2. Whenn= 3 there are two new flows. In one of these the scale varies ast12/7and the free surface has the appearance of a trough filling up. In the other, the free surface resembles flow round the end of a rigid wall; the scale varies ast−4·17.