A class of exact, time-dependent, free-surface flows

Attention is drawn to a class of inviscid irrotational flows which satisfy the conditions at a time-dependent free surface exactly. The flows are related to the ellipsoids of Dirichlet (1860). Depending on a parameter P, the cross-section may take the form of a variable ellipse (P < 0), a hyperbola (P > 0) or a pair of parallel lines (P = 0). The elliptical case was investigated both theoretically and experimentally by Taylor (1960). The hyperbolic case (P > 0) is remarkable in that the flow develops a singularity when the angle between the asymptotes approaches a right-angle. It is suggested that this solution represents a possible instability near the crest of a standing gravity wave of large amplitude. In the intermediate case (P = 0) the solution describes an open-channel flow in which the fluid filaments are stretched uniformly in a horizontal direction. The latter flow is demonstrated experimentally.