Author:
Longuet-Higgins Michael S.
Abstract
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.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Reference12 articles.
1. Gilbarg, D. 1952 Unsteady flow with free boundaries Z. angew. Math. Phys. 3,34–42.
2. Longuet-Higgins, M. S. 1972 A class of exact, time-dependent, free-surface flows J. Fluid Mech. 55,529–543.
3. Gilbarg, D. 1960 Jets and cavities. Handbuch der Physik , vol. IX (ed. S. Flugge ),pp.311–445.Springer. (See especially pp. 15–18, 350–810.)
4. Lamb, H. 1932 Hydrodynamics ,6th edn.Cambridge University Press.
5. Garabedian, P. 1953 Oblique water entry of a wedge Comm. Pure Appl. Math. 6,157–167.
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