Directed fluid sheets

Abstract

This paper is concerned mainly with incompressible inviscid fluid sheets but the incompressible linearly viscous fluid sheet is also considered. Our development is based on a direct formulation using the two dimensional theory of directed media called Cosserat surfaces . The first part of the paper deals with the formulation of appropriate nonlinear equations (which may include the effects of gravity and surface tension) governing the two dimensional motion of incompressible inviscid media for two categories, namely those ( a ) for two dimensional flows confined to a plane perpendicular to a specified direction and ( b ) for propagation of fairly long waves in a stream of variable initial depth. The latter development is a generalization of an earlier direct formulation of a theory of water waves when the fixed bottom of the stream is level (Green, Laws & Naghdi 1974). In the second part of the paper, special attention is given to a demonstration of the relevance and applicability of the present direct formulation to a variety of two dimensional problems of inviscid fluid sheets. These include, among others, the steady motion of a class of two-dimensional flows in a stream of finite depth in which the bed of the stream may change from one constant level to another, the related problem of hydraulic jumps, and a class of exact solutions which characterize the main features of the time-dependent free surface flows in the three dimensional theory of incompressible inviscid fluids.