Conservation of mechanical energy and circulation in the theory of inviscid fluid sheets

Abstract

In the theory of thin fluid sheets, governing equations are derived with specific reference to an assumed simple kinematic structure of the flow. There is a separate set of governing equations associated with each degree of complexity of the kinematic structure, forming a hierarchy of models (Green and Naghdi [3] and Shields and Webster [8]). If one is interested in the velocity profile across the sheet, the kinematic structure can be used again to interpret the variables in the governing equations as an approximate flow. This paper is concerned with the properties of this approximate flow.

Two important consequences of the field equations (Euler's equations) in the classical, three-dimensional theory of ideal fluids are: conservation of mechanical energy, and conservation of circulation (Kelvin's theorem). The research reported herein provides a proof that mechanical energy is exactly conserved for the approximate flow in each level in this hierarchy. Two types of circulation are considered in the approximate flow: “in-sheet” circulation which is computed about circuits lying a fixed fractional distance between the top and bottom surfaces of the sheet, and “cross-sheet” circulation which is computed about circuits lying in a vertical cylindrical surface. It was found that K moments of the in-sheet circulation and K − 1 weighted moments of the cross-sheet circulation are conserved in the Kth level approximate flow.