On the role of control windows in continuum dynamics

Abstract

The general role of control windows is investigated by an assessment of the fundamentals of continuum dynamics in the spacetime context. The momentum rate of change of a continuous body is evaluated with reference to a control window travelling in the spacetime trajectory manifold. The resulting formula involves the rate of change of the momentum in the control window and the rate of outflow of the momentum across the control window boundary. An effective tool for investigating challenging dynamical problems is thus available, and some interesting examples, taken from early and recent contributions, are illustrated. Among these, motions of chains falling under the action of gravity, object of several investigations in the past two centuries, are revisited. Around the middle of the nineteenth century, the problem was inserted as a daring task in the Mathematical Tripos at Cambridge. The choice of a special control window, named the skeleton, in motion with the solid case, provides the proper methodology for the formulation of solid–fluid interaction problems, with applications to turbines, jets, rockets and sprinklers. As a further significant application of travelling control windows, a variational formulation of conservation of mass is developed and shown to yield the notion of mass flow vector field, with applications in problems of fluid flow through a porous solid.