Vortex jumps

It has been supposed so far that the velocity and vorticity fields are continuous and have continuous derivatives. In a real viscous fluid, these assumptions are generally regarded as appropriate, and the velocity field is assumed analytic everywhere, except possibly at an initial instant o, when the speed of boundaries changes in a non-analytic manner. However, if we go to the the limit of vanishingly small viscosity and consider ideal fluids, to which the Helmholtz laws apply, the mathematics allows non-analytic behaviour and we cannot assert on physical grounds that only continuous fields should be considered. Velocity or vorticity fields with singularities or discontinuities are indeed of considerable importance. The singularities are, of course, not arbitrary and must be consistent with integral forms of the Euler equations or equivalently with the conservation of mass, momentum and energy. In particular, the dynamical constraint that pressure is continuous across a surface of discontinuity must be satisfied unless there are also singularities in the external force fields. We suppose in this chapter that the density is uniform and put it equal to unity unless explicitly stated otherwise. Also, external forces are supposed conservative unless non-conservative forces are explicitly introduced.

In the past three decades, the study of vortices and vortex motions – which originated in Helmholtz's great paper of 1858, ‘Uber Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen’ (translated by Tait [1867]), and continued in the brilliant work of Lord Kelvin and others in the nineteenth century, and Prandtl and his Göttingen school in the first half of this century – has received continuing impetus from problems arising in physics, engineering and mathematics. As aptly remarked by Küchemann [1965], vortices are the ‘sinews and muscles of fluid motions’. Aerodynamic problems of stability, control, delta wing aerodynamics, high lift devices, the jumbo jet wake hazard phenomenon, among other concerns, have led to a myriad of studies. Smith [1986] reviews some of this work. The realisation that many problems involving interfacial motion can be cast in the form of vortex sheet dynamics has stimulated much interest. The discovery (rediscovery?) of coherent structures in turbulence has fostered the hope that the study of vortices will lead to models and an understanding of turbulent flow, thereby solving or at least making less mysterious one of the great unsolved problems of classical physics. Vortex dynamics is a natural paradigm for the field of chaotic motion and modern dynamical system theory. It is perhaps not well known that the father of modern dynamics and chaos wrote a monograph on vorticity (Poincaré [1893]).

In this chapter we consider stresses in a fluid. We start by setting the forces resulting from these stresses in their proper perspective, i.e., in relation to body forces, together with which they raise accelerations. This results in a set of momentum equations, which are needed later.

We then consider the relations between the various stress components and proceed to inspect stress in fluids at rest and in moving fluids.

The Momentum Equations

In this section we establish relations between body forces, stresses and their corresponding surface forces and accelerations. Newton's second law of motion is used, and the results are the general momentum equations for fluid flow.

Consider a system consisting of a small cube of fluid, as shown in Fig. 2.1. A system is defined in classical thermodynamics as a given amount of matter with well-defined boundaries. The system always contains the same matter and none may flow through its boundaries.

As a rule a fluid system does not retain its shape, unless, of course, the fluid is at rest. This does not prevent the choice of a system with a certain particular shape, e.g., a cube. The choice means that imaginary surfaces are defined inside the fluid such that at the considered moment they enclose a system of fluid with a given shape. A moment later the system may have a different shape, because the shape is not a property of the fluid or of the location.