Inertial motion of a continuum

Abstract

The Burgers equation ut+uux=νuxx represents pure inertial motion except for the effects of viscosity ν. If ν=0, this equation becomes u̇=0 and describes the inertial motion of a one-dimensional continuum until the time of formation of discontinuities, or ’’shocks’’. The study of pure inertial motion is extended to an arbitrary number of dimensions. Starting from some initial state of motion each parcel of the continuum may or may not have the intrinsic ability to form a shock, this property being a function of the symmetrical part of the velocity gradient matrix in the vicinity of the parcel. This study determines the conditions for which a parcel will, in fact, form a shock, the time that is required, and the temporal development of the full velocity gradient matrix and of coordinate invariants of the flow such as the divergence and the vorticity.