The Fundamental Hydrodynamical Equations and Shock Conditions for Gases

Abstract

1. Fundamental dynamical assumptions. A fluid (liquid or gas) will be considered from our present standpoint as forming a continuous medium. Let the motion of the fluid be represented by equations of the a a -a form x f (X,t) where the x and x are coordinates of the same rectangular system and t denotes the time. A particle, initially at the point x, will be located at the point x at time t in accordance with these relations. It will be assumed that the equations xa = <ii;,t) have a unique inverse at any time t and it suffices for our purpose to suppose that the functions f (x,t) are continuous and have continuous partial derivatives through the third order with respect to the initial coordinates -and the time t. ITe velocity components ua(x,t) are then defined in the usual manner by the equations ua = afa/Ut and these quantities will be continuous in xi and t and will have continuous derivatives through the second order with respect to these variables. Now consider a finite volume V of the fluid and denote by S the boundary or surface of V. We suppose that V is a continuous (1,1) map of a cube and that S is a closed regular surface1; these conditions will be retained under motions of the fluid particles by which V and S are composed in consequence of the above assumptions on the functions f . All volumes and their boundaries in the following discussion are assumed implicitly to be of this type. We make the following dynamical assumptions. (ao) The rate of change of the momentum of V parallel to any fixed direction is equal to the component parallel to this direction of the total external forces acting on V. (/) The rate of change of the angular momentum of V about any fixed line is equal to the moment of the external forces acting on V about