Preamble

Upon modelling molecules as point masses, volumetric densities ρw of mass and pw of momentum are defined as local spatial averages of molecular masses and momenta using a weighting function w which, while possessing certain essential features, is otherwise unspecified and general. Partial (time) differentiation of ρw yields the continuity equation (2.5.16) in which the velocity field vw ≔ pw/ρw. The physical interpretations of ρw,pw and vw depend crucially upon the choice of w. Several physically distinguished classes of weighting function are discussed. Emphasis is placed upon a particular class because the corresponding interpretations of the mass density and velocity fields, and of the boundary, associated with any body are particularly simple. The conceptual problems C.P.1, C.P.2, and C.P.3 listed in Section 3.8 are addressed and completely resolved.

Weighted Averages and the Continuity Equation

The mass density ρ(x, t) at a given location x (a geometrical point) and time t is a local measure of ‘mass per unit volume’. The key questions here are ‘What mass?’ and ‘What volume?’

The mass of any given body of matter derives ultimately from that of its constituent fundamental discrete entities (i.e., electrons and atomic nuclei). While any such fundamental entity could be modelled as a point mass whose location is that of its mass centre, for the purposes of this chapter we adopt a molecular viewpoint. Specifically, we choose here to regard a material system (or body) ℳ to be a fixed, identifiable set of (N, say) molecules modelled as point masses.