Fluid Densities Defined from Probability Density Functions, and New Families of Conservation Laws

Abstract

The mass density, commonly denoted ρ(x,t) as a function of position x and time t, is considered an obvious concept in physics. It is, however, fundamentally dependent on the continuum assumption, the ability of the observer to downscale the mass of atoms present within a prescribed volume to the limit of an infinitesimal volume. In multiphase systems such as flow in porous media, the definition becomes critical, and has been addressed by taking the convolution [ρ](x,t)=∫∫∫V(x,t)w(r,t)ρ(x+r,t)dV(r,t), involving integration of a local density ρ(x+r,t) multiplied by a weighting function w(r,t) over the small local volume V(r,t), where [·] is an expectation and r is a local coordinate. This weighting function is here formally identified as the probability density function p(r|t), enabling the construction of densities from probabilities. This insight is extended to a family of five probability densities derived from p(u,x|t), applicable to fluid elements of velocity u and position x at time t in a fluid flow system. By convolution over a small geometric volume V and/or a small velocimetric domain U, these can be used to define five corresponding fluid densities. Three of these densities are functions of the fluid velocity, enabling a description of fluid flow of higher fidelity than that provided by ρ(x,t) alone. Applying this set of densities within an extended form of the Reynolds transport theorem, it is possible to derive new families of integral conservation laws applicable to different parameter spaces, for the seven common conserved quantities (fluid mass, species mass, linear momentum, angular momentum, energy, charge and entropy). The findings considerably expand the set of known conservation laws for the analysis of physical systems.