New Families of Conservation Laws based on Extended Formulations of the Reynolds Transport Theorem

The Reynolds transport theorem describes the transport of a conserved quantity by a fluid volume, providing all integral conservation laws of fluid mechanics. Recently, extended versions were proved for transformations of a conserved quantity in an n-dimensional manifold or coordinate space, based on general coordinates X, parameters C and tensor field V. These can be used to derive new integral conservation laws in different spaces, including for volumetric, velocimetric and velocivolumetric spaces based on the ordered triples (u,x,t), where u is the Eulerian velocity and x is position. These require different fluid densities, here labelled ρ(x,t) in volumetric space [SI units: kg m^-3], д(u,t) in velocimetric space [kg (m s^-1)^-3] and ζ(u,x,t) in velocivolumetric space [kg (m s^-1)^-3 m^-3]. Such fluid densities can be defined from their underlying probability density functions p(x|t), p(u|t) and p(u,x|t) by convolution. The extended formulation is used to derive 11 tables of conservation laws for different choices of X and C, for the eight common conserved quantities of fluid mechanics (fluid mass, species mass, linear and angular momentum, energy, charge, entropy and probability). The findings considerably expand the set of known conservation laws of fluid mechanics.