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The Reynolds transport theorem occupies a central place in fluid dynamics,providing a generalized integral conservation equation for the transport of anyconserved quantity within a fluid, and connected to its correspondingdifferential equation. Recently, a new generalized framework was presented forthis theorem, enabling parametric transformations between positions on amanifold or in a generalized coordinate space, exploiting the underlyingmultivariate Lie symmetries associated with a conserved quantity. We examinethe implications of this framework for fluid flow systems, within an Eulerianposition-velocity (phase space) description. The analysis invokes a hierarchyof five probability density functions, which by convolution are used to definefive fluid densities and generalized densities appropriate for differentspaces. We obtain 11 formulations of the generalized Reynolds transport theoremfor different choices of the coordinate space, parameter space and density,only the first of which is known. These are used to generate 11 tables ofintegral and differential conservation laws applicable to these systems, foreight important conserved quantities (fluid mass, species mass, linearmomentum, angular momentum, energy, charge, entropy and probability). Thesesubstantially expand the set of conservation laws for the analysis of fluidflow and dynamical systems.

New Classes of Conservation Laws Based on Generalized Fluid Densities and Reynolds Transport Theorems