  The Reynolds transport theorem occupies a central place in fluid dynamics,
providing a generalized integral conservation equation for the transport of any
conserved quantity within a fluid, and connected to its corresponding
differential equation. Recently, a new generalized framework was presented for
this theorem, enabling parametric transformations between positions on a
manifold or in a generalized coordinate space, exploiting the underlying
multivariate Lie symmetries associated with a conserved quantity. We examine
the implications of this framework for fluid flow systems, within an Eulerian
volume-velocity (phase space) description. The analysis invokes a hierarchy of
five probability density functions, which by convolution are used to define
five fluid densities and generalized densities appropriate for different
spaces. We obtain 11 formulations of the generalized Reynolds transport theorem
for different choices of the coordinate space, parameter space and density,
only the first of which is known. These are used to generate 11 tables of
integral and differential conservation laws applicable to these systems, for
eight important conserved quantities (fluid mass, species mass, linear
momentum, angular momentum, energy, charge, entropy and probability). These
substantially expand the set of conservation laws for the analysis of fluid
flow systems.
