Group-invariant solutions and optimal systems for multidimensional hydrodynamics

The group properties of the three-dimensional (3-D), one-temperature hydrodynamic equations, including nonlinear conduction and a thermal source, are presented. A subgroup corresponding to axisymmetric geometry is chosen, and the details of the construction of the one- and two-dimensional optimal systems are shown. The two-dimensional optimal system is used to generate 23 intrinsically different reductions of the 2-D partial differential equations to ordinary differential equations. These ordinary differential equations can be solved to provide analytic solutions to the original partial differential equations. Two example analytic solutions are presented: a 2-D axisymmetric flow with a P2 asymmetry and a 3-D spiraling flow.