Group Invariant Solution And Conservation Law for a Steady Laminar Axisymmetric Free Jet

Abstract

A group invariant solution for a steady laminar axisymmetric free jet is derived. The general form of the group invariant solution is first obtained by considering a linear combination of the Lie point symmetry generators of Prandtl's momentum boundary layer equation for the stream function. The ratios of the constants in the linear combination are determined from conservation of total momentum flux in the downstream direction, from the condition of finite volume flow and finite momentum at the infinitesimal circular orifice and from specification of the arbitrary additive constant in the stream function. An elementary conservation law for the partial differential equation for the stream function is derived. A relation is established between this conservation law and conservation of total momentum flux in the downstream direction by showing that the Lie point symmetry which is associated with the conservation law is the Lie point symmetry that generates the group invariant solution.