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Group Invariant Solution ans 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.
Mathematics Subject Classification (2000): 76D25, 70G65, 76M60.
Key words: Axisymmetric free jet, Lie point symmetries, group invariant solution, conservation law.
Quaestiones Mathematicae 27(2004), 171-183.
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.
Mathematics Subject Classification (2000): 76D25, 70G65, 76M60.
Key words: Axisymmetric free jet, Lie point symmetries, group invariant solution, conservation law.
Quaestiones Mathematicae 27(2004), 171-183.
