Symmetry Solutions of a Third-Order Ordinary Differential Equation which Arises from Prandtl Boundary Layer Equations

Abstract

The similarity solution to Prandtl’s boundary layer equations for two-dimensional and radial flows with vanishing or constant mainstream velocity gives rise to a thirdorder ordinary differential equation which depends on a parameter a. For special values of a the third-order ordinary differential equation admits a three-dimensional symmetry Lie algebra L ₃. For solvable L ₃ the equation is integrated by quadrature. For non-solvable L ₃ the equation reduces to the Chazy equation. The Chazy equation is reduced to a first-order differential equation in terms of differential invariants which is transformed to a Riccati equation. In general the third-order ordinary differential equation admits a two-dimensional symmetry Lie algebra L ₂. For L ₂ the differential equation can only be reduced to a first-order equation. The invariant solutions of the third-order ordinary differential equation are also derived.