Lie group analysis of Prandtl‘s two-dimensional laminar boundary layer equation: analytical and numerical solutions for scaling and non-scaling symmetries

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The Lie point symmetries of Prandtl‘s two-dimensional boundary layer equation expressed in terms of the stream function are derived. The general form of the invariant solutions and boundary conditions, which include slip, suction and blowing at the boundary, are obtained. The analytical solutions for boundary layer flow in convergent and divergent channels generated by Lie point symmetries, which are not scaling symmetries, are investigated. When an ordinary differential equation and some associated boundary conditions are invariant under a scaling transformation, the boundary value problem for the ordinary differential equation can be transformed to an initial value problem which is then solved. This is known as the non-iterative transformation method. The Blasius equation is invariant under a scaling transformation while the Falkner-Skan equation is not invariant. The Blasius and Falkner-Skan equations are ordinary differential equations derived from Prandtl‘s boundary layer partial differential equation for the stream function and describe boundary layer flow over a flat plate and wedge respectively. In the case of the Falkner-Skan equation, which is non-invariant under a scaling transformation method, a modified boundary value problem is derived which is invariant under an extended scaling group. The modified problem is then transformed to an initial value problem.
A research report submitted in partial fulfilment of the requirements for the degree Master of Science to the Faculty of Science, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, 2023
Laminar boundary, Layer equation