The Leading Edge Problem in Fluid Mechanics

TL;DR

This paper investigates the mathematical properties and solutions of the leading edge problem in fluid mechanics, focusing on self-similar equations, their integrability, symmetries, and semi-analytical solutions.

Contribution

It introduces a comprehensive analysis of self-similar momentum equations, including Painlevé tests, Lie symmetries, and semi-analytical solutions using homotopy perturbation methods.

Findings

Painlevé test results on integrability of MODE and MPDE

Lie symmetry analysis identifying infinitesimal operators

Semi-analytical solutions for Falkner-Skan and MODE equations

Abstract

The self-similar momentum ordinary differential equation (MODE) and the self-similar partial differential equation (MPDE) have been derived and the investigation of the integrability of the MODE and the MPDE has been done by performing Painlev\'e test. A detailed discussion of the leading order behavior of the MODE and the MPDE has been presented with the latter being analyzed for the cases in which terms of increasing orders of Reynolds number have been considered. We have provided a brief introduction to Lie point symmetries and have found the Lie infinitesimal operator which when acts on the MPDE to order $\mathcal{O}(R)$ satisfies the Lie symmetry condition. Explicit calculations and expressions for the Lie prolongation terms have been presented. We have also investigated the integrability of various self-similar equations that arise from the generalized self-similar equation for different values of constants $\alpha_{1,2,3}$. Foundational work on transitional boundary solutions has been presented and transition solutions have been found via application of a junction condition at the leading edge-trailing edge boundary domain. A detailed discussion of semi-analytical solutions via the homotopy perturbation method is presented. We find semi-analytical solutions to the Falkner-Skan equation and the MODE by considering a Taylor series expansion as the initial approximation. An algorithmic scheme that involves consideration of a multi-dimensional Taylor expansion as the initial approximation to the MPDE has been presented.