On Recent Developments in the Leading Edge Problem: Self-Similar Solutions to Momentum and Energy Equations of a Flat Plate

TL;DR

This paper reviews recent advances in the leading edge problem, focusing on self-similar solutions to momentum and energy equations, discussing solution methods, boundary conditions, and approximate solutions for heat transfer.

Contribution

It introduces a self-similar approach to the leading edge problem, deriving coupled PDEs, and explores solution strategies including approximations and boundary condition effects.

Findings

Derived a fourth-order momentum PDE and a second-order energy PDE.

Discussed solution methods including approximations and characteristics.

Provided an approximate heat transfer coefficient solution.

Abstract

We provide an overview of the leading edge problem in this paper. We have used a self-similar function having a dependence on both the self-similar variable $\eta$ and Reynold's number R to covert the momentum and energy equations into a fourth-order, non-linear partial differential equation (PDE) and a second-order, non-linear PDE respectively. Attempts have been made to solve the energy equation in a variety of ways, which include solving the PDE approximating the terms of the order $\mathcal{O}(R^{2})$ and solving the PDE via the method of characteristics, but mostly being able to solve the energy PDE sans solving the momentum PDE. The complexities involved in solving the momentum PDE have been discussed and plausible approximate solutions have been given. The importance of boundary conditions and how they influence the solution to the energy PDE has been discussed. We have also shown how the energy PDE can be defined as a well-posed hyperbolic initial-boundary value problem in the leading edge. We conclude the paper by showing an approximate solution to the heat transfer coefficient and plot its characteristic behavior.