New Insights into the Mathematical and Physical Treatment of the Flat Plate Boundary Layer Theory

Abstract

A solution can effectively challenge the core principles of integral calculus, particularly when it comes to ordinary differential equations, as illustrated by Blasius's notable theory of the flat plate boundary layer. A third-order ordinary differential equation is solved using four boundary conditions. While this model is well known for illustrating various aspects of uniform flow over a flat plate, it is classically associated with another flow, supposedly easier to analyze and treat, produced by a towed plate in the same fluid at rest. This association stems from the mistaken belief that both flows are adjusted by changing their velocity reference frame. The classical solution method of the modified system of differential equations does not guarantee the uniqueness of the solution. It estimates the first nonzero term of the series solution as a constant and an unknown, while ignoring the Eulerian reference frame when addressing the continuity equation. A deeper analysis of the issue at hand has paved the way for developing a different model. This new approach acknowledges that the first nonzero term of the series solution is a function of viscosity. This understanding enables the derivation of both nondimensional velocity components that are consistent with Prandtl's boundary layer concepts. With these elements, it becomes easy to address local skin-friction effects and determine the exact vortex equation, thereby enabling exploration of the dynamic actions of both physical quantities in the boundary layer.