Approximate solutions for Dorodnitzyn's gaseous boundary layer limit formula

TL;DR

This paper derives an approximate solution for Dorodnitzyn's gaseous boundary layer limit formula, based on Oleinik's no back-flow condition, providing insights into the boundary layer's behavior in compressible flows.

Contribution

It introduces an approximate analytical solution for Dorodnitzyn's boundary layer limit formula under specific conditions, extending previous theoretical results.

Findings

Derived an approximate velocity solution involving boundary layer parameters.

Connected the solution to the small parameter limit as the domain height-to-length ratio approaches zero.

Provided a formula applicable to compressible boundary layer analysis.

Abstract

Oleinik's \emph{no back-flow} condition ensures the existence and uniqueness of solutions for the Prandtl equations in a rectangular domain $R\subset \mathbb{R}^2$. It also allowed us to find a limit formula for Dorodnitzyn's stationary compre\-ssible boundary layer with constant total energy on a bounded convex domain in the plane $\mathbb{R}^2$. Under the same assumption, we can give an approximate solution $u$ for the limit formula if $|u|<\!\!<\!\!<1$: \[u(z)\cong \delta * c * \left[z+\frac{6}{25}\cdot \frac{1}{2i_0} \cdot \frac{4U^2}{3}z^4\right]+o(z^5),\] that corresponds to an approximate horizontal velocity component when a small parameter $\epsilon$ given by the quotient of the maximum height of the domain divided by its length tends to zero. Here, $c>0$, $\delta$ is the boundary layer's height in Dorodnitzyn's coordinates, $U$ is the \emph{free-stream} velocity at the upper boundary of the domain, and $T_0$ is the absolute surface temperature.