Approximate solutions for Dorodnitzyn's gaseous boundary layer limit formula

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.