Example of a solution for Dorodnitzyn’s limit formula

Abstract

In the present paper, we show an example of a solution for Dorodnitzyn’s gaseous boundary layer limit formula. Oleinik’s no back-flow condition ensures the existence and uniqueness of solutions for the Prandtl equations in a rectangular domain $R\subset R^2$. It also allowed us to find a limit formula for Dorodnitzyn’s stationary compressible boundary layer with constant total energy on a bounded convex domain in the plane $R^2$. Under the same assumption, we can give an approximate solution $u$ for the limit formula if $\left|u\right|<<<1$ such that: $u\left(z\right)\cong \delta \ast c\ast \left(z+\frac{6}{25}\cdot \frac{1}{2i_0}\cdot \frac{4U^2}{3}{z}^4\right)+o\left(z^5\right),$that corresponds to an approximate horizontal velocity component when a small parameter $ϵ$ 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 free-stream velocity at the upper boundary of the domain, and $T_0$ is the absolute surface temperature.