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⊂R2. 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 R2. Under the same assumption, we can give an approximate solution u for the limit formula if |u|<<<1 such that: u(z)≅δ∗c∗z+625⋅12i0⋅4U23z4+o(z5),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, δ is the boundary layer’s height in Dorodnitzyn’s coordinates, U is the free-stream velocity at the upper boundary of the domain, and T0 is the absolute surface temperature.