Exact solution of a differential problem in analytical fluid dynamics: use of Airy’s functions

Treating a boundary value problem in analytical fluid dynamics, translation of 2D steady Navier-Stokes equations to ordinary differential form leads to a second order equation of Riccati type. In the case of a compressible fluid with constant kinematic viscosity along streamlines, it is possible to find an exact solution of the differential problem by rational combination of Airy’s functions and their derivatives. 1 Ordinary differential form of Navier-Stokes equations Suppose to have a 2D steady flow of a fluid. Let Φ : s 7−→ (φ1(s), φ2(s)) = (x, y) (1) an admissible parameterization ([5]) for each streamline, with Φ : [a, b] → R for some suitable values a and b such that Φ(a) is the initial point of the streamline, Φ(b) is the end point in the considered geometrical domain. Then, if v = (v1, v2) is the flow velocity field, by definition of streamline there is a scalar function f = f(s) such that (v1, v2) = f(s)(φ̇1(s), φ̇2(s)) (2)