Some equations of the Fuchs class in hydro- and aeromechanics

Abstract

The analytic theory of linear differential equations of the Fuchs class makes it possible to consider a wide range of two-dimensional steady-state problems of the theory of jets, flow through porous media, gas dynamics, etc. Solutions of problems of finding the function that realizes the conformal mapping of the circular polygons occurring in complex flow velocity domains onto the auxiliary canonical domain reduce to such equations. For polygons with a large number of vertices, in the coefficients of these equations, in addition to the unknown affixes of the vertices, there also appear additional (so-called accessory) parameters, which are not completely determined by the position of the singular points of the equations and the exponents in them [1–3], so that the determination of the parameters is a very complicated task. It is shown that in a whole series of cases in which the polygon has angles that are multiples of π/2 and branch cuts (as is often encountered in hydro-, gas- and aeromechanics), the problem of determining the unknown parameters can be completely solved.