The transformation of cuts in the conformational mapping of circular polygons in problems of liquid and gas mechanics

A linear differential equation of the Fuchs class with six regular singular points is considered, which corresponds to the problem of the conformal mapping of circular hexagons in polar nets with two cuts that are characteristic in the theory of jets and cavitation, the theory of gliding, in gas dynamics, in the theory of the motion of ground waters and in seepage theory. It is shown that, in the case of a fixed parameter, characterizing the ratio of the radii of the circular ares constituting the opposite sides of a polygon on which there are cuts, the configuration and mutual arrangement of the cuts depend considerably not so much on the known properties of theta-functions, on the basis of which special solutions of the equation considered are constructed, but on the ranges of variation in the constants of the conformal mapping occurring in the expression for the mapping functions as well as the modulus of the elliptic functions. It is found that cuts that are different in their configuration and mutual arrangement can correspond to different ranges of variation in these constants in the corresponding region of the velocity hodograph which is evidence of the transformation of the fluid seepage flows depending on the effect of various physical factors.