ON TRANSFORMATIONS OF THE VELOCITY HODOGRAPH IN CERTAIN PROBLEMS OF FLUID MECHANICS

Consideration is given to the fl ow-velocity hodograph representing a circular hexagon in polar grids with two cuts, which is used in the theory of jets and cavitation, gliding theory, gas dynamics, the theory of groundwater motion, and fi ltration theory. It has been shown that in recording the parameter characterizing the ratio of the radii of circles, which are the opposite polygon sides with cuts, the confi guration and relative position of the latter substantially change not so much with the properties of the functions on whose basis particular solutions of the corresponding Fuchsian-type equation are constructed, as with the ranges of conformal-mapping constants involved in the expressions for mapping functions. It has been established that to different ranges of variation in these constants there can correspond cuts varying in confi guration and relative position. This is a manifestation of the transformation of liquid and gas fi ltration fl ows under the infl uence of different physical factors.
   
Author:  É. N. Bereslavskii
Keywords:  Fuchsian differential equations, conformal mapping, region of the velocity hodograph, circular and rectilinear cuts, elliptic Jacobian functions, theta functions
Page:  1221