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 filtration 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 configuration 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 configuration and relative position. This is a manifestation of the transformation of liquid and gas filtration flows under the influence of different physical factors.
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References
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Nauka, Moscow (1971).
W. Koppenfels and F. Stallmann, Praxis der konformen Abbildung [Russian translation], IL, Moscow (1963).
V. V. Golubev, Lectures on the Analytical Theory of Differential Equations [in Russian], Gostekhizdat, Moscow–Leningrad (1950).
L. I. Sedov, Plane Problems of Hydrodynamics and Aerodynamics [in Russian], Nauka, Moscow (1980).
G. Birkhoff and E. Sarantonello, Jets, Wakes, and Caverns [Russian translation], Mir, Moscow (1964).
M. I. Gurevich, The Theory of Jets of a Perfect Fluid [in Russian], Nauka, Moscow (1979).
L. I. Sedov, Application of the theory of functions of a complex variable to certain hydrodynamics problems, Usp. Mat. Nauk, Issue 6, 120–182 (1939).
M. V. Keldysh and L. I. Sedov, Applications of the Theory of Functions of a Complex Variable to Hydrodynamics and Aerodynamics [in Russian], Nauka, Moscow (1964).
G. Yu. Stepanov, Hydrodynamic Theory of Air-Cushion Apparatuses [in Russian], Mashgiz, Moscow (1963).
S. V. Fal′kovich, On the theory of gas jets, Prikl. Mat. Mekh., 21, Issue 4, 458–464 (1957).
P. Ya. Polubarinova-Kochina, Theory of Groundwater Motion [in Russian], 2nd edition, Nauka, Moscow (1977).
P. Ya. Kochina, Hydrodynamic and Filtration Theory, Selected Works [in Russian], Nauka, Moscow (1991).
É. N. Bereslavskii and P. Ya. Kochina, On certain Fuchsian equations in hydro- and aeromechanics, Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 3–7 (1992).
P. Ya. Kochina, É. N. Bereslavskii, N. N. Kochina, Analytical Theory of Fuchs Class Linear Differential and Some Problems of Underground Hydromechanics, Pt. 1, Preprint No. 567 of the Institute for Problems of Mechanics, Russian Academy of Sciences, Moscow (1996).
É. N. Bereslavskii and P. Ya. Kochina, On Fuchsian differential equations occurring in problems of fluid mechanics, Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 9–17 (1997).
E. S. Wittaker and G. N. Watson, A Course of Modern Analysis, Vol. 2, [Russian translation], Fizmatgiz, Moscow (1963).
É. N. Bereslavskii, On Fuchsian differential equations related to conformal mapping of circular polygons in polar grids, Differents. Uravn., 33, No. 3, 296–301 (1997).
É. N. Bereslavskii, On integrals of certain Fuchsian differential equations occurring in problems of fluid mechanics, Differents. Uravn., 48, No. 4, 590–594 (2012).
É. N. Bereslavskii, Mathematical modeling of filtration flows from canals and irrigation channels, Proceedings of the Seminar on Boundary-Value Problems, Kazan, Issue 25 (1990), pp. 70–90.
H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 1 [Russian translation], Nauka, Moscow (1967).
M. A. Lavrentiev and V. V. Shabat, Methods of the Theory of Complex Variable Functions [in Russian], Nauka, Moscow (1987).
É. N. Bereslavskii, On transformations of cuts in conformal mapping of certain circular polygons occurring in problems of flow with free boundaries, Dokl. Ross. Akad. Nauk, 451, No. 2, 160–163 (2013).
É. N. Bereslavskii, On conformal mapping of some circular polygons onto a rectangle, Izv. Vyssh. Uchebn. Zaved., Matematika, No. 5, 3–7 (1980).
É. N. Bereslavskii, On filtration schemes occurring in the problem of flow over a Zhukovskii sheet pile, J. Eng. Phys. Thermophys., 86, Issue 2, 316–326 (2013).
É. N. Bereslavskii, Mathematical modeling of filtration flows from foundation pits fenced with Zhukovskii sheet piles, J. Eng. Phys. Thermophys., 87, Issue 1, 1–14 (2014).
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Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 88, No. 5, pp. 1182–1188, September–October, 2015.
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Bereslavskii, É.N. On Transformations of the Velocity Hodograph in Certain Problems of Fluid Mechanics. J Eng Phys Thermophy 88, 1221–1227 (2015). https://doi.org/10.1007/s10891-015-1303-6
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DOI: https://doi.org/10.1007/s10891-015-1303-6