A functionally-analytic method for modeling axial-symmetric flows of ideal fluid

• Autorzy: Plaksa Sergiy A.

Identyfikatory

DOI

10.1515/dema-2019-0018

• Języki publikacji: EN

Abstrakty

EN

We consider axial-symmetric stationary flows of the ideal incompressible fluid as an important case of potential solenoid fields. We use an integral expression of the Stokes flow function via the corresponding complex analytic function for solving a boundary value problem with respect to a steady streamline of the ideal incompressible fluid along an axial-symmetric body. We describe the solvability of the problem in terms of the singularities of the mentioned complex analytic function. The obtained results are illustrated by concrete examples of modelling of steady axial-symmetric flows.

Słowa kluczowe

EN

axial-symmetric potential flows; Stokes flow function; analytic function; streamline

PL

przepływ Stokesa; funkcja płynięcia; funkcja analityczna; linia prądu

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2019
• Tom: Vol. 52, nr 1
• Strony: 213--224
• Opis fizyczny: Bibliogr. 32 poz., rys.

Twórcy

autor

Plaksa Sergiy A.

• Institute of Mathematics, National Academy of Sciences of Ukraine,Tereshchenkivska Str. 3, 01601, Kyiv, Ukraine

Bibliografia

• [1] Lavrentyev M. A., Shabat B. V., Problems of Hydrodynamics and Theirs Mathematical Models, Nauka, Moscow, 1977 (in Russian)
• [2] Lavrentyev M. A., Shabat B. V., Methods of the Theory of Functions of a Complex Variable, 5rd edition, Nauka, Moscow, 1987 (in Russian)
• [3] Lavrentyev M. A., Cumulative charge and the principles of its operation, Uspekhi Matematicheskikh Nauk, 1957, 12(4), 41–56 (in Russian)
• [4] Batchelor G. K., An Introduction to Fluid Dynamics, Cambridge, 1970
• [5] Vallander S. V., Lectures on hydromechanics, Leningrad University, Leningrad, 1978 (in Russian)
• [6] Loitsyanskii L. G., Mechanics of Liquids and Gases, Nauka, Moscow, 1987 (in Russian)
• [7] Plaksa S. A. Axial-symmetric potential flows, In: Flaut C., Hošková-Mayerova Š., Flaut D. (Eds.), Models and Theories in Social Systems, Springer International Publishing, 2019, 165–195
• [8] Hille E., Phillips R. S., Functional Analysis and Semi-Groups, American Mathematical Society, Providence, R.I., 1957
• [9] Mel’nichenko I. P., Plaksa S. A., Potential fields with axial symmetry and algebras of monogenic functions of vector variable, III, Ukr. Math. J., 1997, 49(2), 253–268
• [10] Mel’nichenko I. P., Plaksa S. A., Commutative algebras and spatial potential fields, Inst. Math. NAS Ukraine, Kyiv, 2008 (in Russian)
• [11] Keldysh M. V., On some cases of degeneration of an equation of elliptic type on the boundary of a domain, Dokl. Akad. Nauk SSSR, 1951, 77(2), 181–183 (in Russian)
• [12] Gilbert R. P., Function theoretic methods in partial differential equations, Academic Press, New York-London, 1969
• [13] Mikhailov L. G., Radzhabov N., An analog of the Poisson formula for second-order equations with singular line, Dokl. Akad.Nauk Tadzh. SSR, 1972, 15(11), 6–9 (in Russian)
• [14] Rutkauskas S., Exact solutions of Dirichlet type problem to elliptic equation, which type degenerates at the axis of cylinder, I, II, Bound. Value Probl., 2016, 2016:183; 2016:182
• [15] Rutkauskas S., On the Dirichlet problem to elliptic equation, the order of which degenerates at the axis of a cylinder, Math. Model. Anal., 2017, 22(5), 717–732
• [16] Whittaker E. T., Watson G. N., A Course of Modern Analysis, vol. 2, Cambridge University Press, Cambridge, 1927
• [17] Bateman H., Partial Differential Equations of Mathematical Physics, Dover, New York, 1944
• [18] Henrici P., Zur Funktionentheory der Wellengleichung, Comment. Math. Helv., 1953, 27(3–4), 235–293
• [19] Mackie A. G., Contour integral solutions of a class of differential equations, J. Ration. Mech. Anal., 1955, 4(5), 733–750
• [20] Krivenkov Yu. P., On one representation of solutions of the Euler–Poisson–Darboux equation, Dokl. Akad. Nauk SSSR, 1957, 116(3), 351–354 (in Russian)
• [21] Radzhabov N. R., Integral representations and their inversion for a generalized Cauchy–Riemann system with singular line, Dokl. Akad. Nauk Tadzh. SSR, 1968, 11(4), 14–18 (in Russian)
• [22] Polozhii G. N., Theory and Application of p-Analytic and (p,q)-Analytic Functions, Naukova Dumka, Kiev, 1973 (in Russian)
• [23] Polozhii G. N., Ulitko A. F., On formulas for an inversion of the main integral representation of p-analiytic function with the characteristic p=xk, Prikl. Mekhanika, 1965, 1(1), 39–51 (in Russian)
• [24] Kapshivyi A. A., On a fundamental integral representation of x-analytic functions and its application to solution of someintegral equations, In: Mathematical Physics, 1972, 12, 38–46 (in Russian)
• [25] Aleksandrov A. Ya., Soloviev Yu. P., Three-dimensional problems of the theory of elasticity, Nauka, Moscow, 1979 (in Russian)
• [26] Plaksa S. A., Dirichlet problem for the Stokes flow function in a simply connected domain of the meridian plane, Ukr. Math. J., 2003, 55(2), 241–281
• [27] Priwalow I. I., Randeigenschaften analytischer Funktionen, Deutsch. Verlag Wissenschaft, Berlin, 1956 (Translated from Russian)
• [28] Weinstein A., Generalized axially symmetric potential theory, Bull. Amer. Math. Soc., 1953, 59(1), 20–38
• [29] Mel’nichenko I. P., Pik E. M., On a method for obtaining axial-symmetric flows, Dopovidi AN Ukr. Ser. A., 1973, 2, 152–155 (in Ukranian)
• [30] Mel’nichenko I. P., Pik E. M., Quaternion equations and hypercomplex potentials in the mechanics of a continuous medium, Soviet Applied Mechanics, 1973, 9(4), 383–387
• [31] Mel’nichenko I. P., Pik E. M., Quaternion potential of the ideal noncomprssible fluid, Prikl. Mechanika, 1975, 11(1), 125–128 (in Russian)
• [32] David G., Operateurs intégraux sur certaines courbes du plan complexe, Ann. Sci. de l’Ecole Normale Supérieure, 4 ser., 1984, 17(1), 157–189

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).