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Semi-analytical study of the Voinovs problem

Published online by Cambridge University Press:  07 March 2018

E. A. KARABUT ,
A. G. PETROV  and
E. N. ZHURAVLEVA
E. A. KARABUT
Affiliation:
Lavrentyev Institute of Hydrodynamics, 630090, Novosibirsk, Russia email: eakarabut@gmail.com Novosibirsk State University, 630090, Novosibirsk, Russia email: zhuravleva_e@mail.ru
A. G. PETROV
Affiliation:
Institute for Problems in Mechanics, 119526, Moscow, Russia email: petrovipmech@gmail.com
E. N. ZHURAVLEVA
Affiliation:
Lavrentyev Institute of Hydrodynamics, 630090, Novosibirsk, Russia email: eakarabut@gmail.com Novosibirsk State University, 630090, Novosibirsk, Russia email: zhuravleva_e@mail.ru
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Abstract

A problem from the class of unsteady plane flows of an ideal fluid with a free boundary is considered. A conformal mapping of the exterior of a unit circle onto the region occupied by the fluid is sought. The solution is constructed in the form of power series in time or Laurent series which are analytically continued with the use of Padé approximants and change of variables of a certain special type. The free boundary shape and the pressure and velocity distributions are found. Singularities of the solution are studied.

Keywords

Conformal mappingPade approximantfree boundary flowideal incompressible fluid
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 2 , April 2019 , pp. 298 - 337
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†The study was supported by Russian Science Foundation, project no. 14-19-01633 at the Ishlinsky Institute for Problems in Mechanics RAS.

References

[1] Aptekarev, A. I., Buslaev, V. I., Martines-Finkelstein, A. & Suetin, S. P. (2011) Padé approximations, continued fractions, and orthogonal polynomials. Russ. Math. Surv. 66 (6), 1049–1131.Google Scholar
[2] Baker, G. R. & Xie, C. (2011) Singularities in the complex physical plane for deep water waves. J. Fluid Mech. 685, 83–116.Google Scholar
[3] Baumel, R. T., Burley, S. K., Freeman, D. F., Gammel, J. L. & Nuttal, J. (1982) The rise of a cylindrical bubble in an inviscid liquid. Can. J. Phys. 60 (7), 999–1007.Google Scholar
[4] Belykh, V. N. (2017) On the evolution of a finite volume of ideal incompressible fluid with a free surface. Dokl. Phys. 62 (4), 213–217.Google Scholar
[5] Belykh, V. N. (2017) Well-posedness of a nonstationary axisymmetric hydrodynamic problem with free surface. Siberian Math. J. 58 (4), 564–577.Google Scholar
[6] Bieberbach, L. (1955) Analytische Fortsetzung, Berlin: Springer-Verlag.Google Scholar
[7] Crew, S. C. & Trinh, P. H. (2016) New singularities for Stokes waves. J. Fluid Mech. 798, 256–283.Google Scholar
[8] Cummings, S. D., Howison, S. D. & King, J. R. (1999) Two-dimensional Stokes and Hele–Shaw flows with free surfases. Eur. J. Appl. Math. 10 (6), 635–680.Google Scholar
[9] Curle, N. (1956) Unsteady two-dimensional flows with free boundaries. Pros. Roy. Soc. London Ser. A. 235 (1202), 375–395.Google Scholar
[10] Dallaston, M. C. & Mc Cue, S. W. (2010) Accurate series solutions for gravity-driven Stokes waves. Phys. Fluids 22 (8), 82–104.Google Scholar
[11] Dorodnitsyn, A. A. (1965) Plane problem of unsteady motions of a heavy fluid. In: Proceedings of International Symposium in Tbilisi “Applications of the function theory in mechanics of continuous media,” Moscow, Nauka, Vol. 2, pp. 171–172 (in Russian).Google Scholar
[12] Dyachenko, A. I. (2001) On the dynamics of an ideal fluid with a free surface. Dokl. Math. 63 (1), 115–118; Translated from Dokl. Akad. Nauk. 376(1), 27–29.Google Scholar
[13] Dyachenko, S. A., Lushnikov, P. M. & Korotkevich, A. O. (2016) Branch cuts of stokes wave on deep water. Part 1: Numerical solution and Pade approximation. Stud. Appl. Math. 137 (4), 419–472.Google Scholar
[14] Dyachenko, A. I. & Zakharov, V. E. (1994) Is free-surface hydrodynamics an integrable system? Phys. Lett. A 190 (2), 144–148.Google Scholar
[15] Galin, L. A. (1945) Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR 47, 246–249 (in Russian).Google Scholar
[16] Gammel, J. L. (1976) The rise of a bubble in a fluid. Lecture Notes Phys. 47, 141–163.Google Scholar
[17] Gaunt, D. S. & Guttman, A. J. (1974) Asymptotic analysis of coefficients. In: Phase Transitions and Critical Phenomena, Vol. 3, Domb, C. and Green, M.S. (eds.), Academic Press, London, pp. 181–243.Google Scholar
[18] Gurevich, M. I. (1965) Theory of Jets in Ideal Fluids, N.Y.: Academic Press.Google Scholar
[19] Jounstone, E. A. & Mackie, A. G. (1973) The use of Lagrangian coordinates in the water entry and related problems. Proc. Camb. Phil. Soc. 74 (3), 529–538.Google Scholar
[20] Karabut, E. A. (1991) Semi-analytical investigation of unsteady free-boundary flows. Int. Ser. Numer. Math. 99, 215–224.Google Scholar
[21] Karabut, E. A. (1996) Asymptotic expansions in the problem of a solitary wave. J. Fluid Mech. 319, 109–123.Google Scholar
[22] Karabut, E. A. (1998) An approximation for the highest gravity waves on water of finite depth. J. Fluid Mech. 372, 45–70.Google Scholar
[23] Karabut, E. A. (2013) Exact solutions of the problem of free-boundary unsteady flows. C. R. Mec. 341 (6), 533–537.Google Scholar
[24] Karabut, E. A. & Kuzhuget, A. A. (2014) Conformal mapping, Padé approximants and example of flow with significant deformation of free boundary. Eur. J. Appl. Math. 25 (6), 729–747.Google Scholar
[25] Karabut, E. A. & Zhuravleva, E. N. (2014) Unsteady flows with a zero acceleration on the free boundary. J. Fluid Mech. 754, 308–331.Google Scholar
[26] Karabut, E. A. & Zhuravleva, E. N. (2016) Reproduction of solutions in the plane problem of motion of a free-boundary fluid. Dokl. Phys. 61 (7), 346–349; Translated from Dokl. Akad. Nauk. 61(3), 295–298.Google Scholar
[27] Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. (1994) Formation of singularities on the free surface of an ideal fluid. Phys. Rev. E. 49 (2), 1283–1290.Google Scholar
[28] Lavrent'ev, M. A. & Shabat, B. V. (1973) Methods of the Function Theory of a Complex Variable, Moscow Nauka (in Russian).Google Scholar
[29] Longuet-Higgins, M. S. (1975) Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. London Ser. A. 342 (1629), 157–174.Google Scholar
[30] Longuet-Higgins, M. S. (1972) A class of exact, time-dependent, free surface flows. J. Fluid Mech. 55 (3), 529–543.Google Scholar
[31] Lushnikov, P. M. (2016) Structure and location of branch point singularities for Stokes waves on deep water. J. Fluid Mech. 800, 557–594.Google Scholar
[32] Makarenko, N. I. & Kostikov, V. K. (2013) Unsteady motion of an elliptic cylinder under a free surface. J. Appl. Mech. Tech. Phys. 54 (3), 367–376.Google Scholar
[33] Menikoff, R. & Zemach, C. (1983) Rayleigh–Taylor instability and the use of conformal maps for ideal fluid flow. J. Comput. Phys. 51 (1), 28–64.Google Scholar
[34] Nuttal, J. (1980) Sets of minimum capacity, Padé approximants and the bubble problem. In: Bardos, C., Bessis, D. (eds) Bifurcation Phenomena in Mathematical Physics and Related Topics. NATO Advanced Study Institutes Series (Series C–Mathematical and Physical Sciences), vol 54. Springer, Dordrecht, pp. 185–201.Google Scholar
[35] Ovsyannikov, L. V. (1967) General equation and examples. In: The Problem of the Unstable Flow with a Free Boundary, Nauka, Novosibirsk, pp. 5–75 (in Russian).Google Scholar
[36] Ovsyannikov, L. V. (1970) On bubble rising. In: Some Problems of Mechanics and Mathematics, Leningrad, Nauka, p. 209 (in Russian).Google Scholar
[37] Ovsyannikov, L. V. (1971) Plane problem of unsteady motion of a fluid with free boundaries. Dynamics of Continuous Media 8, 22–26 (in Russian).Google Scholar
[38] Pearce, G. J. (1978) Transformation methods in the analysis of series for critical properties. Adv. Phys. 27 (1), 89–145.Google Scholar
[39] Pukhnachov, V. V. (1978) On the motion of liquid ellipse. Dynamics of Continuous Media 33, 68–75 (in Russian).Google Scholar
[40] Polubarinova-Kochina, P. Ya. (1945) On the motion of the oil contour. Dokl. Akad. Nauk SSSR 47, 254–257 (in Russian).Google Scholar
[41] Rowe, P. N. & Partridge, B. A. (1964) A note on the initial motion and break-up of two-dimensional air-bubble in water. Chem. Eng. Sci. 19 (1), 81–82.Google Scholar
[42] Shamin, R. V. (2008) Computational Experiments Aimed at Simulating Surface Waves in the Ocean, Moscow, Nauka (in Russian).Google Scholar
[43] Stahl, H. (1997) The convergence of Padé approximants to functions with branch points. J. Approx. Theory 91 (2), 139–204.Google Scholar
[44] Stoker, J. J. (1957) Water Waves. The Mathematical Theory with Applications, Interscience, New York.Google Scholar
[45] Suetin, S. P. (2010) Numerical analysis of some characteristics of the limit cycle of the free van der Pol equation. Sovrem. Probl. Mat. 14, 3–57.Google Scholar
[46] Suetin, S. P. (2015) Distribution of the zeros of Padé polynomials and analytic continuation. Russian Math. Surv. 70 (5), 901–951.Google Scholar
[47] Tanveer, S. (1991) Singularities in water waves and Rayleigh–Taylor instability. Proc. R. Soc. Lond. A. 435 (1893), 137–158.Google Scholar
[48] Trefethen, L. N. (1980) Numerical computation of the Schwarz–Christoffel transformation. SIAM J. Sci. Stat. Comput. 1 (1), 82–102.Google Scholar
[49] Van Dyke, M. (1975) Computer extension of perturbation series in fluid mechanics. SIAM J. Appl. Math. 28 (3), 720–734.Google Scholar
[50] Van Dyke, M. (1978) Semi-analytical applications of the computer. Fluid Dyn. Trans. Warszawa. 9, 305–320.Google Scholar
[51] Van Dyke, M. (1981) Successes and surprises with computer-extended series. Lecture Notes Phys. 141, 405–410.Google Scholar
[52] Voinov, O. V. & Voinov, V. V. (1975) Numerical method for calculating unsteady motions of an incompressible ideal fluid with free surfaces. Dokl. Akad. Nauk. 221 (3), 559–562 (in Russian).Google Scholar
[53] Walters, J. K. & Davidson, J. F. (1962) The initial motion of a gas bubble formed in an inviscid liquid. Part 1. The two-dimensional bubble. J. Fluid Mech. 12 (3), 408–416.Google Scholar
[54] Zakharov, V. E. (2016) Free-surface hydrodynamics in conformal variables: Are equations of free-surface hydrodynamics on deep water integrable? arXiv:1604.04778v1 [math-ph]Google Scholar
[55] Zakharov, V. E., Dyachenko, A. I. & Vasilyev, O. A. (2002) New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface. Europ. J. Mech. B. 21 (3), 283–291.Google Scholar
[56] Zubarev, N. M. & Kuznetsov, E. A. (2014) Singularity formation on a fluid interface during the Kelvin–Helmholtz instability development. J. Exp. Theor. Phys. 119 (1), 169–178.Google Scholar