Skip to main content Accessibility help

Semi-analytical study of the Voinovs problem

  • E. A. KARABUT (a1) (a2), A. G. PETROV (a3) and E. N. ZHURAVLEVA (a1) (a2)


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.



Hide All

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



Hide All
[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), 10491131.
[2] Baker, G. R. & Xie, C. (2011) Singularities in the complex physical plane for deep water waves. J. Fluid Mech. 685, 83116.
[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), 9991007.
[4] Belykh, V. N. (2017) On the evolution of a finite volume of ideal incompressible fluid with a free surface. Dokl. Phys. 62 (4), 213217.
[5] Belykh, V. N. (2017) Well-posedness of a nonstationary axisymmetric hydrodynamic problem with free surface. Siberian Math. J. 58 (4), 564577.
[6] Bieberbach, L. (1955) Analytische Fortsetzung, Berlin: Springer-Verlag.
[7] Crew, S. C. & Trinh, P. H. (2016) New singularities for Stokes waves. J. Fluid Mech. 798, 256283.
[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), 635680.
[9] Curle, N. (1956) Unsteady two-dimensional flows with free boundaries. Pros. Roy. Soc. London Ser. A. 235 (1202), 375395.
[10] Dallaston, M. C. & Mc Cue, S. W. (2010) Accurate series solutions for gravity-driven Stokes waves. Phys. Fluids 22 (8), 82104.
[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).
[12] Dyachenko, A. I. (2001) On the dynamics of an ideal fluid with a free surface. Dokl. Math. 63 (1), 115118; Translated from Dokl. Akad. Nauk. 376(1), 27–29.
[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), 419472.
[14] Dyachenko, A. I. & Zakharov, V. E. (1994) Is free-surface hydrodynamics an integrable system? Phys. Lett. A 190 (2), 144148.
[15] Galin, L. A. (1945) Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR 47, 246249 (in Russian).
[16] Gammel, J. L. (1976) The rise of a bubble in a fluid. Lecture Notes Phys. 47, 141163.
[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. 181243.
[18] Gurevich, M. I. (1965) Theory of Jets in Ideal Fluids, N.Y.: Academic Press.
[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), 529538.
[20] Karabut, E. A. (1991) Semi-analytical investigation of unsteady free-boundary flows. Int. Ser. Numer. Math. 99, 215224.
[21] Karabut, E. A. (1996) Asymptotic expansions in the problem of a solitary wave. J. Fluid Mech. 319, 109123.
[22] Karabut, E. A. (1998) An approximation for the highest gravity waves on water of finite depth. J. Fluid Mech. 372, 4570.
[23] Karabut, E. A. (2013) Exact solutions of the problem of free-boundary unsteady flows. C. R. Mec. 341 (6), 533537.
[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), 729747.
[25] Karabut, E. A. & Zhuravleva, E. N. (2014) Unsteady flows with a zero acceleration on the free boundary. J. Fluid Mech. 754, 308331.
[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), 346349; Translated from Dokl. Akad. Nauk. 61(3), 295–298.
[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), 12831290.
[28] Lavrent'ev, M. A. & Shabat, B. V. (1973) Methods of the Function Theory of a Complex Variable, Moscow Nauka (in Russian).
[29] Longuet-Higgins, M. S. (1975) Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. London Ser. A. 342 (1629), 157174.
[30] Longuet-Higgins, M. S. (1972) A class of exact, time-dependent, free surface flows. J. Fluid Mech. 55 (3), 529543.
[31] Lushnikov, P. M. (2016) Structure and location of branch point singularities for Stokes waves on deep water. J. Fluid Mech. 800, 557594.
[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), 367376.
[33] Menikoff, R. & Zemach, C. (1983) Rayleigh–Taylor instability and the use of conformal maps for ideal fluid flow. J. Comput. Phys. 51 (1), 2864.
[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.
[35] Ovsyannikov, L. V. (1967) General equation and examples. In: The Problem of the Unstable Flow with a Free Boundary, Nauka, Novosibirsk, pp. 575 (in Russian).
[36] Ovsyannikov, L. V. (1970) On bubble rising. In: Some Problems of Mechanics and Mathematics, Leningrad, Nauka, p. 209 (in Russian).
[37] Ovsyannikov, L. V. (1971) Plane problem of unsteady motion of a fluid with free boundaries. Dynamics of Continuous Media 8, 2226 (in Russian).
[38] Pearce, G. J. (1978) Transformation methods in the analysis of series for critical properties. Adv. Phys. 27 (1), 89145.
[39] Pukhnachov, V. V. (1978) On the motion of liquid ellipse. Dynamics of Continuous Media 33, 6875 (in Russian).
[40] Polubarinova-Kochina, P. Ya. (1945) On the motion of the oil contour. Dokl. Akad. Nauk SSSR 47, 254257 (in Russian).
[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), 8182.
[42] Shamin, R. V. (2008) Computational Experiments Aimed at Simulating Surface Waves in the Ocean, Moscow, Nauka (in Russian).
[43] Stahl, H. (1997) The convergence of Padé approximants to functions with branch points. J. Approx. Theory 91 (2), 139204.
[44] Stoker, J. J. (1957) Water Waves. The Mathematical Theory with Applications, Interscience, New York.
[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, 357.
[46] Suetin, S. P. (2015) Distribution of the zeros of Padé polynomials and analytic continuation. Russian Math. Surv. 70 (5), 901951.
[47] Tanveer, S. (1991) Singularities in water waves and Rayleigh–Taylor instability. Proc. R. Soc. Lond. A. 435 (1893), 137158.
[48] Trefethen, L. N. (1980) Numerical computation of the Schwarz–Christoffel transformation. SIAM J. Sci. Stat. Comput. 1 (1), 82102.
[49] Van Dyke, M. (1975) Computer extension of perturbation series in fluid mechanics. SIAM J. Appl. Math. 28 (3), 720734.
[50] Van Dyke, M. (1978) Semi-analytical applications of the computer. Fluid Dyn. Trans. Warszawa. 9, 305320.
[51] Van Dyke, M. (1981) Successes and surprises with computer-extended series. Lecture Notes Phys. 141, 405410.
[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), 559562 (in Russian).
[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), 408416.
[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]
[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), 283291.
[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), 169178.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed