Two-dimensional variational vibroequilibria and Faraday's drops

Zeitschrift fur Angewandte Mathematik und Physik

Volumn 55, Issue 6, 2004, Pages 1015-1033

  Gavrilyuk, Ivan ^a   Lukovsky, Ivan ^b   Timokha, Alexander ^c  

^a Berufsakademie Thuringen Staatl S ^*  (Germany)

^b INSTITUTE OF MATHEMATICS   (Ukraine)

^c FRIEDRICH SCHILLER UNIVERSITY JENA   (Germany)

Author keywords

Flattening drops; Minima principle; Nystr m method; Vibroequilibria

Indexed keywords

EID: 11144254036     PISSN: 00442275     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00033-004-2092-5     Document Type: Article

References (50)

1. A. V. Anilkumar, R. N. Grugel, X. F. Shen and T. G. Wang, Control of thermocapillary convection in a liquid bridge by vibration, J. Appl. Phys. 73 (1993), 4165-4170.
2. R.E. Apfel, Y. Tian, J. Jankovsky, T. Shi, X. Chen, R.G. Holt, E. Trinh, A. Croonguist, K.C. Thornton, A.Jr. Sacco, C. Colemen, F.W. Leslie and D.H. Matthiesen, Free oscillations and surfactant studies of superdeformed drops in microgravity, Phys. Rev. Let. 78 (1998), 1912-1915.
3. K. Beyer, I. Gawrilyuk, M. Günther, I. Lukovsky and A. Timokha, Compressible potential flows with free boundary. Part I: Vibrocapillary equilibria, ZAMM 81 (2001), 261-271.
4. N. K. Bezdenezhnikh, B. A. Briskman, D. V. Lyubimov, A. A. Cherepanov and M. T. Sharov, Control of stability of two-fluid interface by means of vibrations, electric and magnetic fields. In Abstracts of the all-Union Seminar on Hydromechanics and Heat- and Mass Transfer, Chernogolovka 1984, 18-20 (Russian).
5. N. A. Bezdenezhnikh, V. A. Briskman, A. Yu. Lapin, D. V. Lyubimov, T. P. Lyubimova, A. A. Tscherepanov and I. V. Zakharov, The influence of high-frequency tangential vibrations on stability of the fluid interface in microgravity. In Proceedings of IUTAM Symposium "Microgravity Fluid Mechanics", Bremen 1991 Springer, Berlin 1992, 137-144.
6. I. I. Blekhman, Vibrational Mechanics. World Scientific, Singapore 1999
7. O. M. Faltinsen and A. N. Timokha, Adaptive multimodal approach to nonlinear sloshing in a rectangular tank, J. Fluid Mech. 432 (2001), 167-20.
8. O. M. Faltinsen and A. N. Timokha, Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth, J. Fluid Mech. 470 (2002), 319-357.
9. O. M. Faltinsen and A. N. Timokha, Analytically-oriented approaches to two-dimensional fluid sloshing in a rectangular tank (survey), Proceedings of the Institute of Mathematics of the NASU 44 (2002), 321-345.
10. O. M. Faltinsen, O. F. Rognebakke and A. N. Timokha, Resonant three-dimensional nonlinear sloshing in a square base basin, J. Fluid Mech. 487 (2003), 1-42.
11. M. Faraday, On a peculiar class of acoustic figures and on certain forms assumed by groups of particles upon vibrating elastic surfaces, Phil. Trans. Roy. Soc. Lond. 121 (1831), 299-430.
12. G. E. Forsythe, N. A. Malcolm and C. B. Moler, Computer Methods for Mathematical Computation. Prentice-Hall, New York 1977.
13. R. F. Ganiyev, V. D. Lakiza and A. S. Tsapenko, On dynamic behaviour of a free liquid surface under low gravity and vibrational effects, Prikladnaya Mekhanika (Soviet Applied Mechanics) 13 (1977), 102-107 (Russian).
14. I. Gavrilyuk and A.Timokha, A Nyström-Kress scheme for a Dirichlet-Neumann boundary value problem associated with fluid sloshing in a rectangular tank, Proceedings of the Institute of Mathematics of the NASU 47 (2003), 284-301.
15. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston 1985.
16. R. A. Ibrahim, V. N. Pilipchuk and T. Ikeda, Recent advances in liquid sloshing dynamics, Appl. Mech. Rev. 54 (2001), 133-199.
17. A. A. Ivanova, V. G. Kozlov and J. C. Legros, Mean dynamics of two liquid system in a cavity subjected to rotational vibrations. In Proc. Joint 10th Europ. and 6th Russian Symp. Phys. Sci. in Microgravity, St. Petersburg, Russia 1997 Intst. Probl. Mech. RAS, Moscow 1 (1997), 280-283.
18. A. A. Ivanova, V. G. Kozlov, D. V. Lyubomov, T. P. Lyubimova, S. Merady and B. Roux, Structure of averaged flow driven by a vibrating body with a large-curvature edge, Fluid Dyn. 33 (1998), 656-665.
19. A. A. Ivanova, B. G. Kozlov and P. Evoux, Interfacial dynamics of two-liquid system under horizontal vibrations, Izvestiya RAN. Mekhanika Zhidkosti i Gasa (Fluid Dynamics) N 3 (2001), 28-35 (Russian).
20. A. A. Ivanova, V. G. Kozlov and S. I. Tasjkinov, Interfacial dynamics of two-liquid system due to rotational vibrations (experiments), Izvestiya RAN. Mekhanika Zhidkosti i Gasa (Fluid Dynamics) N 6 (2001), 114-121 (Russian).
21. D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software. Prentice-Hall, New York 1988.
22. P. L. Kapitsa, The pendulum with a vibrating point of suspension, Adv. Phy. Sc. 44 (1952), 34-42 (Russian).
23. M. V. Khenner, D. V. Lyubimov, T. S. Belozerova and B. Roux, Stability of plane-parallel vibrational flow in a two-layer system, Eur. J. Mech. B/Fluids 18 (1999), 1085-1101.
24. R. Kress, A Nyström method for boundary integral equations in domains with corners, Numer. Math. 58 (1990), 145-161.
25. M. La Rocca, G. Sciortino and M. Boniforti, Interfacial gravity waves in a two-fluid system, Fluid Dyn. Res. 30 (2002), 31-66.
26. L. D. Landau and E. M. Lifschitz, Mechanik. Akademie-Verlag, Berlin 1962.
27. M. Landrini, G. Grytøyr and O.M. Faltinsen, A B-spline based BEM for unsteady free-surface flows, J. Ship Res. 43 (1999), 13-24.
28. C. P. Lee, A. V. Anilkumar and T. G. Wang, Streaming generated in a liquid bridge due to nonlinear oscillations driven by the vibrations of an endwall, Phys. Fluids 8 (1996), 3234-3246.
29. C. P. Lee, A. V. Anilkumar, A. B. Hmelo and T. G. Wang, Equilibrium of liquid drops under effects of rotation and acoustic flattening: Results from USML-2 experiments in space, J. Fluid Mech. 354 (1998), 43-67.
30. E. G. Lierke (eds.), Acoustic positioning. In Summary Review of Sounding Rocket Experiments in Fluid Science and Material Sciences (post-flight); TEXUS 1 to 20, MASTER 1 and 2, ESA SP-1132, February 1991, 362-365.
31. I. A. Lukovsky, M. Ya. Barnyak and A. N. Komarenko, Approximate Methods of Solving the Problems of the Dynamics of a Limited Liquid Volume. Naukova dumka, Kiev 1984 (Russian).
32. I. A. Lukovsky and A. N. Timokha, Variational Methods in Nonlinear Dynamics of a Limited Liquid Volume. Institute of Mathematics, Kiev 1995 (Russian).
33. I. A. Lukovsky and A. N. Timokha, Nonlinear theory of sloshing in mobile tanks: classical and non-classical problems (survey). In book "Problems of analytical mechanics and its applications", Institute of Mathematics of NASU, Kiev 1999, 169-200 (Russian).
34. I. A. Lukovsky and A. N. Timokha, Steady-state nonlinear sloshing in a rectangular tank: passage to shallow water, Reports of National Academy of Sciences of Ukraine (2000), 48-51.
35. V. D. Lyubimov and A. A. Cherepanov, Development of a steady relief at the interface of fluids in a vibrational field, Fluid Dyn. 21 (1986), 849-854.
36. D. V. Lyubimov, A. A. Cherepanov and V. A. Briskman, Control of the fluid free surface by fluctating fields. In Abstracts of II All-Union Seminar on Hydromechanics and Heat- and Mass Transfer, Perm 1981, Perm 1981, 112-114 (Russian).
37. D. V. Lyubimov, T. P. Lyubimova, R. V. Skuridin, G. Chen and B. Roux, Numerical investigation of meniscus deformation and flow in an isothermal liquid bridge subject to high-frequency vibrations under zero gravity conditions, Computers & Fluids 31 (2002), 663-682.
38. Akisato Mizuno and Yoshihiro Akisato, Analysis of nonlinear wave-making phenomena by means of boundary element method, Res. Repts. Kogakuin Univ. 69 (1990), 15-20 (Japanese).
39. R. Monti (ed.), Fluid Physics in Microgravity. Reading Overseas Publishing Associates 2000.
40. A. D. Myshkis, V. G. Babsky, N. D. Kopachevskii, L. D. Slobozhanin and A. D. Typsov, Low-Gravity Fluid Mechanics. Mathematical Theory of Capillary Phenomena. Springer-Verlag, Berlin Heidelberg 1987.
41. A. D. Myshkis, V. G. Babsky, M. Yu. Zhukov, N. D. Kopachevskii, L. D. Slobozhanin and A. D. Typsov, Methods for Solving Hydromechanic Problems in Zera-Gravity. Naukova dumka, Kiev 1992 (Russian).
42. S. G. Mikhlin, Linear Partial Differential Equations. Vishaya, shkola, Moscow 1977 (Russian).
43. V. G. Nevolin, Parametric excitation of surface waves (survey), Inzhenerno-Physicheskiy Zhurnal 47 (1984), 1028-1042 (Russian).
44. M. Perlin and W. W. Schultz, Capillary effects on surface waves, Annu. Rev. Fluid Mech. 32 (2000), 241-274.
45. A. N. Timokha, Behaviour of the free surface of a fluid in a vibrating bounded vessel. Preprint N 22, Akad. Nauk Ukrainy Inst. Mat. Institute of Mathematics, Kiev 1992 (Russian).
46. A. N. Timokha, Effect of lateral vibration of a tank on free surface of the fluid, Technical Mechanics 5 (1997), 33-41 (Russian).
47. T. Wang, Drop physics module / drop dynamics experiment. In 90-day Science Report of the Second United States Microgravity Laboratory (USML-2), March 1996, 69-71.
48. S. Wanis, A. Sercovich and N. Komerath, Acoustic shaping in microgravity: higher order surface shapes, AIAA Paper N 99-0954 (1999).
49. G. H. Wolf, The dynamic stabilisation of the Rayleigh-Taylor instability and the corresponding dynamic equilibrium, Z. Physik 227 (1969), 291-300.
50. G. H. Wolf, Dynamic stabilisation of the interchange instability of a liquid-gas interface, Phys. Rev. Lett. 24 (1970), 444-446.