×

Nonlinear oscillations in three-armed tubes. (English) Zbl 1040.76007

Summary: We consider nonlinear gravitational oscillations of inviscid liquid in arrangements of three tubes joined at their bases, for which the dynamical system is four-dimensional and conservative. Though the problem of two joined tubes was solved in 1738, that of three tubes appears to have remained unstudied. We consider both weakly-nonlinear theory, which gives rise to coupled amplitude evolution equations; and also full numerical solutions. In this way, an understanding is reached of the strengths and limitations of the weakly-nonlinear approximation. Both the weakly-nonlinear approximation and the full system display amplitude modulations on a slow timescale; but only the full system captures a narrow region of chaos.

MSC:

76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
76M20 Finite difference methods applied to problems in fluid mechanics
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D. Bernoulli, Hydrodynamica, sive de viribus et motibus fluidorum commentarii, Joh. Reinholdi Dulseckeri, Argentorati (Strasbourg), 1738; D. Bernoulli, Hydrodynamica, sive de viribus et motibus fluidorum commentarii, Joh. Reinholdi Dulseckeri, Argentorati (Strasbourg), 1738
[2] Gradshteyn, I. S.; Ryzhik, I. W., Tables of Integrals, Series, and Products (1965), Academic Press: Academic Press New York
[3] (Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1965), Dover: Dover New York) · Zbl 0171.38503
[4] Nayfeh, A. H., Perturbaton Methods (1973), Wiley: Wiley New York
[5] Nagata, M., Nonlinear Faraday resonance in a box with square base, J. Fluid Mech., 209, 265-284 (1989) · Zbl 0681.76019
[6] Umeki, M., Faraday resonance in rectangular geometry, J. Fluid Mech., 227, 161-192 (1991) · Zbl 0722.76014
[7] Douady, S.; Fauve, S.; Thual, O., Oscillatory phase modulation of parametrically forced surface waves, Europhys. Lett., 10, 4, 309-315 (1989)
[8] Danglemayr, G.; Knobloch, E., Hopf bifurcation with broken symmetry, Nonlinearity, 4, 399-427 (1991) · Zbl 0731.58051
[9] Golubitsky, M.; Stewart, I.; Schaeffer, D. G., Singularities and Groups in Bifurcation Theory, Vol. 2 (1988), Springer: Springer New York · Zbl 0691.58003
[10] Craik, A. D.D., Wave Interactions and Fluid Flows (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0581.76002
[11] Decent, S. P.; Craik, A. D.D., Hysteresis in Faraday resonance, J. Fluid Mech., 293, 237-268 (1995) · Zbl 0866.76014
[12] Hénon, M.; Heiles, C., The applicability of the third integral of motion: some numerical experiments, Astronomical J., 69, 73-79 (1964)
[13] Umeki, M.; Kambe, T., Nonlinear dynamics and chaos in parametrically excited surface waves, J. Phys. Soc. Japan, 58, 140-154 (1989)
[14] Craik, A. D.D., Nonlinear interaction of standing waves with Faraday excitation, (Debnath, L.; Riahi, D. N., Nonlinear Instability, Chaos and Turbulence, Vol. 1, Chapter 4 (1998), WIT Press: WIT Press Southampton, UK), 91-128 · Zbl 0982.76045
[15] Miles, J. W.; Henderson, D. M., Parametrically forced surface waves, Ann. Rev. Fluid Mech., 22, 143-165 (1990)
[16] Forster, G. K.; Craik, A. D.D., Second-harmonic resonance with Faraday excitation, Wave Motion, 26, 361-377 (1997) · Zbl 0930.76011
[17] Decent, S. P.; Craik, A. D.D., On limit cycles arising from the parametric excitation of standing waves, Wave Motion, 25, 275-294 (1997) · Zbl 0936.76501
[18] Decent, S. P.; Craik, A. D.D., Sideband instability and modulations of Faraday waves, Wave Motion, 30, 43-55 (1999) · Zbl 1067.76545
[19] Martel, C.; Knobloch, E.; Vega, J. M., Dynamics of counterpropagating waves in parametrically forced systems, Phys. D, 137, 94-123 (2000) · Zbl 0949.35019
[20] Craik, A. D.D.; Okamoto, H.; Allen, H. R., Second-harmonic resonance with parametric excitation and damping, (Lumley, J. L., Fluid Mechanics and the Environment: Dynamical Approaches (2001), Springer: Springer Berlin), 63-89 · Zbl 1051.76005
[21] Tanigawa, H.; Hirata, K.; Yano, H., Nonlinear oscillations of liquid column in a U-tube, Nagare, 17, 368-372 (1998), (in Japanese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.