- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Modal coupling of nonlinear inviscid axisymmetric droplet shape oscillations

Schahin Akbari1,2,*, Kilian Vinzenz Wilhelm1,2, Dominik Plümacher1, Florian Kummer1,2, Yongqi Wang1, and Martin Oberlack1,2

  • *Contact author: akbari@fdy.tu-darmstadt.de
PDF

Phys. Rev. Fluids 11, 033602 – Published 5 March, 2026

DOI: https://doi.org/10.1103/5wrs-k8z4

Abstract

We investigate the nonlinear axisymmetric shape oscillations of inviscid droplets, focusing on the modal coupling between different oscillation modes. The research builds upon the geometrically exact nonlinear theory for droplet shape oscillations by Plümacher et al. [Phys. Fluids 32, 067104 (2020)], which allows for large deformations and superimposed initial deformations. The theory rests on the potential flow assumption and the usual momentum jump condition at the droplet surface including the surface tension. Using the unified transform method of A. S. Fokas [Proc. R. Soc. London Ser. A 453, 1411 (1997)], the governing equations are transformed into a system of integrodifferential equations defined on a unit sphere. These equations are then solved numerically with a Galerkin method that uses spherical harmonics and exhibits overall spectral convergence. The frequency shift and the extended time spent in the prolate state within an oscillation period are analyzed for different initial deformation modes and show good agreement with existing literature. The results demonstrate that for large initial deformations, the modal coupling significantly influences the oscillation dynamics, particularly in the interaction between even and odd modes. To study modal coupling in detail, superimposed initial deformations are investigated in which a large-amplitude even mode is superimposed with various individual small-amplitude odd modes. It is shown that the initially excited large-amplitude even mode remains nearly unaffected by the superimposed small-amplitude odd modes. The initially superimposed small-amplitude odd mode, however, is altered by the large-amplitude even mode without being significantly amplified or damped. This nonlinear modal coupling is not present in single even initial deformation modes, where it is shown that an even initial deformation mode cannot excite odd modes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (39)

  1. K. V. Beard, H. T. Ochs, and R. J. Kubesh, Natural oscillations of small raindrops, Nature (London) 342, 408 (1989).
  2. O. A. Basaran, T. C. Scott, and C. H. Byers, Drop oscillations in liquid-liquid systems, AIChE J. 35, 1263 (1989).
  3. J. R. Carruthers and L. R. Testardi, Materials processing in the reduced-gravity environment of space, Annu. Rev. Mater. Sci. 13, 247 (1983).
  4. F. Savart, Mémoire sur la constitution des veines liquides lancées par des orifices circulaires en mince paroi, Annales de chimie et de physique 53, 337 (1833).
  5. J. Plateau, Statique Expérimentale Et Théorique Des Liquides Soumis Aux Seules Forces Moléculaires (Gauthier-Villars, Paris; Trübner & Co., Londres; Gand, F. Clemm, 1873), Vol. 1.
  6. J. W. Strutt (Lord Rayleigh), VI. On the capillary phenomena of jets, Proc. R. Soc. London 29, 71 (1879).
  7. H. Lamb, On the oscillations of a viscous spheroid, Proc. London Math. Soc. s1-13, 51 (1881).
  8. H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, UK, 1932).
  9. S. Chandrasekhar, The oscillations of a viscous liquid globe, Proc. London Math. Soc. s3-9, 141 (1959).
  10. W. H. Reid, The oscillations of a viscous liquid drop, Quart. Appl. Math. 18, 86 (1960).
  11. C. A. Miller and L. E. Scriven, The oscillations of a fluid droplet immersed in another fluid, J. Fluid Mech. 32, 417 (1968).
  12. G. B. Foote, A numerical method for studying liquid drop behavior: Simple oscillation, J. Comput. Phys. 11, 507 (1973).
  13. C. T. Alonso, The dynamics of colliding and oscillating drops, in Proceedings of the International Colloquium on Drops and Bubbles: Their Science and the Systems They Model, Volume I, edited by D. J. Collins, M. S. Plesset, and M. M. Saffren (California Institute of Technology and Jet Propulsion Laboratory, La Cañada Flintridge, CA, 1974).
  14. A. Prosperetti, Free oscillations of drops and bubbles: The initial-value problem, J. Fluid Mech. 100, 333 (1980).
  15. E. Trinh and T. G. Wang, Large-amplitude free and driven drop-shape oscillations: Experimental observations, J. Fluid Mech. 122, 315 (1982).
  16. J. A. Tsamopoulos and R. A. Brown, Nonlinear oscillations of inviscid drops and bubbles, J. Fluid Mech. 127, 519 (1983).
  17. T. G. Wang, A. V. Anilkumar, and C. P. Lee, Oscillations of liquid drops: results from USML-1 experiments in space, J. Fluid Mech. 308, 1 (1996).
  18. R. Natarajan and R. A. Brown, Quadratic resonance in the three‐dimensional oscillations of inviscid drops with surface tension, Phys. Fluids 29, 2788 (1986).
  19. R. Natarajan and R. A. Brown, Third-order resonance effects and the nonlinear stability of drop oscillations, J. Fluid Mech. 183, 95 (1987).
  20. T. S. Lundgren and N. N. Mansour, Oscillations of drops in zero gravity with weak viscous effects, J. Fluid Mech. 194, 479 (1988).
  21. E. Becker, W. J. Hiller, and T. A. Kowalewski, Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets, J. Fluid Mech. 231, 189 (1991).
  22. T. W. Patzek, Jr., R. E. Benner, O. A. Basaran, and L. E. Scriven, Nonlinear oscillations of inviscid free drops, J. Comput. Phys. 97, 489 (1991).
  23. O. A. Basaran, Nonlinear oscillations of viscous liquid drops, J. Fluid Mech. 241, 169 (1992).
  24. S. Meradji, T. Lyubimova, D. Lyubimov, and B. Roux, Numerical simulation of a liquid drop freely oscillating, Cryst. Res. Technol. 36, 729 (2001).
  25. W. R. Smith, Modulation equations for strongly nonlinear oscillations of an incompressible viscous drop, J. Fluid Mech. 654, 141 (2010).
  26. D. Zrnić and G. Brenn, Weakly nonlinear shape oscillations of inviscid drops, J. Fluid Mech. 923, A9 (2021).
  27. D. Zrnić and G. Brenn, Weakly nonlinear shape oscillations of viscoelastic drops, Proc. R. Soc. A 480, 20230887 (2024).
  28. M. Smuda, F. Kummer, M. Oberlack, D. Zrnić, and G. Brenn, From weakly to strongly nonlinear viscous drop shape oscillations: An analytical and numerical study, Phys. Rev. Fluids 9, 063601 (2024).
  29. D. Plümacher, M. Oberlack, Y. Wang, and M. Smuda, On a non-linear droplet oscillation theory via the unified method, Phys. Fluids 32, 067104 (2020).
  30. A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. London Ser. A 453, 1411 (1997).
  31. A. S. Fokas, On the integrability of linear and nonlinear partial differential equations, J. Math. Phys. 41, 4188 (2000).
  32. A. S. Fokas and B. Pelloni, Unified Transform for Boundary Value Problems: Applications and Advances, edited by B. Pelloni and A. S. Fokas (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2014).
  33. A. S. Fokas and B. Pelloni, Method for solving moving boundary value problems for linear evolution equations, Phys. Rev. Lett. 84, 4785 (2000).
  34. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover Publications, Mineola, NY, 2001).
  35. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed., Springer Series in Computational Mathematics, Vol. 8 (Springer, Berlin, 1993).
  36. L. Armijo, Minimization of functions having Lipschitz continuous first partial derivatives, Pac. J. Math. 16, 1 (1966).
  37. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5wrs-k8z4 for the oscillation videos in Figs. 5, 6, 7, 8, 9, 18, 19, 22, and 23.
  38. S. Akbari, M. Oberlack, F. Kummer, K. V. Wilhelm, Y. Wang, and D. Plümacher, Modal coupling of non-linear inviscid axisymmetric droplet shape oscillations, https://tudatalib.ulb.tu-darmstadt.de/handle/tudatalib/4674.
  39. Edited by F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, UK, 2010).

Outline

Information

Phys. Rev. Fluids 11, 033602– Published 5 March, 2026
  • Received 15 July 2025
  • Accepted 4 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation