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Nonlinear mode coupling and excitation in nonaxisymmetric droplet shape oscillations

Schahin Akbari*,†, Mostafa Noori†, Yongqi Wang, and Martin Oberlack†

  • *Contact author: akbari@fdy.tu-darmstadt.de
  • †Also at Technical University of Darmstadt, Centre for Computational Engineering, Dolivostraße 15, 64293 Darmstadt, Germany.
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Phys. Rev. Fluids 11, 083602 – Published 14 August, 2026

DOI: https://doi.org/10.1103/6r6m-rt5q

Abstract

Nonaxisymmetric droplet shape oscillations are investigated by extending the numerical framework developed in Akbari et al. [Phys. Rev. Fluids 11, 033602 (2026)], which is based on a Galerkin method using spherical harmonics, to fully three-dimensional configurations. The study focuses on the role of shape symmetry and mode excitation of inviscid droplet shape oscillation. The numerical results confirm that the oscillation frequency is independent of the azimuthal wave number, in agreement with classical theoretical predictions. The discrete geometric symmetries of both the initial droplet deformation and the initial velocity potential are demonstrated to be preserved throughout the oscillation. Consequently, only modes compatible with these symmetries are excited. When the initial velocity potential breaks the symmetries of the droplet deformation, leading to a reduction in these symmetries, additional modes are excited compared to those for a zero initial velocity. A similar result is observed in the droplet shape oscillations of a single initial deformation mode and a superposition of two initial deformation modes. In the former case, only modes that satisfy the symmetries of the initial deformation are excited. In the latter case, the superposition reduces the overall symmetry of the droplet shape, leading to additional mode excitation through symmetry breaking. These results highlight the fundamental role of symmetry preservation in nonlinear droplet oscillations. By exploiting these symmetry properties, the computational effort could be reduced without compromising numerical accuracy, as modes that do not occur can be excluded from the solution approach prior to the simulation.

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Phys. Rev. Fluids 11, 083602– Published 14 August, 2026
  • Received 5 February 2026
  • Accepted 22 July 2026
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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.

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