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Impact of the history force on the motion of droplets in shaken liquids

Frederik R. Gareis and Walter Zimmermann

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Phys. Rev. Fluids 11, 043604 – Published 22 April, 2026

DOI: https://doi.org/10.1103/y1zd-lpyw

Abstract

Droplets, solid particles, and gas bubbles in unsteady flows experience the Basset–Boussinesq history force (BBH) in addition to steady viscous drag, added mass, and buoyancy. Although physically relevant, the BBH term is often neglected because its inclusion is analytically and numerically demanding. To assess when this approximation fails, we revisit unsteady Stokes flows around spherical droplets of finite viscosity and derive, from first principles, the velocity fields and hydrodynamic forces, including both the classical rigid-particle limit and the free-slip (zero-viscosity) bubble limit. The resulting expressions also encompass cases with time-dependent bubble radii. We further illustrate how the BBH force arises from transient, diffusion-driven vortex structures around accelerating particles. Applying these results to droplets or particles in horizontally shaken liquids (periodically accelerated flows), we find that in the transition regime between the quasisteady Stokes limit and the inertia-dominated regime, BBH can lead to a reduction of the droplet deflection amplitude by more than 60% compared to predictions that neglect memory effects. We also derive a characteristic scaling of the displacement amplitude in the low-frequency limit, providing an unambiguous, experimentally verifiable signature of the BBH. For light particles and gas bubbles, the BBH contribution becomes more significant (relative to the other hydrodynamic forces) compared to that of heavy particles, such as droplets in air.

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Phys. Rev. Fluids 11, 043604– Published 22 April, 2026
  • Received 27 June 2025
  • Accepted 26 February 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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