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The influence of an insoluble surfactant on capillary oscillations of a bubble in a liquid

Published online by Cambridge University Press:  06 November 2025

Tatyana Lyubimova Open the ORCID record for Tatyana Lyubimova [Opens in a new window] ,
Vladimir Konovalov Open the ORCID record for Vladimir Konovalov [Opens in a new window] ,
Evgeny Borzenko Open the ORCID record for Evgeny Borzenko [Opens in a new window]  and
Alexander Nepomnyashchy
Tatyana Lyubimova*
Affiliation:
Institute of Continuous Media Mechanics UB RAS, 1, Koroleva Street, Perm 614013, Russia
Vladimir Konovalov
Affiliation:
Institute of Continuous Media Mechanics UB RAS, 1, Koroleva Street, Perm 614013, Russia
Evgeny Borzenko
Affiliation:
Tomsk State University, Tomsk, 36, Lenin Ave., Tomsk 634050 Russia
Alexander Nepomnyashchy
Affiliation:
Technion–Israel Institute of Technology, Haifa 32000, Israel
*
Corresponding author: Tatyana Lyubimova, lubimova@psu.ru
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Abstract

This study is devoted to the analysis of capillary oscillations of a gas bubble in a liquid with an insoluble surfactant adsorbed on the surface. The influence of the Gibbs elasticity, the viscosities of the liquid and gas, as well as the shear and dilatational surface viscosities, on the damping of free oscillations is examined. Dependences of the frequency shift and the damping rate on the parameters of the problem are determined. In the limit of small viscosities and neglecting the surfactant surface diffusion, a simplified dispersion relation is obtained, which includes finite parameters of surface viscosities and Gibbs elasticity. From this relation, conditions are identified under which the damping of capillary oscillations can occur with a small frequency. Numerical solutions of the full dispersion relation demonstrate that a non-oscillatory regime is impossible for the considered configuration. An additional mode associated with Gibbs elasticity is discovered, characterized as a rule by low natural frequency and damping rate. Approximate relations for the complex natural frequency of bubble oscillations in a low-viscosity liquid in the presence of a surfactant are derived, including an estimate of the contribution of the gas inside the bubble to viscous dissipation. An original Lagrangian–Eulerian method is proposed and used to perform direct numerical simulations based on the full nonlinear Navier–Stokes equations and natural boundary conditions at the interface, accounting for shear and dilatational viscosities. The numerical data on the damping process confirm the results of the linear theory.

JFM classification

Drops and Bubbles: Bubble dynamics Waves/Free-surface Flows: Capillary waves Mathematical Foundations: Computational methods

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1022 , 10 November 2025 , A45
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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