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Natural oscillations of sessile drops with a central bubble on a solid wall

Published online by Cambridge University Press:  14 March 2025

Fei Zhang Open the ORCID record for Fei Zhang [Opens in a new window] ,
Chunyu Zhang Open the ORCID record for Chunyu Zhang [Opens in a new window] ,
Shuguang Zhao ,
Yingjie Yu Open the ORCID record for Yingjie Yu [Opens in a new window] ,
Jingwei Chen ,
Hang Ding Open the ORCID record for Hang Ding [Opens in a new window]  and
Xinping Zhou Open the ORCID record for Xinping Zhou [Opens in a new window]
Fei Zhang
Affiliation:
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China
Chunyu Zhang
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, PR China
Shuguang Zhao
Affiliation:
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China
Yingjie Yu
Affiliation:
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China
Jingwei Chen
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, PR China
Hang Ding
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, PR China
Xinping Zhou*
Affiliation:
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China State Key Laboratory of Intelligent Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, PR China
*
Corresponding author: Xinping Zhou, [email protected]
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Abstract

We investigate the natural oscillations of sessile drops with a central trapped bubble on a plane using linear potential flow theory, considering both free and pinned contact lines. The system is governed by the contact angle $\alpha$ and the ratio $\tau$ of inner to outer contact line radii. For bubble-containing (BC) hemispherical drops with free contact lines (referred to as free BC semi-drops), the modes mirror half of those in concentric spherical BC drops due to plane symmetry. These modes are labelled ‘plus’ (with greater inner surface deformation) and ‘minus’ (with greater outer surface deformation). As $\tau \to 0$, minus modes converge to those of bubble-free drops. Results show that varying $\alpha$ from $90^\circ$ or pinning the contact line in free BC semi-drops alters the topology of spectral lines, turning original crossings of spectral lines between minus and plus modes into avoided crossings. This shift causes minus and plus modes to form spectral trends with avoided crossings, maintaining their original spectral shapes. In an avoided crossing, two coupled modes cannot be classified as plus or minus due to their comparable inner and outer surface deformations, resulting in mode beating when both are excited, as confirmed by our direct numerical simulations. This study on the impact of inner bubbles on the spectrum may help in predicting bubble size in opaque sessile drops.

JFM classification

Interfacial Flows (free surface): Capillary flows Waves/Free-surface Flows: Capillary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1007 , 25 March 2025 , A18
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Supplementary material: File

Zhang et al. supplementary material movie 1

Mode beating of two coupled modes with equal outer amplitudes of 0.05.
Download Zhang et al. supplementary material movie 1(File)
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Supplementary material: File

Zhang et al. supplementary material movie 2

Mode beating of two coupled modes with unequal outer amplitudes of 0.05 and $0.05/3$.
Download Zhang et al. supplementary material movie 2(File)
File 2.2 MB