Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T09:37:49.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Oscillations of a partially wetting bubble

Published online by Cambridge University Press:  21 July 2022

D. Ding  and
J.B. Bostwick Open the ORCID record for J.B. Bostwick [Opens in a new window]
D. Ding
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA
J.B. Bostwick*
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the linear stability of a compressible sessile bubble in an ambient fluid that partially wets a planar solid support, where the gas is assumed to be an ideal gas that obeys the adiabatic law. The frequency spectrum is computed from an integrodifferential boundary value problem and depends upon the wetting conditions through the static contact angle $\alpha$, the dimensionless equilibrium bubble pressure $\varPi$, and the contact-line dynamics that we assume to be either (i) pinned or (ii) freely moving with fixed contact angle. Corresponding mode shapes are defined by the polar-azimuthal mode number pair $[k,\ell ]$ with $k+\ell =\mathbb {Z}^{+}_{even}$. We report instabilities to the (i) $[0,0]$ breathing mode associated with volume change, and (ii) $[1,1]$ mode that is linked to horizontal centre-of-mass motion of the bubble. Stability diagrams and instability growth rates are computed, and the respective instability mechanisms are revealed through an energy analysis. The zonal $\ell =0$ modes are associated with volume change, and we show that there is a complex dependence between the classical volume and shape change modes for wetting conditions that differ from neutral wetting $\alpha =90^\circ$. Finally, we show how the classical frequency degeneracy for the Rayleigh–Lamb modes of the free bubble splits for the azimuthal modes $\ell \neq 0,1$.

JFM classification

Drops and Bubbles: Bubble dynamics Interfacial Flows (free surface): Contact lines
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 945 , 25 August 2022 , A24
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Allen, J.S. & Roy, R.A. 2000 Dynamics of gas bubbles in viscoelastic fluids. 1. Linear viscoelasticity. J. Acoust. Soc. Am. 107 (6), 3167–3178.CrossRefGoogle Scholar
Anna, S.L. 2016 Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 48 (1), 285–309.CrossRefGoogle Scholar
Ardron, K.H. & Giustini, G. 2021 On the wetting behavior of surfaces in boiling. Phys. Fluids 33 (11), 111302.CrossRefGoogle Scholar
Asaki, T.J. & Marston, P.L. 1995 Equilibrium shape of an acoustically levitated bubble driven above resonance. J. Acoust. Soc. Am. 97 (4), 2138–2143.CrossRefGoogle Scholar
Benilov, E.S. & Cummins, C.P. 2013 The stability of a static liquid column pulled out of an infinite pool. Phys. Fluids 25 (11), 112105.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2009 Capillary oscillations of a constrained liquid drop. Phys. Fluids 21 (3), 032108.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2010 Stability of constrained cylindrical interfaces and the torus lift of Plateau–Rayleigh. J. Fluid Mech. 647, 201–219.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2014 Dynamics of sessile drops. Part 1. Inviscid theory. J. Fluid Mech. 760, 5–38.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2015 Stability of constrained capillary surfaces. Annu. Rev. Fluid Mech. 47, 539–568.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2016 Response of driven sessile drops with contact-line dissipation. Soft Matt. 12 (43), 8919–8926.CrossRefGoogle ScholarPubMed
Chang, C.-T., Bostwick, J.B., Daniel, S. & Steen, P.H. 2015 Dynamics of sessile drops. Part 2. Experiment. J. Fluid Mech. 768, 442–467.CrossRefGoogle Scholar
Chang, C.-T., Bostwick, J.B., Steen, P.H. & Daniel, S. 2013 Substrate constraint modifies the Rayleigh spectrum of vibrating sessile drops. Phys. Rev. E 88 (2), 023015.CrossRefGoogle ScholarPubMed
Davis, S.H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98 (2), 225–242.CrossRefGoogle Scholar
Douglas, Z., Boziuk, T.R., Smith, M.K. & Glezer, A. 2012 Acoustically enhanced boiling heat transfer. Phys. Fluids 24 (5), 052105.CrossRefGoogle Scholar
Dupré, A. & Dupré, P. 1869 Théorie mécanique de la chaleur. Gauthier-Villars.Google Scholar
Fayzrakhmanova, I.S., Straube, A.V. & Shklyaev, S. 2011 Bubble dynamics atop an oscillating substrate: interplay of compressibility and contact angle hysteresis. Phys. Fluids 23 (10), 102105.CrossRefGoogle Scholar
Feng, Z.C. & Leal, L.G. 1997 Nonlinear bubble dynamics. Annu. Rev. Fluid Mech. 29 (1), 201–243.CrossRefGoogle Scholar
Gelderblom, H., Zijlstra, A.G., van Wijngaarden, L. & Prosperetti, A. 2012 Oscillations of a gas pocket on a liquid-covered solid surface. Phys. Fluids 24 (12), 122101.CrossRefGoogle Scholar
Guédra, M., Cleve, S., Mauger, C., Blanc-Benon, P. & Inserra, C. 2017 Dynamics of nonspherical microbubble oscillations above instability threshold. Phys. Rev. E 96 (6), 063104.CrossRefGoogle ScholarPubMed
Guédra, M., Cleve, S., Mauger, C. & Inserra, C. 2020 Subharmonic spherical bubble oscillations induced by parametric surface modes. Phys. Rev. E 101 (1), 011101.CrossRefGoogle ScholarPubMed
Hocking, L.M. 1987 The damping of capillary–gravity waves at a rigid boundary. J. Fluid Mech. 179, 253–266.CrossRefGoogle Scholar
Hua, C. & Johnsen, E. 2013 Nonlinear oscillations following the Rayleigh collapse of a gas bubble in a linear viscoelastic (tissue-like) medium. Phys. Fluids 25 (8), 083101.CrossRefGoogle Scholar
Kawchuk, G.N., Fryer, J., Jaremko, J.L., Zeng, H., Rowe, L. & Thompson, R. 2015 Real-time visualization of joint cavitation. PloS one 10 (4), e0119470.CrossRefGoogle ScholarPubMed
Keller, J.B. & Miksis, M. 1980 Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68 (2), 628–633.CrossRefGoogle Scholar
Ko, S.H., Lee, S.J. & Kang, K.H. 2009 A synthetic jet produced by electrowetting-driven bubble oscillations in aqueous solution. Appl. Phys. Lett. 94 (19), 194102.CrossRefGoogle Scholar
Lamb, H. 1924 Hydrodynamics. University Press.Google Scholar
Longuet-Higgins, M.S. 1989 Monopole emission of sound by asymmetric bubble oscillations. Part 1. Normal modes. J. Fluid Mech. 201, 525–541.CrossRefGoogle Scholar
Lv, P., Peñas, P., The, H.L., Eijkel, J., Van Den Berg, A., Zhang, X. & Lohse, D. 2021 Self-propelled detachment upon coalescence of surface bubbles. Phys. Rev. Lett. 127 (23), 235501.CrossRefGoogle ScholarPubMed
Maassen, K.F., Brown, J.S., Choi, H., Thompson, L.L. & Bostwick, J.B. 2020 Acoustic analysis of ultrasonic assisted soldering for enhanced adhesion. Ultrasonics 101, 106003.CrossRefGoogle ScholarPubMed
MacRobert, T.M. 1967 Spherical Harmonics. Pergamon Press.Google Scholar
Maksimov, A. 2020 Splitting of the surface modes for bubble oscillations near a boundary. Phys. Fluids 32 (10), 102104.CrossRefGoogle Scholar
Maksimov, A.O. & Polovinka, Y.A. 2013 Volume oscillations of a constrained bubble. Phys. Fluids 25 (6), 062104.CrossRefGoogle Scholar
Marin, A., Rossi, M., Rallabandi, B., Wang, C., Hilgenfeldt, S. & Kähler, C.J. 2015 Three-dimensional phenomena in microbubble acoustic streaming. Phys. Rev. Appl. 3 (4), 041001.CrossRefGoogle Scholar
Marmottant, P., Raven, J.P., Gardeniers, H.J.G.E., Bomer, J.G. & Hilgenfeldt, S. 2006 Microfluidics with ultrasound-driven bubbles. J. Fluid Mech. 568, 109–118.CrossRefGoogle Scholar
Marmottant, P., Van Der Meer, S., Emmer, M., Versluis, M., De Jong, N., Hilgenfeldt, S. & Lohse, D. 2005 A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture. J. Acoust. Soc. Am. 118 (6), 3499–3505.CrossRefGoogle Scholar
McCraney, J., Kern, V., Bostwick, J., Daniel, S. & Steen, P. 2022 Oscillations of drops with mobile contact lines on the International Space Station: Elucidation of terrestrial inertial droplet spreading. Phys. Rev. Lett. (accepted).Google Scholar
Mei, C.C. & Zhou, X. 1991 Parametric resonance of a spherical bubble. J. Fluid Mech. 229, 29–50.CrossRefGoogle Scholar
Pereiro, I., Khartchenko, A.F., Petrini, L. & Kaigala, G.V. 2019 Nip the bubble in the bud: a guide to avoid gas nucleation in microfluidics. Lab on a Chip 19 (14), 2296–2314.CrossRefGoogle ScholarPubMed
Plesset, M.S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9 (1), 145–185.CrossRefGoogle Scholar
Prosperetti, A. 2011 Advanced Mathematics for Applications. Cambridge University Press.CrossRefGoogle Scholar
Prosperetti, A. 2012 Linear oscillations of constrained drops, bubbles, and plane liquid surfaces. Phys. Fluids 24 (3), 032109.CrossRefGoogle Scholar
Rallabandi, B., Wang, C. & Hilgenfeldt, S. 2014 Two-dimensional streaming flows driven by sessile semicylindrical microbubbles. J. Fluid Mech. 739, 57–71.CrossRefGoogle Scholar
Rayleigh, Lord, et al. 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29 (196–199), 71–97.Google Scholar
Sakakeeny, J., Deshpande, C., Deb, S., Alvarado, J.L. & Ling, Y. 2021 A model to predict the oscillation frequency for drops pinned on a vertical planar surface. J. Fluid Mech. 928, A28.CrossRefGoogle Scholar
Shaffer, J., Maassen, K., Wilson, C., Tilton, P., Thompson, L., Choi, H. & Bostwick, J. 2019 Development of an open-sourced automated ultrasonic-assisted soldering system. J. Manuf. Process. 47, 284–290.CrossRefGoogle Scholar
Sharma, S. & Wilson, D.I. 2021 On a toroidal method to solve the sessile-drop oscillation problem. J. Fluid Mech. 919, A39.CrossRefGoogle Scholar
Sharp, J.S. 2012 Resonant properties of sessile droplets; contact angle dependence of the resonant frequency and width in glycerol/water mixtures. Soft Matt. 8 (2), 399–407.CrossRefGoogle Scholar
Sharp, J.S., Farmer, D.J. & Kelly, J. 2011 Contact angle dependence of the resonant frequency of sessile water droplets. Langmuir 27 (15), 9367–9371.CrossRefGoogle ScholarPubMed
Shaw, S.J. 2006 Translation and oscillation of a bubble under axisymmetric deformation. Phys. Fluids 18 (7), 072104.CrossRefGoogle Scholar
Shaw, S.J. 2017 Nonspherical sub-millimeter gas bubble oscillations: parametric forcing and nonlinear shape mode coupling. Phys. Fluids 29 (12), 122103.CrossRefGoogle Scholar
Shklyaev, S. & Straube, A.V. 2008 Linear oscillations of a compressible hemispherical bubble on a solid substrate. Phys. Fluids 20 (5), 052102.CrossRefGoogle Scholar
Steen, P.H., Chang, C.-T. & Bostwick, J.B. 2019 Droplet motions fill a periodic table. Proc. Natl Acad. Sci. USA 116 (11), 4849–4854.CrossRefGoogle ScholarPubMed
Vejrazka, J., Vobecka, L. & Tihon, J. 2013 Linear oscillations of a supported bubble or drop. Phys. Fluids 25 (6), 062102.CrossRefGoogle Scholar
Vergniolle, S. & Brandeis, G. 1996 Strombolian explosions. 1. A large bubble breaking at the surface of a lava column as a source of sound. J. Geophys. Res. 101 (B9), 20433–20447.CrossRefGoogle Scholar
Vogel, T. 1982 Symmetric unbounded liquid bridges. Pac. J. Maths 103 (1), 205–241.CrossRefGoogle Scholar
Volk, A. & Kähler, C.J. 2018 Size control of sessile microbubbles for reproducibly driven acoustic streaming. Phys. Rev. Appl. 9 (5), 054015.CrossRefGoogle Scholar
Wang, C., Rallabandi, B. & Hilgenfeldt, S. 2013 Frequency dependence and frequency control of microbubble streaming flows. Phys. Fluids 25 (2), 022002.CrossRefGoogle Scholar
Yang, X. & Church, C.C. 2005 A model for the dynamics of gas bubbles in soft tissue. J. Acoust. Soc. Am. 118 (6), 3595–3606.CrossRefGoogle Scholar
Young, T. 1805 III. An essay on the cohesion of fluids. Phil. Trans. R. Soc. Lond. 95, 65–87.Google Scholar
Zhang, C., Adera, S., Aizenberg, J. & Chen, Z. 2021 Why are water droplets highly mobile on nanostructured oil-impregnated surfaces? ACS Appl. Mater. Interfaces 13 (13), 15901–15909.CrossRefGoogle ScholarPubMed