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Damped shape oscillations of a viscous compound droplet suspended in a viscous host fluid

Published online by Cambridge University Press:  01 December 2021

Fang Li Open the ORCID record for Fang Li [Opens in a new window] ,
Xie-Yuan Yin  and
Xie-Zhen Yin
Fang Li*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, PR China
Xie-Yuan Yin
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, PR China
Xie-Zhen Yin
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, PR China
*
†Email address for correspondence: fli6@ustc.edu.cn
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Abstract

A study of small-amplitude shape oscillations of a viscous compound droplet suspended in a viscous host fluid is performed. A generalized eigenvalue problem is formulated and is solved by using the spectral method. The effects of the relevant non-dimensional parameters are examined for three cases, i.e. a liquid shell in a vacuum and a compound droplet in a vacuum or in a host fluid. The fundamental mode $l=2$ is found to be dominant. There exist two oscillatory modes: the in phase and the out of phase. In most situations, the interfaces oscillate in phase rather than out of phase. For the in-phase mode, in the absence of the host, as the viscosity of the core or the shell increases, the damping rate increases whereas the oscillation frequency decreases; when the viscosity exceeds a critical value, the mode becomes aperiodic with the damping rate bifurcating into two branches. In addition, when the tension of the inner interface becomes smaller than some value, the in-phase mode turns aperiodic. In the presence of the unbounded host fluid, there exists a continuous spectrum. The viscosity of the host may decrease or increase the damping rate of the in-phase mode. The mechanism behind it is discussed. The density contrasts between fluids affect oscillations of the droplet in a complicated way. Particularly, sufficiently large densities of the core or the host lead to the disappearance of the out-of-phase mode. The thin shell approximation predicts well the oscillation of the compound droplet when the shell is thin.

JFM classification

Drops and Bubbles: Drops Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 931 , 25 January 2022 , A33
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abbasi, M.S., Song, R. & Lee, J. 2019 Breakups of an encapsulated surfactant-laden aqueous droplet under a DC electric field. Soft Matt. 15, 8905–8911.CrossRefGoogle Scholar
Anilkumar, A.V., Hmelo, A.B. & Wang, T.G. 2001 Core centering of immiscible compound drops in capillary oscillations: experimental observations. J. Colloid Interface Sci. 242, 465–469.CrossRefGoogle Scholar
Arcidiacono, S., Poulikakos, D. & Ventikos, Y. 2004 Oscillatory behavior of nanodroplets. Phys. Rev. E 70, 011505.CrossRefGoogle ScholarPubMed
Basaran, O.A., Scott, T.C., Byers, C.H. 1989 Drop oscillations in liquid-liquid systems. AIChE J. 35, 1263–1270.CrossRefGoogle Scholar
Brenn, G. & Teichtmeister, S. 2013 Linear shape oscillations and polymeric time scales of viscoelastic drops. J. Fluid Mech. 733, 504–527.CrossRefGoogle Scholar
Chandrasekhar, S. 1959 The oscillations of a viscous liquid globe. Proc. Lond. Math. Soc. 9, 141–149.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Chauhan, A., Maldarelli, C., Papageorgiou, D.T. & Rumschitzki, D.S. 2000 Temporal instability of compound threads and jets. J. Fluid Mech. 420, 1–25.CrossRefGoogle Scholar
Duangsuwan, W., Tüzün, U. & Sermon, P.A. 2009 Configurations and dynamics of single air/alcohol gas-liquid compound drops in vegetable oil. Chem. Engng Sci. 64, 3147–3158.CrossRefGoogle Scholar
Egry, I. 2005 The oscillation spectrum of a compound drop. J. Mater. Sci. 40, 2239–2243.CrossRefGoogle Scholar
Foroushan, H.K. & Jakobsen, H.A. 2020 On the dynamics of fluid partical breakage induced by hydrodynamic instabilities: a review of modelling approaches. Chem. Engng Sci. 219, 115575.CrossRefGoogle Scholar
García, F.J. & González, H. 2008 Normal-mode linear analysis and initial conditions of capillary jets. J. Fluid Mech. 602, 81–117.CrossRefGoogle Scholar
Hoath, S.D., Hsiao, W.K., Martin, G.D., Jung, S., Butler, S.A., Morrison, N.F., Harlen, O.G., Yang, L.S., Bain, C.D. & Hutchings, I.M. 2015 Oscillations of aqueous PEDOT:PSS fluid droplets and the properties of complex fluids in drop-on-demand inkjet printing. J. Non-Newtonian Fluid Mech. 223, 28–36.CrossRefGoogle Scholar
Kawano, S., Hashimoto, H., Ihara, A. & Azima, T. 1997 Small-amplitude oscillations of encapsulated liquid drop interfaces. JSME Intl J. 40, 33–41.CrossRefGoogle Scholar
Khismatullin, D.B. & Nadim, A. 2001 Shape oscillations of a viscoelastic drop. Phys. Rev. E 63, 061508.CrossRefGoogle ScholarPubMed
Kremer, J., Kilzer, A. & Petermann, M. 2018 Simultaneous measurement of surface tension and viscosity using freely decaying oscillations of acoustically levitated droplets. Rev. Sci. Instrum. 89, 015109.CrossRefGoogle ScholarPubMed
Lalanne, B. & Masbernat, O. 2020 Determination of interfacial concentration of a contaminated droplet from shape oscillation damping. Phys. Rev. Lett. 124, 194501.CrossRefGoogle ScholarPubMed
Lamb, H. 1881/82 On the oscillations of a viscous spheroid. Proc. Lond. Math. Soc. 13, 51–70.CrossRefGoogle Scholar
Landman, K.A. 1985 Stability of a viscous compound fluid drop. AIChE J. 31, 567–573.CrossRefGoogle Scholar
Lee, C.P. & Wang, T.G. 1988 The centering dynamics of a thin liquid shell in capillary oscillations. J. Fluid Mech. 188, 411–435.CrossRefGoogle Scholar
Li, F., Yin, X.Y. & Yin, X.Z. 2008 Instability of a viscous coflowing jet in a radial electric field. J. Fluid Mech. 596, 285–311.CrossRefGoogle Scholar
Li, F., Yin, X.Y. & Yin, X.Z. 2019 Small-amplitude shape oscillation and linear instability of an electrically charged viscoelastic liquid droplet. J. Non-Newtonian Fluid Mech. 264, 85–97.CrossRefGoogle Scholar
Li, F., Yin, X.Y. & Yin, X.Z. 2020 Shape oscillations of a viscoelastic droplet suspended in a viscoelastic host liquid. Phys. Rev. Fluids 5, 033610.CrossRefGoogle Scholar
Liu, B., Sumanasekara, U.R. & Bhattacharya, S. 2019 Damped interfacial oscillation of a particle-embedded viscous drop. Phys. Fluids 31, 053303.CrossRefGoogle Scholar
Liu, M., Zheng, Y., Li, J., Chen, S., Liu, Y., Li, J., Li, B. & Zhang, Z. 2017 Effects of molecular weight of PVA on formation, stability and deformation of compound droplets for ICF polymer shells. Nucl. Fusion 57, 016018.CrossRefGoogle Scholar
Lyell, M.J. & Wang, T.G. 1986 Oscillations of a viscous compound drop. Phys. Fluids 29, 3481–3483.CrossRefGoogle Scholar
Lyubimov, D.V., Konovalov, V.V., Lyubimova, T.P. & Egry, I. 2011 Small amplitude shape oscillations of a spherical liquid drop with surface viscosity. J. Fluid Mech. 677, 204–217.CrossRefGoogle Scholar
Lyubimov, D.V., Konovalov, V.V., Lyubimova, T.P. & Egry, I. 2012 Oscillations of a liquid spherical drop encapsulated by a non-concentric spherical layer of dissimilar liquid. Eur. J. Mech. (B/Fluids) 32, 80–87.CrossRefGoogle Scholar
Miller, C.A. & Scriven, L.E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 417–435.CrossRefGoogle Scholar
Montanero, J.M. & Gañán-Calvo, A.M. 2020 Dripping, jetting and tip streaming. Rep. Prog. Phys. 83, 097001.CrossRefGoogle ScholarPubMed
Prosperetti, A. 1980 a Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100, 333–347.CrossRefGoogle Scholar
Prosperetti, A. 1980 b Normal-mode analysis for the oscillations of a viscous liquid drop in an immiscible liquid. J. Méc. 19, 149–182.Google Scholar
Rayleigh, L. 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. A 29, 71–97.Google Scholar
Reid, W.H. 1960 The oscillations of a viscous liquid drop. Q. Appl. Maths 18, 86.CrossRefGoogle Scholar
Saffren, M., Elleman, D.D. & Rhim, W.K. 1981 Normal modes of a compound drop. In Proceedings of the Second International Colloquium on Drops and Bubbles (ed. D.H. Le Croissette). Jet Propulsion Laboratory Publication 82-7.Google Scholar
Santra, S., Das, S. & Chakraborty, S. 2020 Electrically modulated dynamics of a compound droplet in a confined microfluidic environment. J. Fluid Mech. 882, A23.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Shao, X., Fredericks, S.A., Saylor, J.R. & Bostwick, J.B. 2019 Elastocapillary transition in gel drop oscillations. Phys. Rev. Lett. 123, 188002.CrossRefGoogle ScholarPubMed
Shen, J. & Li, X. 1996 Instability of an annular viscous liquid jet. Acta Mechanica 114, 167–183.CrossRefGoogle Scholar
Shiryaev, A. 2020 Oscillations of an inviscid encapsulated drop. Fluid Dyn. Mater. Process. 16, 761–771.CrossRefGoogle Scholar
Shusser, M. & Weihs, D. 2010 Compound drops as spherical shell vortices. Fluid Dyn. Res. 42, 025502.CrossRefGoogle Scholar
Staat, H.J.J., van der Bos, A., van den Berg, M., Reinten, H., Wijshoff, H., Versluis, M. & Lohse, D. 2017 Ultrafast imaging method to measure surface tension and viscosity of inkjet-printed droplets in flight. Exp. Fluids 58, 2.CrossRefGoogle Scholar
Vian, A., Reuse, B. & Amstad, E. 2018 Scalable production of double emulsion drops with thin shells. Lab on a Chip 18, 1936–1942.CrossRefGoogle ScholarPubMed
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