Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T19:20:18.338Z Has data issue: false hasContentIssue false
  • English
  • Français

Detailed finer features in spectra of interfacial waves for characterization of a bubble-laden drop

Published online by Cambridge University Press:  17 October 2017

Udugama R. Sumanasekara  and
Sukalyan Bhattacharya Open the ORCID record for Sukalyan Bhattacharya [Opens in a new window]
Udugama R. Sumanasekara
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA
Sukalyan Bhattacharya*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA
*
†Email address for correspondence: s.bhattacharya@ttu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This article describes unexplored details of the intriguing spectral manifestation of the small-amplitude waves at the surfaces of a bubble-laden drop. Its natural frequencies of interfacial pulsation reveal a non-trivial variation with the position of the cavity inside the liquid. This configurational dependence of spectra is calculated for arbitrary location of the void by using a novel approach under low capillary number and low Bond number limits. The analysis is based on expansion in two sets of basis functions where their mutual transformations are utilized to enforce interfacial boundary conditions. The obtained results quantify a few important features which have both scientific and technological significance. For a concentric geometry, the inherent azimuthal degeneracy makes the frequencies for a number of vibrational modes exactly the same. For an eccentric position of the bubble, however, this degeneracy disappears, creating small deviations in the spectral values corresponding to different azimuthal modes. Such behaviour is akin to fine-structure split in an atomic system, where different quantum numbers ensure small deviation in energy levels of the states. The formulated mathematical procedure can determine the individual frequency values for the interfacial oscillation even if these are grouped closely together in bands. The paper shows how the number of fine structures inside a band and their specific values can be exploited to predict the size and position of the cavity in an opaque drop without any direct visualization of its interior.

JFM classification

Drops and Bubbles: Bubble dynamics Waves/Free-surface Flows: Capillary waves Drops and Bubbles: Drops and Bubbles
Type
Papers
Information
Journal of Fluid Mechanics , Volume 831 , 25 November 2017 , pp. 698 - 718
Check for updates
Copyright
© 2017 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

Apfel, R. E., Tian, Y., Jankovsky, J., Shi, T., Chen, X., Holt, R. G., Trinh, E., Croonquist, A., Thornton, K. C., Sacco, A. et al. 1997 Free oscillations and surfactant studies of superdeformed drops in microgravity. Phys. Rev. Lett. 78, 1912–1915.CrossRefGoogle Scholar
Basaran, O. A. 1992 Nonlinear oscillations of viscous-liquid drops. J. Fluid Mech. 241, 169.Google Scholar
Bhattacharya, S. 2008 Cooperative motion of spheres arranged in periodic grids between two parallel walls. J. Chem. Phys. 128, 074709.Google Scholar
Bhattacharya, S. 2016 Interfacial wave dynamics of a drop with an embedded bubble. Phys. Rev. E 93, 023119.Google Scholar
Bhattacharya, S., Mishra, C. & Bhattacharya, S. 2010 Analysis of general creeping motion of a sphere inside a cylinder. J. Fluid Mech. 642, 295–328.CrossRefGoogle Scholar
Bostwick, J. B. & Steen, P. H. 2009 Capillary oscillations of a constrained liquid drop. Phys. Fluids 21, 032108.CrossRefGoogle Scholar
Bostwick, J. B. & Steen, P. H. 2013 Coupled oscillations of deformable spherical-cap droplets. Part 1. Inviscid motions. J. Fluid Mech. 714, 312–335.Google Scholar
Chandrasekhar, S. 1959 The oscillation of a viscous liquid globe. Proc. Lond. Math. Soc. 9, 141.CrossRefGoogle Scholar
Epstein, P. S. & Plesset, M. S. 1950 On the stability of gas bubbles in liquid–gas solutions. J. Chem. Phys. 18, 1505.Google Scholar
Iyer, V. A., Abraham, J. & Magi, V. 2002 Exploring injected droplet size effects on steady liquid penetration in a Diesel spray with a two-fluid model. Intl J. Heat Mass Transfer 45, 519.CrossRefGoogle Scholar
Jiri, V., Lucie, V. & Jaroslav, T. 2013 Linear oscillations of a supported bubble or drop. Phys. Fluids 25, 062102.Google Scholar
Kawano, S., Shirai, A. & Nagasaka, S. 2007 Deformations of thin liquid spherical shells in liquid–liquid–gas systems. Phys. Fluids 19, 012105.CrossRefGoogle Scholar
Landman, K. A. 1985 Stability of viscous compound fluid drop. AIChE J. 31, 567–573.Google Scholar
Lauterborn, W. & Ohl, C. D. 1997 Cavitation bubble dynamics. In Ultrasonic Sonochemistry.Google Scholar
Lundgren, T. S. & Mansour, N. N. 1988 Oscillations of drops in zero gravity with weak viscous effects. J. Fluid Mech. 194, 479–510.Google Scholar
Lyubimov, D. V., Konovalov, V. V., Lyubimova, T. P. & Egry, I. 2012 Oscillation 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 Oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 217.Google Scholar
Minemawari, H., Yamada, T., Matsui, H. & Tsutsumi, J. 2011 Inkjet printing of single crystal films. Nature 475, 364.Google Scholar
Navardi, S. & Bhattacharya, S. 2010a A new lubrication theory to derive far-field axial pressure-difference due to force singularities in cylindrical or annular vessels. J. Math. Phys. 51, 043102.Google Scholar
Navardi, S. & Bhattacharya, S. 2010b Axial pressure-difference between far-fields across a sphere in viscous flow bounded by a cylinder. Phys. Fluids 22, 103306.Google Scholar
Navardi, S. & Bhattacharya, S. 2010c Effect of confining conduit on effective viscosity of dilute colloidal suspension. J. Chem. Phys. 132, 114114.Google Scholar
Navardi, S. & Bhattacharya, S. 2013 General methodology to evaluate two-particle hydrodynamic friction inside cylinder-bound viscous fluid. Comput. Fluids 76, 149–169.CrossRefGoogle Scholar
Obreschkow, D., Kobel, P., Dorsaz, N., Bosset, A. De., Nicollier, C. & Farhat, M. 2006 Cavitation bubble dynamics inside liquid drops in microgravity. Phys. Rev. Lett. 97, 094502.Google Scholar
Patzer, J. F. & Homsy, G. M. 1975 Hydrodynamic stability of thin spherically concentric fluid shells. J. Colloid Interface Sci. 51, 499–508.Google Scholar
Pelekasis, N. A., Tsamopoulos, J. A. & Manolis, G. D. 1991 Nonlinear oscillations of liquid shells in zero gravity. J. Fluid Mech. 230, 541–582.CrossRefGoogle Scholar
Pelosi, A. D. & Gany, A. 2012 Modeling the combustion of a solid fuel containing a liquid oxidizer droplet. J. Propul. Power 28, 1379.CrossRefGoogle Scholar
Plesset, M. S. 1949 The dynamics of cavitation bubbles. Trans. ASME J. Appl. Mech. 16, 228.Google Scholar
Prosperetti, A. A. 1980 Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100, 333.Google Scholar
Rayleigh, J. W. S. 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29, 71.Google Scholar
Rayleigh, J. W. S. 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94–98.Google Scholar
Saffren, M., Elleman, D. D. & Rhim, W. K. 1982 Proceedings of the second international colloquium on drops and bubbles. Jet Propul. 82‐7, 7–14.Google Scholar
Sangani, A. S. & Mo, G. B. 1996 An O (N) algorithm for Stokes and Laplace interactions of particles. Phys. Fluids 8, 1990–2010.Google Scholar
Schwabe, M., Rubin-Zuzic, M., Zhdanov, S., Ivlev, A. V., Thomas, H. M. & Morfill, G. E. 2009 Formation of bubbles, blobs and surface cusps in complex plasmas. Phys. Rev. Lett. 102, 255005.Google Scholar
Shaw, S. J. 2006 Translation and oscillation of a bubble under axisymmetric deformation. Phys. Fluids 18, 072104.CrossRefGoogle Scholar
Strani, M. & Sebetta, F. 1984 Free-vibrations of a drop in partial contact with a solid support. J. Fluid Mech. 141, 233–247.Google Scholar
Strani, M. & Sebetta, F. 1988 Viscous oscillations of a supported drop in an immiscible fluid. J. Fluid Mech. 189, 397–421.Google Scholar
Taller, D., Go, D. B. & Chang, H. C. 2012 Self-similar micron-size and nanosize drops of liquid generated by surface acoustic waves. Phys. Rev. Lett. 109, 224301.CrossRefGoogle ScholarPubMed
Trinh, E. H., Marston, P. L. & Robey, J. L. 1988 Acoustic measurement of the surface tension of levitated drops. J. Colloid Interface Sci. 124, 95.Google Scholar
Tsamopoulos, J. A. & Brown, R. A. 1983 Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519–537.CrossRefGoogle Scholar
Tsamopoulos, J. A. & Brown, R. A. 1987 Dynamic centering of liquid shells. Phys. Fluids 30, 27.Google Scholar
Yunker, P. J., Still, T., Lohr, M. A. & Yodh, A. G. 2011 Suppression of the coffee-ring effect by shape-dependent capillary interactions. Nature 476, 308.Google Scholar