Hostname: page-component-669899f699-cf6xr Total loading time: 0 Render date: 2025-04-30T10:47:53.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Oscillations of a ring-constrained charged drop

Published online by Cambridge University Press:  30 June 2021

Brayden W. Wagoner ,
Vishrut Garg Open the ORCID record for Vishrut Garg [Opens in a new window] ,
Michael T. Harris ,
Doraiswami Ramkrishna  and
Osman A. Basaran Open the ORCID record for Osman A. Basaran [Opens in a new window]
Brayden W. Wagoner
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN47907, USA
Vishrut Garg
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN47907, USA
Michael T. Harris
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN47907, USA
Doraiswami Ramkrishna
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN47907, USA
Osman A. Basaran*
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN47907, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Free drops of uncharged and charged inviscid, conducting fluids subjected to small-amplitude perturbations undergo linear oscillations (Rayleigh, Proc. R. Soc. London, vol. 29, no. 196–199, 1879, pp. 71–97; Rayleigh, Philos. Mag., vol. 14, no. 87, 1882, pp. 184–186). There exist a countably infinite number of oscillation modes, $n=2, 3, \ldots$, each of which has a characteristic frequency and mode shape. Presence of charge ($Q$) lowers modal frequencies and leads to instability when $Q>Q_R$ (Rayleigh limit). The $n=0$ and $n=1$ modes are disallowed because they violate volume conservation and cause centre of mass (COM) motion. Thus, the first mode to become unstable is the $n=2$ prolate–oblate mode. For free drops, there is a one-to-one correspondence between mode number and shape (Legendre polynomial $P_n$). Recent research has shifted to studying oscillations of spherical drops constrained by solid rings. Pinning the drop introduces a new low-frequency mode of oscillation ($n=1$), one associated primarily with COM translation of the constrained drop. We analyse theoretically the effect of charge on oscillations of constrained drops. Using normal modes and solving a linear operator eigenvalue problem, we determine the frequency of each oscillation mode. Results demonstrate that for ring-constrained charged drops (RCCDs), the association between mode number and shape is lost. For certain pinning locations, oscillations exhibit eigenvalue veering as $Q$ increases. While slightly charged RCCDs pinned at zeros of $P_2$ have a first mode that involves COM motion and a second mode that entails prolate–oblate oscillations, the modes flip as $Q$ increases. Thereafter, prolate–oblate oscillations of RCCDs adopt the role of being the first mode because they exhibit the lowest vibration frequency. At the Rayleigh limit, the first eigenmode – prolate–oblate oscillations – loses stability while the second – involving COM motion – remains stable.

JFM classification

Drops and Bubbles: Drops Interfacial Flows (free surface): Capillary flows Drops and Bubbles: Electrohydrodynamic effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 921 , 25 August 2021 , A19
Check for updates
Copyright
© The Author(s), 2021. 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.)

Article purchase

Temporarily unavailable

Footnotes

‡

Present address: Air Products and Chemicals Inc., Allentown, Pennsylvania 18195, USA

References

REFERENCES

Adornato, P.M. & Brown, R.A. 1983 Shape and stability of electrostatically levitated drops. Proc. R. Soc. A 389 (1796), 101–117.Google Scholar
Anthony, C.R., Kamat, P.M., Harris, M.T. & Basaran, O.A. 2019 Dynamics of contracting filaments. Phys. Rev. Fluids 4 (9), 093601.CrossRefGoogle Scholar
Apfel, R.E., et al. 1997 Free oscillations and surfactant studies of superdeformed drops in microgravity. Phys. Rev. Lett. 78 (10), 1912–1915.CrossRefGoogle Scholar
Barmatz, M.B., Trinh, E.H., Wang, T.G., Elleman, D.D. & Jacobi, N. 1983 Acoustic system for material transport. US Patent 4 393 708.Google Scholar
Basaran, O.A. 1992 Nonlinear oscillations of viscous liquid drops. J. Fluid Mech. 241, 169–198.CrossRefGoogle Scholar
Basaran, O.A. & DePaoli, D.W. 1994 Nonlinear oscillations of pendant drops. Phys. Fluids 6 (9), 2923–2943.CrossRefGoogle Scholar
Basaran, O.A., Gao, H. & Bhat, P.P. 2013 Nonstandard inkjets. Ann. Rev. Fluid Mech. 45, 85–113.CrossRefGoogle Scholar
Basaran, O.A., Patzek, T.W., Benner, R.E. Jr. & Scriven, L.E. 1995 Nonlinear oscillations and breakup of conducting, inviscid drops in an externally applied electric field. Ind. Engng Chem. Res. 34 (10), 3454–3465.CrossRefGoogle Scholar
Basaran, O.A., Scott, T.C. & Byers, C.H. 1989 Drop oscillations in liquid-liquid systems. AIChE J. 35 (8), 1263–1270.CrossRefGoogle Scholar
Basaran, O.A. & Scriven, L.E. 1989 a Axisymmetric shapes and stability of charged drops in an external electric field. Phys. Fluids A 1 (5), 799–809.CrossRefGoogle Scholar
Basaran, O.A. & Scriven, L.E. 1989 b Axisymmetric shapes and stability of isolated charged drops. Phys. Fluids A 1 (5), 795–798.CrossRefGoogle Scholar
Basaran, O.A. & Scriven, L.E. 1990 Axisymmetric shapes and stability of pendant and sessile drops in an electric field. J. Colloid Interface Sci. 140 (1), 10–30.CrossRefGoogle Scholar
Basaran, O.A. & Wohlhuter, F.K. 1992 Effect of nonlinear polarization on shapes and stability of pendant and sessile drops in an electric (magnetic) field. J. Fluid Mech. 244, 1–16.CrossRefGoogle Scholar
du Bois, J.L., Adhikari, S. & Lieven, N. 2009 Eigenvalue curve veering in stressed structures: an experimental study. J. Sound Vib. 322 (4–5), 1117–1124.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. 2013 a Coupled oscillations of deformable spherical-cap droplets. Part 1. Inviscid motions. J. Fluid Mech. 714, 312–335.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2013 b Coupled oscillations of deformable spherical-cap droplets. Part 2. Viscous motions. J. Fluid Mech. 714, 336–360.CrossRefGoogle Scholar
Brazier-Smith, P.R. 1971 Stability and shape of isolated and pairs of water drops in an electric field. Phys. Fluids 14 (1), 1–6.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
Brocca, P., Saponaro, A., Introini, B., Rondelli, V., Pannuzzo, M., Raciti, D., Corti, M. & Raudino, A. 2019 Protein adsorption at the air–water interface by a charge sensing interferometric technique. Langmuir 35 (49), 16087–16100.CrossRefGoogle Scholar
Busse, F.H. 1984 Oscillations of a rotating liquid drop. J. Fluid Mech. 142, 1–8.CrossRefGoogle Scholar
Castrejón-Pita, J.R., Baxter, W.R.S., Morgan, J., Temple, S., Martin, G.D. & Hutchings, I.M. 2013 Future, opportunities and challenges of inkjet technologies. Atom. Sprays 23 (6), 541–565.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydromagnetic and Hydrodynamic Stability. Oxford University Press.Google Scholar
Claassen, R.W. 1962 Vibrations of a rectangular cantilever plate. J. Aerosp. Sci. 29 (11), 1300–1305.CrossRefGoogle Scholar
Collins, R.T., Jones, J.J., Harris, M.T. & Basaran, O.A. 2008 Electrohydrodynamic tip streaming and emission of charged drops from liquid cones. Nat. Phys. 4 (2), 149–154.CrossRefGoogle Scholar
Collins, R.T., Sambath, K., Harris, M.T. & Basaran, O.A. 2013 Universal scaling laws for the disintegration of electrified drops. Proc. Natl Acad. Sci. 110 (13), 4905–4910.CrossRefGoogle ScholarPubMed
DePaoli, D.W., Basaran, O.A., Feng, J.Q. & Scott, T.C. 1995 Forced oscillations of pendant drops. Sep. Sci. Tech. 30 (7–9), 1189–1202.CrossRefGoogle Scholar
Feng, J.Q. & Beard, K.V. 1990 Small-amplitude oscillations of electrostatically levitated drops. Proc. R. Soc. A 430 (1878), 133–150.Google Scholar
Fernández de La Mora, J. 2007 The fluid dynamics of taylor cones. Annu. Rev. Fluid Mech. 39, 217–243.CrossRefGoogle Scholar
Harris, M.T., Scott, T.C. & Byers, C.H. 1992 Method and apparatus for the production of metal oxide powder. US Patent 5 122 360.Google Scholar
Hirsa, A.H., López, C.A., Laytin, M.A., Vogel, M.J. & Steen, P.H. 2005 Low-dissipation capillary switches at small scales. App. Phys. Lett. 86 (1), 014106.CrossRefGoogle Scholar
Lalanne, B., Masbernat, O. & Risso, F. 2020 Determination of interfacial concentration of a contaminated droplet from shape oscillation damping. Phys. Rev. Lett. 124 (19), 194501.CrossRefGoogle ScholarPubMed
Lamb, H. 1932 Hydrodynamics, 6th edn, Dover.Google Scholar
Leissa, A.W. 1974 On a curve veering aberration. Z. Angew. Math. Phys. 25 (1), 99–111.CrossRefGoogle Scholar
Lopez, J.M. & Hirsa, A.H. 2000 Surfactant-influenced gas–liquid interfaces: nonlinear equation of state and finite surface viscosities. J. Colloid Interface Sci. 229 (2), 575–583.CrossRefGoogle ScholarPubMed
López, C.A. & Hirsa, A.H. 2008 Fast focusing using a pinned-contact oscillating liquid lens. Nat. Phot. 2 (10), 610–613.CrossRefGoogle Scholar
Lundgren, T.S. & Mansour, N.N. 1988 Oscillations of drops in zero gravity with weak viscous effects. J. Fluid Mech. 194, 479–510.CrossRefGoogle Scholar
Lyubimov, D.V., Lyubimova, T.P. & Shklyaev, S.V. 2006 Behavior of a drop on an oscillating solid plate. Phys. Fluids 18 (1), 012101.CrossRefGoogle Scholar
Marston, P.L. 1980 Shape oscillation and static deformation of drops and bubbles driven by modulated radiation stresses–theory. J. Acoust. Soc. Am. 67 (1), 15–26.CrossRefGoogle Scholar
Matsumoto, T., Fujii, H., Ueda, T., Kamai, M. & Nogi, K. 2004 Oscillating drop method using a falling droplet. Rev. Sci. Instrum. 75 (5), 1219–1221.CrossRefGoogle Scholar
Matsumoto, T., Fujii, H., Ueda, T., Kamai, M. & Nogi, K. 2005 Measurement of surface tension of molten copper using the free-fall oscillating drop method. Meas. Sci. Technol. 16 (2), 432–437.CrossRefGoogle Scholar
Michael, D.H. 1981 Meniscus stability. Ann. Rev. Fluid Mech. 13 (1), 189–216.CrossRefGoogle Scholar
Miksis, M.J. 1981 Shape of a drop in an electric field. Phys. Fluids 24 (11), 1967–1972.CrossRefGoogle Scholar
Miller, C.A. & Scriven, L.E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32 (3), 417–435.CrossRefGoogle Scholar
Notz, P.K. & Basaran, O.A. 2004 Dynamics and breakup of a contracting liquid filament. J. Fluid Mech. 512, 223–256.CrossRefGoogle Scholar
Patzek, T.W., Basaran, O.A., Benner, R.E. & Scriven, L.E. 1995 Nonlinear oscillations of two-dimensional, rotating inviscid drops. J. Comput. Phys. 116 (1), 3–25.CrossRefGoogle Scholar
Patzek, T.W., Benner, R.E. Jr., Basaran, O.A. & Scriven, L.E. 1991 Nonlinear oscillations of inviscid free drops. J. Comput. Phys. 97 (2), 489–515.CrossRefGoogle Scholar
Perkins, N.C. & Mote, C.D. Jr. 1986 Comments on curve veering in eigenvalue problems. J. Sound Vib. 106 (3), 451–463.CrossRefGoogle Scholar
Prosperetti, A. 1977 Viscous effects on perturbed spherical flows. Q. Appl. Maths 34 (4), 339–352.CrossRefGoogle Scholar
Prosperetti, A. 1980 Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100 (2), 333–347.CrossRefGoogle Scholar
Prosperetti, A. 2012 Linear oscillations of constrained drops, bubbles, and plane liquid surfaces. Phys. Fluids 24 (3), 032109.CrossRefGoogle Scholar
Przyborowski, M., Hibiya, T., Eguchi, M. & Egry, I. 1995 Surface tension measurement of molten silicon by the oscillating drop method using electromagnetic levitation. J. Cryst. Growth 151 (1–2), 60–65.CrossRefGoogle Scholar
Ptasinski, K.J. & Kerkhof, P.J.A.M. 1992 Electric field driven separations: phenomena and applications. Sep. Sci. Tech. 27 (8–9), 995–1021.CrossRefGoogle Scholar
Ramalingam, S.K. & Basaran, O.A. 2010 Axisymmetric oscillation modes of a double droplet system. Phys. Fluids 22 (11), 112111.CrossRefGoogle Scholar
Ramalingam, S., Ramkrishna, D. & Basaran, O.A. 2012 Free vibrations of a spherical drop constrained at an azimuth. Phys. Fluids 24 (8), 082102.CrossRefGoogle Scholar
Ramkrishna, D. & Amundson, N.R. 1985 Linear Operator Methods in Chemical Engineering with Applications to Transport and Chemical Reaction Systems. Prentice Hall.Google Scholar
Rayleigh, Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29 (196–199), 71–97.Google Scholar
Rayleigh, Lord 1882 XX. On the equilibrium of liquid conducting masses charged with electricity. Philos. Mag. 14 (87), 184–186.CrossRefGoogle Scholar
Sambath, K. & Basaran, O.A. 2014 Electrohydrostatics of capillary switches. AIChE J. 60 (4), 1451–1459.CrossRefGoogle Scholar
Scott, T.C. & Wham, R.M. 1988 Surface area generation and droplet size control in solvent extraction systems utilizing high intensity electric fields. US Patent 4 767 515.Google Scholar
Scott, T.C. & Wham, R.M. 1989 An electrically driven multistage countercurrent solvent extraction device: the emulsion-phase contactor. Ind. Engng Chem. Res. 28 (1), 94–97.CrossRefGoogle Scholar
Strani, M. & Sabetta, F. 1984 Free vibrations of a drop in partial contact with a solid support. J. Fluid Mech. 141, 233–247.CrossRefGoogle Scholar
Strani, M. & Sabetta, F. 1988 Viscous oscillations of a supported drop in an immiscible fluid. J. Fluid Mech. 189, 397–421.CrossRefGoogle Scholar
Taylor, G.I. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. A 280 (1382), 383–397.Google Scholar
Theisen, E.A., Vogel, M.J., Lopez, C.A., Hirsa, A.H. & Steen, P.H. 2007 Capillary dynamics of coupled spherical-cap droplets. J. Fluid Mech. 580, 495–505.CrossRefGoogle Scholar
Trinh, E. & Wang, T.G. 1982 Large-amplitude free and driven drop-shape oscillations: experimental observations. J. Fluid Mech. 122, 315–338.CrossRefGoogle Scholar
Trinh, E., Zwern, A. & Wang, T.G. 1982 An experimental study of small-amplitude drop oscillations in immiscible liquid systems. J. Fluid Mech. 115, 453–474.CrossRefGoogle Scholar
Tsamopoulos, J.A., Aklas, T.R. & Brown, R.A. 1985 Dynamics of charged drop break-up. Proc. R. Soc. Lond. A 401 (1820), 67–88.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. 1984 Resonant oscillations of inviscid charged drops. J. Fluid Mech. 147, 373–395.CrossRefGoogle Scholar
Vlahovska, P.M., Blawzdziewicz, J. & Loewenberg, M. 2009 Small-deformation theory for a surfactant-covered drop in linear flows. J. Fluid Mech. 624, 293–337.CrossRefGoogle Scholar
Wang, F., Contò, F.P., Naz, N., Castrejón-Pita, J.R., Castrejón-Pita, A.A., Bailey, C.G., Wang, W., Feng, J.J. & Sui, Y. 2019 A fate-alternating transitional regime in contracting liquid filaments. J. Fluid Mech. 860, 640–653.CrossRefGoogle Scholar
Wee, H., Wagoner, B.W., Garg, V., Kamat, P.M. & Basaran, O.A. 2021 Pinch-off of a surfactant-covered jet. J. Fluid Mech. 908, A38.CrossRefGoogle Scholar
Wee, H., Wagoner, B.W., Kamat, P.M. & Basaran, O.A. 2020 Effects of surface viscosity on breakup of viscous threads. Phys. Rev. Lett. 124 (20), 204501.CrossRefGoogle ScholarPubMed
Wilkes, E.D. & Basaran, O.A. 1997 Forced oscillations of pendant (sessile) drops. Phys. Fluids 9 (6), 1512–1528.CrossRefGoogle Scholar
Wilkes, E.D. & Basaran, O.A. 1999 Hysteretic response of supported drops during forced oscillations. J. Fluid Mech. 393, 333–356.CrossRefGoogle Scholar
Wohlhuter, F.K. & Basaran, O.A. 1992 Shapes and stability of pendant and sessile dielectric drops in an electric field. J. Fluid Mech. 235, 481–510.CrossRefGoogle Scholar
Wohlhuter, F.K. & Basaran, O.A. 1993 Effects of physical properties and geometry on shapes and stability of polarizable drops in external fields. J. Magn. Magn. Mater. 122 (1–3), 259–263.CrossRefGoogle Scholar
Zhang, X., Harris, M.T. & Basaran, O.A. 1994 Measurement of dynamic surface tension by a growing drop technique. J. Colloid Interface Sci. 168 (1), 47–60.CrossRefGoogle Scholar