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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: obasaran@purdue.edu
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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

‡

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

References

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