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Linear stability of a rotating liquid column revisited

Published online by Cambridge University Press:  06 January 2022

Pulkit Dubey Open the ORCID record for Pulkit Dubey [Opens in a new window] ,
Anubhab Roy Open the ORCID record for Anubhab Roy [Opens in a new window]  and
Ganesh Subramanian Open the ORCID record for Ganesh Subramanian [Opens in a new window]
Pulkit Dubey
Affiliation:
Integrated Applied Mathematics Program, University of New Hampshire, Durham, NH03824, USA Engineering Mechanics Unit, JNCASR, Bangalore, Karnataka560064, India
Anubhab Roy
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, Tamil Nadu600036, India
Ganesh Subramanian*
Affiliation:
Engineering Mechanics Unit, JNCASR, Bangalore, Karnataka560064, India
*
†Email address for correspondence: [email protected]
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Abstract

We revisit the somewhat classical problem of the linear stability of a rigidly rotating liquid column in this article. Although the literature pertaining to this problem dates back to 1959, the relation between inviscid and viscous stability criteria has not yet been clarified. While the viscous criterion for stability, given by $We < n^2 + k^2 -1$, is both necessary and sufficient, this relation has only been shown to be sufficient in the inviscid case. Here, $We = \rho \varOmega ^2 a^3 / \gamma$ is the Weber number and measures the relative magnitudes of the centrifugal and surface tension forces, with $\varOmega$ being the angular velocity of the rigidly rotating column, $a$ the column radius, $\rho$ the density of the fluid and $\gamma$ the surface tension coefficient; $k$ and $n$ denote the axial and azimuthal wavenumbers of the imposed perturbation. We show that the subtle difference between the inviscid and viscous criteria arises from the surprisingly complicated picture of inviscid stability in the $We$–$k$ plane. For all $n > 1$, the viscously unstable region, corresponding to $We > n^2 + k^2-1$, contains an infinite hierarchy of inviscidly stable islands ending in cusps, with a dominant leading island. Only the dominant island, now infinite in extent along the $We$ axis, persists for $n=1$. This picture may be understood, based on the underlying eigenspectrum, as arising from the cascade of coalescences between a retrograde mode, that is the continuation of the cograde surface-tension-driven mode across the zero Doppler frequency point, and successive retrograde Coriolis modes constituting an infinite hierarchy.

JFM classification

Waves/Free-surface Flows: Capillary waves Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A55
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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