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Faraday waves on a cylindrical fluid filament – generalised equation and simulations

Published online by Cambridge University Press:  19 October 2018

Sagar Patankar ,
Palas Kumar Farsoiya  and
Ratul Dasgupta Open the ORCID record for Ratul Dasgupta [Opens in a new window]
Sagar Patankar
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
Palas Kumar Farsoiya
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
Ratul Dasgupta*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
*
†Email address for correspondence: [email protected]
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Abstract

We perform linear stability analysis of an interface separating two immiscible, inviscid, quiescent fluids subject to a time-periodic body force. In a generalised, orthogonal coordinate system, the time-dependent amplitude of interfacial perturbations, in the form of standing waves, is shown to be governed by a generalised Mathieu equation. For zero forcing, the Mathieu equation reduces to a (generalised) simple harmonic oscillator equation. The generalised Mathieu equation is shown to govern Faraday waves on four time-periodic base states. We use this equation to demonstrate that Faraday waves and instabilities can arise on an axially unbounded, cylindrical capillary fluid filament subject to radial, time-periodic body force. The stability chart for solutions to the Mathieu equation is obtained through numerical Floquet analysis. For small values of perturbation and forcing amplitude, results obtained from direct numerical simulations (DNS) of the incompressible Euler equation (with surface tension) show very good agreement with theoretical predictions. Linear theory predicts that unstable Rayleigh–Plateau modes can be stabilised through forcing. This prediction is borne out by DNS results at early times. Nonlinearity produces higher wavenumbers, some of which can be linearly unstable due to forcing and thus eventually destabilise the filament. We study axisymmetric as well as three-dimensional perturbations through DNS. For large forcing amplitude, localised sheet-like structures emanate from the filament, suffering subsequent fragmentation and breakup. Systematic parametric studies are conducted in a non-dimensional space of five parameters and comparison with linear theory is provided in each case. Our generalised analysis provides a framework for understanding free and (parametrically) forced capillary oscillations on quiescent base states of varying geometrical configurations.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Faraday waves Instability: Parametric instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 857 , 25 December 2018 , pp. 80 - 110
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Copyright
© 2018 Cambridge University Press 

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