Floquet analysis on a viscous cylindrical fluid surface subject to a time-periodic radial acceleration
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O R I G I NA L A RT I C L E
Dilip Kumar Maity
Floquet analysis on a viscous cylindrical fluid surface subject to a time-periodic radial acceleration
Received: 16 January 2020 / Accepted: 29 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Parametrically excited standing waves are observed on a cylindrical fluid filament. This is the cylindrical analog of the Faraday instability in a flat surface or spherical droplet. Using Floquet theory, a linear stability analysis is carried out on a viscous cylindrical fluid surface, which is subjected to a time-periodic radial acceleration. Viscosity of the fluid has a significant impact on the critical forcing amplitude as well as the dispersion relation of the non-axisymmetric patterns. The effect of viscosity on the threshold of the pattern with azimuthal wavenumber m = 1 shows a different dependence from m > 1. It is also observed that the effect of viscosity is greater on the threshold with higher m. Keywords Faraday instability · Capillary wave · Cylindrical fluid surface · Floquet analysis 1 Introduction Interfacial instability occurs on the bounding surface of a fluid due to the application of a time-periodic vertical acceleration. This phenomenon is known as the Faraday instability. This type of parametrically excited standing wave pattern was first observed by Faraday [1] in 1831 [see for a review [2]]. The excited surface waves have a frequency either half [1] or equal to [3,4] the forcing frequency. Experiments [1,5,6] with low viscous fluid have shown square patterns near onset. Benjamin and Ursell [7] first proposed a theory in which they converted the linear inviscid dynamical equations to the Mathieu equation. They showed that the fluid surface is unstable inside the tongue-like boundaries which are plotted in the parameter space of the axial wavenumber k and the forcing amplitude a. Later, experiments with viscous fluid showed stripes [8], regular triangles [9], and patterns of competing hexagons and triangles [10]. These experimental observations show that viscosity has a significant role in the Faraday instability. Kumar [4] and Kumar and Tuckerman [11] converted the linear viscous Navier–Stokes equations to an eigenvalue problem using Floquet theory. Both the references [4,11] present linear stability curves for the eigenfunctions of the Laplacian on the surface. On the other hand, the instabilities on curved fluid surfaces [12–33] are of great interest in the scientific community. These have a wide application in measuring surface tension, studying pattern formation, microfluidic devices, controlling jet breakup, fluid atomization, coating, and drug delivery. A sphere and a cylinder are the two most basic curved geometric objects. Fluid versions of these objects are liquid drops and cylindrical jets or filaments, which are formed due to surface tension. Kelvin [12], Rayleigh [13], and Lamb [14] derived the natural frequencies of a spherical drop when it is slightly perturbed from its equilibrium shape. The generalization
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