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Capillary surfaces in and around exotic cylinders with application to stability analysis

Published online by Cambridge University Press:  12 November 2019

Fei Zhang Open the ORCID record for Fei Zhang [Opens in a new window]  and
Xinping Zhou Open the ORCID record for Xinping Zhou [Opens in a new window]
Fei Zhang
Affiliation:
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China
Xinping Zhou*
Affiliation:
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China
*
†Email address for correspondence: xpzhou08@hust.edu.cn
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Abstract

A capillary surface in or around exotic cylinders cannot locate itself, since the configurations of the exotic cylinders with a variable radius permit an entire continuum of equilibrium menisci, all of which have the same potential energy. The ‘exotic’ property indicates that all the menisci have the smallest eigenvalues $\unicode[STIX]{x1D706}_{1}=0$ for the corresponding Sturm–Liouville problems without a volume constraint for stability analysis. Three types of exotic cylinders are addressed and the Sturm–Liouville problems with $\unicode[STIX]{x1D706}=0$ for stability analysis are solved numerically. Notably, the two-dimensional cases can be solved analytically. In the method of Slobozhanin & Alexander (Phys. Fluids, vol. 15, 2003, pp. 3532–3545), the stability of the meniscus is determined by comparing the boundary parameter $\unicode[STIX]{x1D712}_{1}$ and the critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$, which is derived directly from the solution of the Sturm–Liouville problem with $\unicode[STIX]{x1D706}=0$. Results validate that the exotic cylinders have the boundary parameters $\unicode[STIX]{x1D712}_{1}=\unicode[STIX]{x1D712}_{1}^{\ast }$. Motivated by this observation, a new way to determine the critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$ under pressure disturbances for stability analysis is proposed without solving the Sturm–Liouville problem with $\unicode[STIX]{x1D706}=0$.

JFM classification

Interfacial Flows (free surface): Capillary flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 882 , 10 January 2020 , A28
Copyright
© 2019 Cambridge University Press 

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