Linear stability analysis of capillary instabilities for concentric cylindrical shells
Name
Johnson_Linear Stability.pdf
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2.18 MB
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Author(s)
Liang, Xiangdong
Deng, D. S.
Nave, Jean-Christophe
Johnson, Steven G.
Date Issued
August 2011
Journal
Journal of Fluid Mechanics
Publisher
Cambridge University Press
Citation
Liang, X. et al. “Linear Stability Analysis of Capillary Instabilities for Concentric Cylindrical Shells.” Journal of Fluid Mechanics 683 (2011) : 235-262.
Version
Author's final manuscript
Abstract
Motivated by complex multi-fluid geometries currently being explored in fibre-device manufacturing, we study capillary instabilities in concentric cylindrical flows of $N$ fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier–Stokes problem. Generalizing previous work by Tomotika ($N= 2$), Stone & Brenner ($N= 3$, equal viscosities) and others, we present a full linear stability analysis of the growth modes and rates, reducing the system to a linear generalized eigenproblem in the Stokes case. Furthermore, we demonstrate by Plateau-style geometrical arguments that only axisymmetric instabilities need be considered. We show that the $N= 3$ case is already sufficient to obtain several interesting phenomena: limiting cases of thin shells or low shell viscosity that reduce to $N= 2$ problems, and a system with competing breakup processes at very different length scales. The latter is demonstrated with full three-dimensional Stokes-flow simulations. Many $N\gt 3$ cases remain to be explored, and as a first step we discuss two illustrative $N\ensuremath{\rightarrow} \infty $ cases, an alternating-layer structure and a geometry with a continuously varying viscosity.
MIT Department
Massachusetts Institute of Technology. Department of Chemical Engineering
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1017/jfm.2011.260
