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Linear stability analysis of capillary instabilities for concentric cylindrical shells

Published online by Cambridge University Press:  19 August 2011

X. Liang ,
D. S. Deng ,
J.-C. Nave  and
Steven G. Johnson
X. Liang*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
D. S. Deng
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
J.-C. Nave
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 2K6, Canada
Steven G. Johnson
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

Motivated by complex multi-fluid geometries currently being explored in fibre-device manufacturing, we study capillary instabilities in concentric cylindrical flows of 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 (), Stone & Brenner (, 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 case is already sufficient to obtain several interesting phenomena: limiting cases of thin shells or low shell viscosity that reduce to 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 cases remain to be explored, and as a first step we discuss two illustrative cases, an alternating-layer structure and a geometry with a continuously varying viscosity.

JFM classification

Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 683 , 25 September 2011 , pp. 235 - 262
Copyright
Copyright © Cambridge University Press 2011

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