Linear stability analysis of capillary instabilities for concentric cylindrical shells

• Main Authors: Liang, Xiangdong (Contributor), Deng, D. S. (Contributor), Nave, Jean-Christophe (Author), Johnson, Steven G. (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Chemical Engineering (Contributor), Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Cambridge University Press, 2011-09-21T17:22:25Z.
• Subjects: Article
• Online Access: Get fulltext

 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.