Linear stability analysis of capillary instabilities for concentric cylindrical shells

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 3-dimensional Stokes-flow simulations. Many $N > 3$ cases remain to be explored, and as a first step we discuss two illustrative $N \to \infty$ cases, an alternating-layer structure and a geometry with a continuously varying viscosity.