Instability of two annular layers or a liquid thread bounded by an elastic membrane

Abstract

The instability of an annular layer coated on the interior side of an outer circular tube and surrounding another annular layer coated on the exterior side of an inner circular tube, is studied in the absence of an imposed flow due to a pressure gradient or boundary motion. As the radius of the inner cylinder tends to vanish and the radius of the outer cylinder tends to infinity, the inner layer reduces to a liquid thread suspended in a quiescent infinite ambient fluid. The fluids are separated by a membrane that exhibits constant surface tension and develops elastic tensions due to deformation from the unstressed cylindrical shape. The surface tension is responsible for the Rayleigh capillary instability, but the elastic tensions resist the deformation and slow down or even prevent the growth of small perturbations. In the first part of this paper, we formulate the linear stability problem for axisymmetric perturbations, and derive a nonlinear eigenvalue system whose solution produces the complex phase velocity of the normal modes. When inertial effects are negligible, there are two normal modes; one is stable under any conditions, and the second may be unstable when the interfacial elasticity is sufficiently small compared to surface tension, and the wavelength of the perturbation is sufficiently long. Stability graphs are presented to illustrate the properties of the normal modes and their dependence on the ratio of the viscosity of the outer to inner fluid, the interfacial elasticity, and the ratios of the cylinders' radii to the interface radius. The results show that as the interfacial elasticity tends to vanish, the unconditionally stable mode becomes physically irrelevant by requiring extremely large ratios of axial to lateral displacement of material points along the trace of the membrane in an azimuthal plane. In the second part of this paper, we investigate the nonlinear instability of an infinite thread in the limit of vanishing Reynolds numbers by dynamical simulation based on a boundary-integral method. In the problem formulation, the elastic tensions derive from a constitutive equation for a thin sheet of an incompressible isotropic elastic solid described by Mooney's constitutive law. The numerical results suggest that the interfacial elasticity ultimately restrains the growth of disturbances and leads to slowly evolving periodic shapes, in agreement with laboratory observations.