Effect of surfactants on the instability of a liquid thread or annular layer: Part I: Quiescent fluids

Abstract

The effect of an insoluble surfactant on the capillary instability of an annular layer that lines the interior surface of a circular tube and surrounds another annular layer that lines the exterior surface of an inner circular tube is considered. As the radius of the inner cylinder tends to vanish or the radius of the outer cylinder tends to infinity, we obtain either an annular layer coated on the interior or exterior surface of a circular tube, or an infinite thread suspended in a quiescent infinite ambient fluid. In the first part of this paper, a linear stability analysis is carried out for axisymmetric perturbations in the absence or presence of fluid inertia, resulting in a nonlinear algebraic eigenvalue problem whose solution produces the complex phase velocity. When the fluid inertia is negligible, there are two normal modes; one is stable under any conditions, and the second is unstable when the wave length of the perturbation is longer than the circumferential length of the unperturbed interface. Stability graphs are presented to illustrate the properties of the normal modes and their dependence on the ratio of the viscosities of the outer and inner fluid, the surfactant diffusivity, the sensitivity of the surface tension to the surfactant concentration, and the ratio of the cylinder to the thread radius. In all cases, the presence of a surfactant reduces the growth rate of the unstable normal mode but is not able to stabilize the interface. As the surfactant diffusivity is raised, or the surface tension becomes insensitive to the surfactant concentration, the unconditionally stable mode becomes physically irrelevant by requiring an extremely large amplitude of the perturbation in the surfactant concentration, yielding well-known results for uniform surface tension. In the second part of this paper, the nonlinear growth of the instability of an infinite thread is studied under conditions of Stokes flow by dynamical simulation, assuming a linear relationship between the surface tension and the surfactant concentration. The numerical results reveal that the presence of a surfactant may have a significant effect on the shapes of developing interfacial structures, and that a similarity solution adequately describes the behavior of the thread close to the time of breakup. In the third part of this paper, the instability of an annular layer coated on the interior of a cylindrical tube is considered, with particular reference to bronchial airway collapse. Numerical simulations reveal that the qualitative features of evolution are insensitive to the presence of the surfactant and to the wave length of the perturbation, although both significantly affect the growth rate of the instability. A comparison of the predictions of a thin-layer flow model with the results of the full linear stability theory and with boundary-integral simulations illustrates the capabilities and limitations of the asymptotic approach.