On the impact of viscosity ratio on falling two-layer viscous film flow inside a tube

A two-layer falling film consisting of two immiscible viscous fluids with identical density but different viscosity lining the interior of a vertical tube is studied using a long-wave asymptotic model. Linear stability analysis of the model shows an unstable `free-surface mode' consisting largely of perturbations to the free surface, and an `interfacial mode' that can either be stable for all wavenumbers, unstable for a band of long-wave wavenumbers, or unstable only to a band of wavenumbers bounded away from zero. These instabilities grow outside the linear regime and either saturate as a series of waves or continue to grow so that the free surface at a wave crest tends to the center of the tube in finite time, indicating the formation of a plug. Families of traveling wave solutions are found by continuation from Hopf bifurcations that arise due to the instability of one or both modes. The free-surface traveling waves have a turning point that indicates a critical thickness required for plug formation to occur; decreasing the viscosity of the outer layer decreases this critical thickness so that plugs form more readily. The impact on plug formation of fixing the volume flux of each layer, rather than the layer thicknesses, is discussed as well.