Impact of viscosity ratio on falling two-layer viscous film flow inside a tube

H. Reed Ogrosky
Phys. Rev. Fluids 6, 104005 – Published 13 October 2021

Abstract

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. In this setup there is both an interface between the two layers and a free surface where the inner layer meets the passive air core. The model consists of a set of coupled partial differential equations describing the evolution of both surfaces, and the impact of the viscosity ratio on both the linear and nonlinear dynamics of the model is studied. 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 wave numbers, unstable for a band of long-wave wave numbers, or unstable only to a band of wave numbers 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 of fixing the volume flux of each layer, rather than the layer thicknesses, is discussed. Lastly, the model is extended to account for pressure-driven airflow in the core region, and it is shown that the airflow destabilizes the free-surface mode but can have a stabilizing impact on the interfacial mode. The significance of this study for applications, including human airways, is briefly discussed.

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  • Received 30 December 2020
  • Accepted 27 September 2021

DOI:https://doi.org/10.1103/PhysRevFluids.6.104005

©2021 American Physical Society

Physics Subject Headings (PhySH)

Fluid Dynamics

Authors & Affiliations

H. Reed Ogrosky

  • Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, Virginia 23284, USA

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Vol. 6, Iss. 10 — October 2021

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