Date of Award

Fall 2020

Degree Type

Open Access Dissertation

Degree Name

Mathematics, PhD

Program

Institute of Mathematical Sciences

Advisor/Supervisor/Committee Chair

Marina Chugunova

Dissertation or Thesis Committee Member

Ali Nadim

Dissertation or Thesis Committee Member

Alfonso Castro

Terms of Use & License Information

Creative Commons Attribution-Share Alike 3.0 License
This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.

Rights Information

© Copyright Yadong Ruan, 2020

Abstract

In this work, we consider the dynamics of falling liquid films in various geometries. We first ex- amine the dynamics of a thin film formed by a distributed liquid source on a vertical solid wall. The mathematical model is derived using the lubrication approximation and includes the effects of gravity, upward airflow and surface tension. When surface tension is neglected, a critical source strength is found below which the film flows entirely upward due to the airflow, and above which some of the flow is carried downward by gravity. In both cases, a steady state is established over the region where the finite source is located. Shock waves that propagate in both directions away from the source region are analyzed. Numerical simulations are included to validate the analytical results. For models including surface tension, numerical simulations are carried out. The presence of surface tension, even when small, causes a dramatic change in the film profiles and the speed and structure of the shock waves. These are studied in more detail by examining the traveling wave solutions away from the source region. Next, we present several analytical results pertaining to the thin film equation when it includes a source term. The existence of weak solutions, the long-time behavior of solutions for a constant initial condition, and the general qual- itative behavior of solutions are all considered. The thin film equation with a source is a highly simplified version of the model derived earlier in the thesis. Finally, we consider a separate model describing the axisymmetric flow corresponding to a falling liquid film around a vertical circular fiber. Recent experimental results have shown that a film exiting a nozzle at the top and falling down a vertical fiber can give rise to individual “droplets,” i.e., thicker liquid regions, separated by much thinner zones. The droplets that traverse the circular fiber may exhibit several distinct regimes. Depending on nozzle diameter and flow rate, they may appear as uniformly distributed uniformly sized droplets, as large droplets separated by a series of small droplets in between, or as non-uniformly distributed non-uniformly sized droplets. We present and qualitatively analyze a

novel mathematical model of such flows to supplement this experimental analysis, one capable of showing the convective regime where faster moving droplets collide and sometimes merge with slower moving ones initially, but with a steady travelling state emerging eventually. While previ- ous models of such flows have focused on the slow laminar viscous regime, our model assumes high Reynolds number flow and takes the flow profile to be a plug-flow, but with a thin boundary layer near the fiber providing the drag force on the film. We compare these models and provide various simulations using both inflow-outflow and periodic boundary conditions. We also ana- lyze the linear stability of an initially uniform state and show that there exists a finite range of wavenumbers, including a unique wavenumber with the maximum growth rate, for which the uniform system is unstable.

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