Defense Date
2024
Document Type
Thesis
Degree Name
Master of Science
Department
Mathematical Sciences
First Advisor
Harold Ogrosky
Abstract
Lung fluid dynamics is a well studied topic and relevant in the field of research surrounding patients with Chronic Obstructive Pulmonary Disease (COPD) and cystic fibrosis. However, the literature on the mathematical description of the net pulmonary fluid transport in the case of constricting and expanding airways is scarce. In this paper we aim to address how breathing contributes to airways clearance in patients with pulmonary diseases such as COPD and CF in which cilia do not function properly. We begin with the flow of a fluid-filled tube with rigid walls of fixed radius and the effects of gravity being considered negligible, and derive the flux for its cross-sectional area. We then extend this concept into two fluids, a core fluid with low viscosity (air), and a thin-film around the walls of the tube with high viscosity and a completely flat surface, and derive the net flux of the film along a cross section of the tube. Next, the effects of a time-dependent radius were included by first assuming a fixed “average” inhalation and exhalation value; we allowed this piecewise-constant function to represent a test function for the core flux. Setting the tube wall to be narrower on average during exhalation and wider on average upon inhalation it was found that the net clearance of the thin film was indeed positive. The same result was found for sinusoidal breathing rates. Reintroducing gravity into the system, it was determined that the sign of the net flux was dependent on the responsiveness of the core flow with respect to the motion of the walls. Our final complexity added was the introduction of small amplitude long wave disturbances in the free surface. Assuming laminar flow $(\text{R}_{\text{e}}\ll 1)$, linear stability analysis on the system was performed. It was shown that for walls that are stationary, the perturbations create an unstable system, and that given enough time, plugs will form under the right conditions. We continued further and assumed that the walls were allowed to move as a function of time and found that the system was still unstable, with the growth rates modified by the periodic changes in tube radius.
Rights
© Robert E. Hicks
Is Part Of
VCU University Archives
Is Part Of
VCU Theses and Dissertations
Date of Submission
12-13-2024