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Newtonian and Non-Newtonian Flows in a Simple Model of the Human Trachea (Under the Direction of Roberto Camassa)

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Newtonian and Non-Newtonian Flows in a Simple Model of the Human Trachea (Under the Direction of Roberto Camassa) NEWTONIAN AND NON-NEWTONIAN FLOWS IN A SIMPLE MODEL OF THE HUMAN TRACHEA Jeffrey Eric Olander A dissertation submitted to the faculty at the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics and Astronomy in the College of Arts and Sciences. Chapel Hill 2020 Approved by: Roberto Camassa M. Gregory Forest Richard Superfine Amy Oldenburg Fabian Heisch c 2020 Jeffrey Eric Olander ALL RIGHTS RESERVED ii ABSTRACT Jeffrey Eric Olander: Newtonian and Non-Newtonian Flows in a Simple Model of the Human Trachea (Under the direction of Roberto Camassa) The dynamics of fluid flow in a simple model of the human trachea are studied both experimentally and theoretically. Two configurations of this model are presented. In the first, viscous Newtonian liquids are driven upward in a glass tube by a constant flow of air. In the second, viscous Newtonian and non-Newtonian liquids are pumped into a glass tube and allowed to flow under gravity. In all experiments, the liquids are initially flat and thin moving films that eventually develop periodic traveling waves. These flows are modeled by the low-Reynolds Navier-Stokes equations; the non- Newtonian stresses are described by the Upper Convected Maxwell constitutive law. Linear stability analyses for these models are carried out, and results are compared with experiments. For the air flow case, wave tracking and dyed oil experiments are presented to show the existence of trapped cores of liquid that are carried up a 1:0 cm I.D. tube by the air. Two previously-derived models for the liquid (from different descriptions of the coupling air flow) are presented and compared with linear wave experiments. It is shown that the simpler, locally-Poiseuille description of the air flow is able to qualitatively describe features of the experiments such as changes in liquid layer thickness and wave velocity. The more accurate, multi-scale approach to the air-liquid problem improves upon these descriptions as well as giving a much better prediction of interfacial wavelength in the linear regime. For the Newtonian gravity-driven case, experiments in 1:0 cm, 0:59 cm, and 0:34 cm I.D. tubes show that falling wave instabilities can either saturate or grow unchecked into liquid plugs separated by large bullet-shaped bubbles that move in lock-step with the plugs. A long wave criterion for the transition of waves from convective to absolute instabilities is given and shown to accurately capture experimental observations. A similar result is obtained when sample numerical data is compared to experimental plug formation data. iii For the non-Newtonian gravity-driven case, it is shown that PIB(polyisobutylene)-PB(polybutene) Boger fluids, prepared from a recipe, experimentally reduce the growth of interfacial instabilities compared with equally-viscous falling Newtonian films in a 1:0 cm I.D. tube. All fluids were tested in a cone-and-plate rheometer, and sample results and their measured relaxation times are given. The Boger fluids’ relaxation times were found to be unexpectedly very small, indicating that viscoelasticity may play a minor role in accounting for experimental observations. Instead, a change in surface tension may be responsible, and some corroborating measurements are shown. A small-amplitude, linear Weissenberg model is derived. It predicts an elasticity-enhanced Rayleigh-Plateau instability, which is stabilized by drops in surface tension, i.e., increases in the nondimensional Bond number. When measured relaxation times and surface tensions are inserted into the model, it is found to qualitatively match the experimental plug formation trend. iv To Mom and Austin, my loving family v ACKNOWLEDGMENTS First I need to thank my committee for their saintly patience over the past several months as I’ve scheduled and rescheduled my defense again and again. I really mean it this time. To my advisors Roberto, Greg, and Rich thank you for the academic guidance and support you’ve so collegiately given. You’ve always made me feel like I’m a worthy researcher, even when I have had my doubts. To Reed, no words can quite express how amazing a colleague you’ve been. I’ve always felt that I got the luckiest deal one could hope for getting to work on this project with you. Even these many years after you graduated and moved on to your own career, you’ve never failed to give me hours of your time to field my many questions and give me such warm support. I don’t know that I could have made it here without you; certainly I wouldn’t have made it with my sanity. It’s been such an honor to work with you. Those who know me know that I must co-manage lives as a researcher and a person with a physical disability. My disability means that I need lots of help every day getting around and doing my work. Over the years I have had wonderful assistants who were my ‘+1’ wherever I went on campus. Two of them have been with me longer than any others. Mary, thanks for being my reliable, kind companion during those early years of lab work and teaching. I think we both learned a lot! Stephanie, we get to finish this ride together. Thanks for being there and being my friend. I’ve also been incredibly fortunate to have students and friends at UNC to help me with my research and dissertation. Jessica, Jeeho, Gerard, Matt, and other undergrad researchers gave me their time, creativity, and assistance in the lab. These last few years, I have switched mostly into writing mode. Eshika, you have been an incredible help to me as a scribe this whole time. I’m very lucky to have gotten to work with you throughout this dissertation. All these pages of words and equations wouldn’t be here if it weren’t for you. To all the folks at BeAM, thanks for giving me an outlet to explore my creative side and hone my maker skills. A special shout-out to Drew for being a great boss and mentor. My grad friends, Lori, Andy, and Holly, thanks for giving me those excuses to take much needed vi breaks from the lab to grab lunch or a drink and chat. Lori, we’ve known each other since NCSU, and it’s been amazing to have your friendship all these years. To Warren "Whiskey Walrus", being best friends with you has no doubt been the best thing that has come out of grad school. From the beginning we’ve had the greatest adventures together, and I’ve lost count of how many things we’ve broken–I mean “taken apart”–in the name of Figuring Things Out. You have taken this conceptual physicist and turned him into an engineer. It’s been real, it’s been fun, and yes, it’s been real fun. To Dr. Castlebury, thank you for being my steadfast supporter, thoughtful sounding board, and good friend for so many years. When I needed a guide or a kick in the pants, you were always there for me. I look forward to sharing the next exciting phase of my journey with you. Finally, to my Mom and brother Austin, you are my whole life. This accomplishment would be nothing without you to share it with. I love you both so much. I’m more grateful than words can express to be a part of our little family. This has been a 10+ year journey, and there have been many others who’ve been there for me along the way. To all of you–thank you! vii TABLE OF CONTENTS LIST OF FIGURES . ix LIST OF TABLES . x CHAPTER 1: INTRODUCTION . 1 1.1 Introduction . 1 1.2 Experimental background . 2 1.2.1 Core-annular flows . 2 1.2.2 Gravity-driven annular flows . 4 1.2.3 Boger fluid review . 5 1.3 Theoretical background . 7 1.3.1 Air-driven flow . 7 1.3.2 Gravity-driven flow . 8 1.3.3 Viscoelastic modeling . 10 CHAPTER 2: EXPERIMENTAL SETUPS AND MODELING OVERVIEW . 11 2.1 Air-driven Newtonian flow . 11 2.2 Gravity-driven Newtonian flow . 13 2.3 Gravity-driven non-Newtonian flow . 16 2.3.1 Viscoelastic modeling . 16 CHAPTER 3: EXPERIMENTS . 18 3.1 Air-driven Newtonian flow . 18 3.1.1 Basic procedure . 18 3.1.2 Mean thicknesses . 18 3.1.3 Wave tracking . 20 3.1.4 Mass transport mechanism . 22 viii 3.1.5 Linear-wave experiment for long wave stability analysis comparison . 28 3.2 Gravity-driven Newtonian flow . 30 3.2.1 Film thicknesses and volume-averages . 32 3.2.2 Absolute and convective instabilities . 35 3.2.3 Waves vs. plugs . 35 3.3 Gravity-driven non-Newtonian flow . 38 3.3.1 Preparing the test liquids . 38 3.3.2 Boger fluids and their rheology . 39 3.3.3 Waves vs. plugs . 40 3.3.4 Surface tension . 41 3.3.5 Absolute versus convective instabilities . 43 CHAPTER 4: MODELING . 46 4.1 Air-driven Newtonian flow . 46 4.1.1 Nondimensionalization in the long wave limit . 49 4.1.2 Turbulence closure model for gas: leading order long wave model . 52 4.1.3 Turbulence closure model for gas: leading order thin film model . 55 4.1.4 Multi-scale approach: long wave model . 56 4.1.5 Linear stability analysis . 62 4.2 Gravity-driven Newtonian flow . 65 4.2.1 Long wave asymptotic model . 65 4.2.2 Long wave asymptotic model: leading order and first order .
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