University of Kentucky UKnowledge Theses and Dissertations--Mechanical Engineering Mechanical Engineering 2020 MODELING THIN FLUID FILM ON A ROTARY BELL Mark Doerre University of Kentucky, [email protected] Author ORCID Identifier: https://orcid.org/0000-0003-1105-3522 Digital Object Identifier: https://doi.org/10.13023/etd.2020.148 Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Doerre, Mark, "MODELING THIN FLUID FILM ON A ROTARY BELL" (2020). Theses and Dissertations-- Mechanical Engineering. 150. https://uknowledge.uky.edu/me_etds/150 This Doctoral Dissertation is brought to you for free and open access by the Mechanical Engineering at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Mechanical Engineering by an authorized administrator of UKnowledge. For more information, please contact [email protected]. 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Mark Doerre, Student Dr. Kozo Saito, Major Professor Dr. Alexandre Martin, Director of Graduate Studies MODELING THIN FLUID FILM ON A ROTARY BELL ________________________________________ DISSERTATION ________________________________________ A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the College of Engineering at the University of Kentucky By Mark Doerre Lexington, Kentucky Co-Directors: Dr. Kozo Saito, TVA Professor of Mechanical Engineering and Dr. Nelson Akafuah, Professor of Mechanical Engineering Lexington, Kentucky 2020 Copyright © Mark Doerre 2020 https://orcid.org/0000-0003-1105-3522 ABSTRACT OF DISSERTATION MODELING THIN FLUID FILM ON A ROTARY BELL A component of the mission in the Institute of Research for Technology Development at the University of Kentucky is advancing research and development and bringing it to the factory floor for continuous improvement. This dissertation delves into the art and science of rotational fluid mechanics in the context of rotary bell atomizers. One outcome proves that an approximation for calculating fluid film thicknesses on high-speed spinning surfaces inferred while working in cylindrical and spherical coordinate systems can be applied to an arbitrary bell profile. However, the analytical limits of this approximation were not investigated. In all cases, a restriction exists that the bell profile curvature is much greater than the fluid film thickness. The validity of this approximation was supported by findings in publications employing curvilinear coordinates created by axisymmetric revolution of a planar curve. This validation enabled rigorous analyses of bell profiles beyond the common cylindrical or spherical profiles. A curvilinear system containing a coordinate of arc length along the bell profile is the arbitrary case, and the most reduced high-speed case collapses to an approximation from an observed pattern in simpler coordinate systems. The jump to varying curvature curvilinear coordinates requires additional mathematics to calculate metric tensor coefficients and spatial derivatives of directional unit vectors, and to develop lengthy vector invariants. Another outcome was to explain the underlying symmetry in the reduced order solution for a coupled rotary system that included centrifugal and Coriolis effects on a conical rotary bell. KEYWORDS: curvilinear, metric tensor, vector invariant, Navier-Stokes, rotary bell Mark Doerre 5/3/2020 Date MODELING THIN FLUID FILM ON A ROTARY BELL By Mark Doerre Dr. Kozo Saito Co-Director of Thesis Dr. Nelson Akafuah Co-Director of Thesis Dr. Alexandre Martin Director of Graduate Studies 5/3//2020 Date DEDICATION To Mom (3/31/2013) and Dad (4/2/2019) And My wife, Ollie And Samantha & Paul, Richard & Maddie, Lily and Gabby And John, George, Debbie, Barbara, George III, Theresa, Rick, Cailyn, Mallory, James And James Jr. and Olivia ACKNOWLEDGMENTS I would like to thank the University of Kentucky and Toyota for establishing the IR4TD Toyota Paint Lab. Professor Dr. Kozo Saito is owed a debt of gratitude that cannot be repaid. However, I can pay it forward by using a lifetime of engineering to educate the next generation, and to engage in service for students, scholars, and practitioners in rotating thin film research and development. Professor Dr. Nelson Akafuah has been invaluable in teaching the art of publication and academic writing; thank you. His story as a “kid from Ghana” that used the pensions of family members as collateral to pay for his education, to having his PhD interrupted by Hurricane Katrina, to transferring to UK and repeating his qualifying exams, and finally finishing his doctorate at U.K is an inspiration. Thank you to Professor Dr. John Wenk for the formalism of your teachings in ME640, Foundations of Solid Mechanics, and ME601, Theory of Plasticity. Learning the art and science of indicial notation was essential to understanding the mathematics contained in this dissertation. This dissertation used those teachings to extrapolate from Cartesian coordinate systems, and into a surface following varying curvature curvilinear coordinate system. Thank you to Dr. John Stencel for reviewing manuscripts and sharing wisdom that can only come from years of experience. Thank you to Research Professor Dr. Ahmad Salaimeh for his guidance and expertise in the area of automotive paint processes. iii Thank you to Adnan Darwish Ahmad - I have known you the longest as a graduate student and colleague - for friendship and all the discussions and cultural exchanges between the USA and Middle East. Thank you to Essa Khalid, Ahmad Abubaker, Masoud Arabghahestani, Helmut Koch, Dr. Zhaojin Diao, and Ricky Green, “you all” made our office a great place to work and learn. Thank you to my committee members: Dr. Saito Mechanical Engineering Dr. Akafuah Mechanical Engineering Dr. Bailey Mechanical Engineering Dr. Salaimeh Mechanical Engineering Dr. Perry Mathematics Dr. Harik Civil Engineering - External Examiner iv TABLE OF CONTENTS ACKNOWLEDGMENTS ............................................................................................... iii LIST OF TABLES ......................................................................................................... viii LIST OF FIGURES ......................................................................................................... ix NOMENCLATURE .......................................................................................................... x INTRODUCTION................................................................................. 1 1.1 Summary of Chapters ................................................................................................................... 1 1.2 Map of Dissertation .................................................................................................................... 11 PATENT HISTORY ........................................................................... 14 2.1 Introduction ................................................................................................................................ 14 2.2 US Patent Database .................................................................................................................... 15 LITERATURE REVIEW .................................................................. 21 3.1 Introduction ................................................................................................................................ 21 3.2 Literature Review ....................................................................................................................... 23 PRELIMINARIES .............................................................................. 42 4.1 Introduction ................................................................................................................................ 42 4.2 Non-Inertial Rotating Frame - Cylindrical Coordinate System .................................................. 43 4.3 Thin Film Approximation – Cylindrical Coordinate System ....................................................