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Numerical Studies of Liquid-Liquid Segmented Flows in Square Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2015 Numerical Studies of Liquid-Liquid Segmented Flows in Square Microchannels Using a Front- Tracking Algorithm Eamonn Daire Walker Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Mechanical Engineering Commons Recommended Citation Walker, Eamonn Daire, "Numerical Studies of Liquid-Liquid Segmented Flows in Square Microchannels Using a Front-Tracking Algorithm" (2015). LSU Doctoral Dissertations. 3923. https://digitalcommons.lsu.edu/gradschool_dissertations/3923 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. NUMERICAL STUDIES OF LIQUID-LIQUID SEGMENTED FLOWS IN SQUARE MICROCHANNELS USING A FRONT-TRACKING ALGORITHM A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfilment of the requirements for the degree of Doctor of Philosophy in The Department of Mechanical and Industrial Engineering by Eamonn Daire Walker B.S., Louisiana State University, 2003 May 2016 Acknowledgments The author would like to thank Professors Dimitris E. Nikitopoulos at LSU and Damien Tromeur-Dervout at the Université Lyon 1 for their tremendous support over the course of his graduate studies; Dr. Dorel Moldovan and Dr. Sunggook Park for their patience while serving on his advisory committee; Dr. Gretar Tryggvason for graciously sharing his research code; Dr. Namwon Kim, Dr. Sai Rao, and Mr. Jace Boudreaux, whose work and assistance at LSU were instrumental to this research; and finally the author’s immediate and extended family, whose support made his studies possible. The research presented in this dissertation has been funded in part by the LSU IGERT on Multi- scale Computations of Fluid Dynamics (NSF Award Number 0504507) and by the Department of Mechanical and Industrial Engineering at LSU. Portions of this research were conducted with high performance computational resources provided by the Louisiana Optical Network Initiative (http://www.loni.org) and by Louisiana State University (http://www.hpc.lsu.edu). ii Table of Contents Acknowledgments........................................................................................................................... ii Nomenclature ...................................................................................................................................v Abstract .......................................................................................................................................... vi Chapter 1. Introduction ...................................................................................................................1 1.1. Background ......................................................................................................................1 1.2. Pressure-driven segmented flows ....................................................................................2 1.2.1. Non-dimensionalization ....................................................................................... 4 1.2.2. Droplet velocity .................................................................................................... 6 1.2.3. Pressure drop ........................................................................................................ 8 1.2.4. Periodic pressure field ........................................................................................ 13 1.3. Flows in a rotating reference frame ...............................................................................14 1.3.1. Non-dimensionalization ..................................................................................... 17 1.3.2. Competition of forces ......................................................................................... 18 1.4. Past work ........................................................................................................................20 Chapter 2. Numerical Approach ...................................................................................................23 2.1. Problem setup .................................................................................................................23 2.2. Front-tracking/front-capturing method ..........................................................................24 2.3. Code solution algorithm .................................................................................................26 2.3.1. Predictor-corrector algorithm ............................................................................. 26 2.3.2. Poisson solver .................................................................................................... 28 2.3.3. Parallelisation ..................................................................................................... 30 2.4. Numerical method performance.....................................................................................30 Chapter 3. Code Modifications .....................................................................................................39 3.1. Near-wall modifications .................................................................................................39 3.1.1. Wall non-penetration code ................................................................................. 40 3.1.2. Near-wall surface tension footprint modification .............................................. 40 3.1.3. Wall slip modification ........................................................................................ 41 3.2. Addition of Coriolis terms for a rotating reference frame .............................................42 3.3. Schwarz-Aitken acceleration of parallel solution of Poisson equation..........................44 3.3.1. Schwarz domain decomposition ........................................................................ 44 3.3.2. Aitken acceleration of the local boundary conditions ........................................ 46 3.3.3. Results ................................................................................................................ 47 3.4. Front smoothing .............................................................................................................50 Chapter 4. Parametric Studies – Part I: Pressure Drop .................................................................53 4.1. Problem setup and analysis ............................................................................................53 4.1.1. Simulation parameters ........................................................................................ 54 4.1.2. Parametrization summary .................................................................................. 54 4.2. Droplet size ....................................................................................................................55 4.3. Capillary flows ...............................................................................................................57 iii 4.3.1. Droplet spacing .................................................................................................. 58 4.3.2. Reynolds number ............................................................................................... 60 4.3.3. Capillary number ................................................................................................ 61 4.4. Plug flows ......................................................................................................................63 4.4.1. Plug spacing ....................................................................................................... 63 4.4.2. Reynolds number ............................................................................................... 65 4.4.3. Capillary number ................................................................................................ 66 4.5. Predictive models ...........................................................................................................68 Chapter 5. Parametric Studies – Part II: Droplet Velocity and Geometry ....................................73 5.1. Mobility ..........................................................................................................................73 5.1.1. Droplet size ........................................................................................................ 73 5.1.2. Spacing ............................................................................................................... 74 5.1.3. Reynolds number ............................................................................................... 76 5.1.4. Capillary number ................................................................................................ 77 5.2. Droplet shape .................................................................................................................78 Chapter 6. Flows in a Rotating Reference Frame .........................................................................86 6.1. Problem setup and analysis ............................................................................................87 6.2. Isolated Coriolis Effect Study
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