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<ul><li><p>MicrofluidMechanics</p><p>http://dx.doi.org/10.1036/0071443223</p></li><li><p>Nanoscience and Technology SeriesOmar Manasreh, Series Editor</p><p>MICHAEL H. PETERS Molecular Thermodynamics and TransportKENNETH GILLEO MEMS/MOEM PackagingNICOLAE O. LOBONTIU Mechanical Design of MicroresonatorsROBERTOPAIELLA IntersubbandTransitions inQuantumStructuresJOSEPH H. KOO Polymer NanocompositesJENS W. TOMM AND JUAN JIMENEZ Quantum-Well High-Power</p><p>Laser Arrays</p></li><li><p>MicrofluidMechanics</p><p>Principles and Modeling</p><p>William W. Liou</p><p>Yichuan FangDepartment of Mechanical and</p><p>Aeronautical EngineeringWestern Michigan University</p><p>Kalamazoo, Michigan</p><p>McGraw-HillNew York Chicago San Francisco Lisbon London Madrid</p><p>Mexico City Milan New Delhi San Juan SeoulSingapore Sydney Toronto</p><p>http://dx.doi.org/10.1036/0071443223</p></li><li><p>Copyright 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in theUnited States of America. Except as permitted under the United States Copyright Act of 1976, no partof this publication may be reproduced or distributed in any form or by any means, or stored in a data-base or retrieval system, without the prior written permission of the publisher. </p><p>0-07-158888-4</p><p>The material in this eBook also appears in the print version of this title: 0-07-144322-3.</p><p>All trademarks are trademarks of their respective owners. Rather than put a trademark symbol afterevery occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. </p><p>McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact GeorgeHoare, Special Sales, at george_hoare@mcgraw-hill.com or (212) 904-4069. </p><p>TERMS OF USE </p><p>This is a copyrighted work and The McGraw-Hill Companies, Inc. (McGraw-Hill) and its licensorsreserve all rights in and to the work. Use of this work is subject to these terms. Except as permittedunder the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may notdecompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon,transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it withoutMcGraw-Hills prior consent. You may use the work for your own noncommercial and personal use;any other use of the work is strictly prohibited. Your right to use the work may be terminated if youfail to comply with these terms. </p><p>THE WORK IS PROVIDED AS IS. McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETE-NESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED,INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY ORFITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operationwill be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damagesresulting therefrom. McGraw-Hill has no responsibility for the content of any information accessedthrough the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for anyindirect, incidental, special, punitive, consequential or similar damages that result from the use of orinability to use the work, even if any of them has been advised of the possibility of such damages. Thislimitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. </p><p>DOI: 10.1036/0071443223</p><p>http://dx.doi.org/10.1036/0071443223</p></li><li><p>We hope you enjoy thisMcGraw-Hill eBook! If</p><p>youd like more information about this book,its author, or related books and websites,please click here.</p><p>Professional</p><p>Want to learn more?</p><p>http://dx.doi.org/10.1036/0071443223</p></li><li><p>Contents</p><p>Preface ix</p><p>Chapter 1. Introduction 1</p><p>References 4</p><p>Chapter 2. Basic Kinetic Theory 5</p><p>2.1 Molecular Model 52.2 Micro and Macroscopic Properties 62.3 Binary Collisions 11</p><p>2.3.1 Kinematics 112.3.2 Dynamics and postcollision properties 142.3.3 Molecular force field models 19</p><p>2.4 Statistical Gas Properties 232.5 Position and Velocity Distribution Functions 272.6 Boltzmann Equation and Maxwellian Distribution Function 30</p><p>Appendix 2A: Some Useful Integrals 38References 38</p><p>Chapter 3. Microfluid Flow Properties 39</p><p>3.1 Fundamental Flow Physics 393.2 Surface Phenomena 433.3 Basic Modeling Approaches 44</p><p>References 45</p><p>Chapter 4. Moment Method: Navier-Stokes and Burnett Equations 47</p><p>4.1 Introduction 474.2 Moment Equations 484.3 The Chapman-Enskog Expansion 54</p><p>4.3.1 The Krook equation 554.3.2 The Boltzmann equation 58</p><p>v</p><p>For more information about this title, click here</p><p>http://dx.doi.org/10.1036/0071443223</p></li><li><p>vi Contents</p><p>4.4 Closure Models 594.4.1 First-order modeling 594.4.2 Second-order modeling 62</p><p>4.5 A Numerical Solver for the Burnett Equations 674.5.1 Numerical method 684.5.2 An example of NB2D results 71</p><p>Appendix 4A: Coefficients for Maxwellian Molecules 74Appendix 4B: First-Order and Second-Order Metrics of Transformation 75Appendix 4C: Jacobian Matrices 79References 81</p><p>Chapter 5. Statistical Method: Direct Simulation Monte Carlo Methodand Information Preservation Method 83</p><p>5.1 Conventional DSMC 835.1.1 Overview 835.1.2 Methodology 865.1.3 Binary elastic collisions 895.1.4 Collision sampling techniques 915.1.5 Cell schemes 925.1.6 Sampling of macroscopic properties 94</p><p>5.2 DSMC Accuracy and Approximation 965.2.1 Relationship between DSMC and Boltzmann equation 965.2.2 Computational approximations and input data 98</p><p>5.3 Information Preservation Method 995.3.1 Overview 995.3.2 IP governing equations 1025.3.3 DSMC modeling for the information preservation equations 1035.3.4 Energy flux model for the preserved temperature 1055.3.5 IP implementation procedures 107</p><p>5.4 DSMC-IP Computer Program and Applications 1135.4.1 IP1D program 1135.4.2 IP1D applications to microCouette flows 116</p><p>5.5 Analysis of the Scatter of DSMC and IP 119Appendix 5A: Sampling from a Probability Distribution Function 124Appendix 5B: Additional Energy Carried by Fast Molecules Crossinga Surface 127</p><p>Appendix 5C: One-Dimensional DSMC-IP Computer Program 129References 154</p><p>Chapter 6. Parallel Computing and Parallel Direct SimulationMonte Carlo Method 157</p><p>6.1 Introduction 1576.2 Cost-Effective Parallel Computing 158</p><p>6.2.1 Parallel architecture 1586.2.2 Development of HPCC 1596.2.3 Beowulf system 1626.2.4 Parallel programming 164</p></li><li><p>Contents vii</p><p>6.3 Parallel Implementation of DSMC 1656.3.1 Data distribution 1676.3.2 Data communication 1686.3.3 A parallel machine 170</p><p>6.4 Parallel Performance of PDSMC 171References 173</p><p>Chapter 7. GasSurface Interface Mechanisms 175</p><p>7.1 Introduction 1757.2 Phenomenological Modeling 179</p><p>7.2.1 Specular and diffusive reflection models of Maxwell 1797.2.2 Cercignani, Lampis, and Lord model 180</p><p>7.3 DSMC Implementation 1827.3.1 Specular reflection 1837.3.2 Diffusive reflection 1837.3.3 CLL model 185</p><p>7.4 Wall-Slip Models for Continuum Approaches 187References 191</p><p>Chapter 8. Development of Hybrid Continuum/Particle Method 193</p><p>8.1 Overview 1938.2 Breakdown Parameters 1948.3 Hybrid Approaches for Microfluid Flow 1968.4 Development of Additional Hybrid Approaches 1968.5 Remarks 198</p><p>References 198</p><p>Chapter 9. Low-Speed Microflows 201</p><p>9.1 Introduction 2019.2 Analytical Flow Solutions 201</p><p>9.2.1 MicroCouette flows 2019.2.2 MicroPoiseuille flows 204</p><p>9.3 Numerical Flow Simulations 2079.3.1 Subsonic flow boundary conditions 2079.3.2 MicroCouette flows 2109.3.3 MicroPoiseuille flows 2139.3.4 Patterned microchannel flow 2269.3.5 Microchannels with surface roughness 230</p><p>References 235</p><p>Chapter 10. High-Speed Microflows 237</p><p>10.1 Introduction 23710.2 High-Speed Channel Flows 237</p><p>References 248</p></li><li><p>viii Contents</p><p>Chapter 11. Perturbation in Microflows 249</p><p>11.1 Introduction 24911.2 Forced MicroCouette Flows 250</p><p>11.2.1 Flows in two-dimensional planes 25311.2.2 Three-dimensional flows 260</p><p>11.3 MicroRayleigh-Benard Flows 30211.3.1 Two-dimensional flows 30711.3.2 Three-dimensional flows 318References 334</p><p>Index 337</p></li><li><p>Preface</p><p>This book is written as a textbook for an analytical-oriented microfluidflow course with graduate students who, preferably, have already takenan advanced fluid mechanics course. The derivations of the equationsare presented, whenever possible, in a fairly detailed manner. Theintent is that, even without the help of an instructor, students can self-navigate through the materials with confidence and come away with asuccessful learning experience. Similarly, practicing engineers who areinterested in the subject should also be able to pick up the book and fol-low the flow of the contents without much difficulty. Some backgroundin modern numerical computation tools will help the readers gettingthe full benefit of the two computer programs, NB2D and DSMC-IP, as-sociated with the book. In fact, it is highly recommended that readersdo make use of these Fortran programs.The book begins with an introduction to the kinetic theory of gas and</p><p>the Boltzmann equation to build the foundation to the later mathemat-ical modeling approaches. With the dilute gas assumption, the natureof the micro gas flows allows the direct application of the Chapman-Enskog theory, which then brings in the modeling equations at thevarious orders of the Knudsen number in Chapter 4. The direct simu-lation Monte Carlo (DSMC) method and the information preservation(IP) method are described as the numerical tools to provide solutions tothe Boltzmann equation when the Knudsen number is high. The laterchapters cover the hybrid approaches and the important surface mech-anisms. Some examples of micro gas flows at high and low speeds areshown. One interesting aspect of micro gas flows that is yet to see ex-tensive examination in the literature is the characteristics of the flowdisturbances at microscales. Chapter 11 provides a detailed descriptionof some of our preliminary studies in this area.Although a part of the content of the book has been used in a one-</p><p>semester graduate level microfluid dynamics course at WesternMichigan University, the book is best used as a textbook for a two-semester course. Chapters 1 to 4 provide the introductory content for</p><p>ix</p><p>Copyright 2006 by The McGraw-Hill Companies, Inc. Click here for terms of use. </p></li><li><p>x Preface</p><p>the basic mathematical and the physical aspects of micro gas flows. Thecomputer programNB2Dmay be used, for example, as project exercises.The second part would emphasize the DSMC and the IP solution meth-ods, and their parallelization. The computer program DSMC-IP can beused for term project type of assignments. In the situation where theanalyticalmicrofluid course is preceded by another experiment-orientedcourse on microfabrication or microengineering, and the studentshave already had somewhat extensive knowledge of micro gas flows,the instructor may wish to concentrate on Chapters 2, 4, and 5 in thelectures and leave the rest as reading assignments. When the book isused in a 16-to-20-hour short-course setting, the instructor may wish tohighlight the materials from Chapters 2, 4, 5, and 7. It might be a goodidea to provide opportunities to run at least one of the two computerprograms onsite. Chapters 1, 3, and 10 can be assigned as overnightreading materials.The book does not contain extensive updates and details on the cur-</p><p>rent engineering microfluidic devices. We feel that the books focus ison the fundamental aspects of mirofluid flows and there is a myriad ofreadily available information on the technologies and the many differ-ent microfluidic device applications that have been cleverly designedand painstakingly manufactured by experts in the field. Also, sincenew devices are being brought to light almost daily, we feel that whatis current at the time of this writing may become outdated within a fewyears.A portion of the work the authors have accomplished at Western</p><p>Michigan University has been performed under the support of NASALangley Research Center. Near the completion of the manuscript, thesecond author moved to the Georgia Institute of Technology.We appreciate the help of Dr. James Moss for reviewing a part of the</p><p>manuscript. Thanks should go to those at McGraw Hill who worked onthe book and to those who reviewed it. We should also thank Jin Suand Yang Yang for their work on the codes. The first author (WWL)would like to thank the unconditional support and patience of his wife,Shiou-Huey Lee, during this writing and the love from his children,Alex and Natalie.</p><p>William W. LiouKalamazoo, Michigan</p><p>Yichuan FangAtlanta, Georgia</p></li><li><p>MicrofluidMechanics</p></li><li><p>This page intentionally left blank </p></li><li><p>Chapter</p><p>1Introduction</p><p>Microelectromechanical systems (MEMS) are considered one of themajor advances of industrial technologies in the past decades. MEMStechnology was derived initially from the integrated circuit (IC) fab-rication technologies. Now, microfabrication is a diverse spectrum ofprocessing techniques that involve a wide range of disciplines fromchemical sciences to plastic molding. As the name suggests, MEMS cov-ers micron-sized, electrically and/or mechanically driven devices. Com-pared with the conventional mechanical or electrical systems, theseMEMS devices are five to six orders of magnitude smaller in size. Infact, these dimensions are in the same range as the average diameterof human hair (about 50 m, 50 106m, or 50 m). An MEMS de-vice can be a single piece of hardware that produces outputs directlybased on the inputs from external sources. The outputs can be me-chanical and fluidic movement, electrical charges, analog signals, anddigital signals. Often several microcomponents are integrated, such asthe lab-on-a-chip device, which performs the multistage processing ofthe inputs and produces several different types of outputs, all in onesingle miniature device. The small sizes of MEMS make them portableand implantable. The manufacturing cost of MEMS is far from pro-hibitive because of the wide use of the batch-processing technologiesthat grow out of the well-developed IC industry. MEMS, therefore, of-fer opportunities to many areas of application, such as biomedical andinformation technology, that were thought not achievable using conven-tional devices. Estimates of the potential commercial market size wereas high as billions of U.S. dollars by 2010.Since the early work of Tai et al. (1989) and Mehregany et al. (1990)</p><p>on the surface-machined micromotors, there has been an explosivegrowth of the number and the types of potential application of MEMS.</p><p>1</p><p>Copyright 2006 by The McGraw-Hill Companies, Inc. Click here f..</p></li></ul>