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The Microchannel Flow of a Micropolar Fluid Guohua Liu Louisiana Tech University

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The Microchannel Flow of a Micropolar Fluid Guohua Liu Louisiana Tech University Louisiana Tech University Louisiana Tech Digital Commons Doctoral Dissertations Graduate School Fall 1999 The microchannel flow of a micropolar fluid Guohua Liu Louisiana Tech University Follow this and additional works at: https://digitalcommons.latech.edu/dissertations Part of the Fluid Dynamics Commons, and the Other Mechanical Engineering Commons Recommended Citation Liu, Guohua, "" (1999). Dissertation. 176. https://digitalcommons.latech.edu/dissertations/176 This Dissertation is brought to you for free and open access by the Graduate School at Louisiana Tech Digital Commons. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Louisiana Tech Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE MICRO-CHANNEL FLOW OF A MICRO-POLAR FLUID by Guohua Liu, M.S. A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy COLLEGE OF ENGINEERING AND SCIENCE LOUISIANA TECH UNIVERSITY November 1999 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9947114 UMI Microform 9947114 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LOUISIANA TECH UNIVERSITY THE GRADUATE SCHOOL October 26. 1999____________ Date We hereby recommend that the dissertation prepared under our supervision by Guohua Liu____________________________________________________________ entitled The Micro-Channel Flow of A M irm-Pnlar FlniH _________________________ be accepted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy eryispr of Dissertation Research of Department Applied Computational Analysis and Modeling Department Recommendation concurred in: Advisory Committee Approved: lies Director of the Graduate School GS Form 13 2/97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Micro-channel flows have been computed to investigate the influence of Navier- Stokes formulation for the slip-flow boundary condition, and a micro-polar fluid model, respectively. The results of the slip boundary condition show that the current methodology is valid for slip-flow regime (i.e., for values of Knudsen number less than approximately 0.1). Drag reduction phenomena apparent in some micro-channels can be explained by slip-flow theory. These results are in agreement with some computations and experiments. An ad hoc micro-polar fluid model is developed to investigate the influence of micro effects, such as micro-gyration, in micro-scale flows. The foundation of the ad hoc micro-polar fluid is based on Eringen’s micro simple fluid, and is simplified for incompressible, two-dimensional, iso-thermal, and micro-isotropic case. Our model contains two material constants, p. and k , one scale parameter, mxKn, and one boundary condition parameter n. The number of parameters is significantly reduced from general micro-polar fluid model and makes the theory practical. The scale parameter mxKn introduces the Knudsen number into the micro-polar fluid dynamics by statistical explanation. Therefore, the effect of rarefaction can be accounted into the model by modeling this parameter. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The parameter p. is classical bulk viscosity. The vortex viscosity k is related to micro-gyration, and needs modeling at current time. It affects the flow field in two aspects, by modifying the apparent viscosity and by introducing the effect of micro- gyration. In the simplest case of fully-developed channel flow, the overall effect is equivalent to lessen the Reynolds number by (1 +/c/2). The current micro-polar fluid model explains the drag increase phenomenon in some micro-channel flows from both experimental and computational data. This result is exactly opposite to that predicted by slip-flow theory. The existence of micro-effect needs to be taken into account for the micro-scale flow. A projection method is used as a numerical technique for both models to solve the difficulty of implicit pressure equation, with the help of staggered grids. An explicit Euler scheme is used for solving the steady flow. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPROVAL FOR SCHOLARLY DISSEMINATION The author grants to the Prescott Memorial Library o f Louisiana Tech University the right to reproduce, by appropriate methods, upon request, any or all portions of this Dissertation. It is understood that “proper request” consists o f the agreement, on the part o f the requesting party, that said reproduction is for his personal use and that subsequent reproduction will not occur without written approval o f the author of this Dissertation. Further, any portions of the Dissertation used in books, papers, and other works must be appropriately referenced to this Dissertation. Finally, the author o f this Dissertation reserves the right to publish freely, in the literature, at any time, any or all portions o f this Dissertation. Author Date GS Form 14 2/97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ABSTRACT ............................................................................................................................. iii LIST OF TABLES................................................................................................................. viii LIST OF FIGURES ............................................................................................................... ix LIST OF EQUATIONS .......................................................................................................xii NOMENCLATURE ............................................................................................................xvii ACKNOWLEDGMENTS .................................................................................................. xxi CHAPTER I INTRODUCTION 1.1 General Overview .............................................................................................1 1.2 Classical Fluid Dynamics for Micro-Scale Flow ..........................................5 1.3 Development of Micro-Polar Fluid M odel ....................................................9 1.4 Objectives ........................................................................................................15 1.5 Organization of the Dissertation ....................................................................15 CHAPTER 2 SLIP-FLOW MODEL 2.1 Introduction ................................................................................................... 17 2.2 Governing Equations and Boundary Conditions .......................................18 CHAPTER 3 MICRO-POLAR FLUID MODEL 3.1 Introduction ....................................................................................................22 3.2 General Micro-Polar Fluid Model ...............................................................24 3.3 Ad hoc Two-Dimensional Micro-Polar Fluid Model ................................. 27 3.3.1 Basic Assumptions and Formulations..........................................................27 3.3.2 Relations among the Material Constants .....................................................29 3.3.3 Non-Dimensionalized Version .....................................................................32 3.3.4 Boundary Conditions ..................................................................................... 34 3.3.5 Unidirectional Flow Solution .......................................................................
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