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Experimental Study of the Fluid Drag on a Torus at Low Reynolds Number. Dharmaratne Amarakoon Louisiana State University and Agricultural & Mechanical College

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Experimental Study of the Fluid Drag on a Torus at Low Reynolds Number. Dharmaratne Amarakoon Louisiana State University and Agricultural & Mechanical College Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1982 Experimental Study of the Fluid Drag on a Torus at Low Reynolds Number. Dharmaratne Amarakoon Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Amarakoon, Dharmaratne, "Experimental Study of the Fluid Drag on a Torus at Low Reynolds Number." (1982). LSU Historical Dissertations and Theses. 3743. https://digitalcommons.lsu.edu/gradschool_disstheses/3743 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 Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality o f the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. t.The sign or “target” for pages apparently lacking from the document photographed is "Missing Page(s)” . If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image o f the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame. 3. When a map, drawing or chart, etc., is part of the material being photographed, a definite method of “sectioning” the material has been followed. It is customary to begin filming at the upper left hand comer o f a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again-beginning below the first row and continuing on until complete. 4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases the best available copy has been filmed. University Microhms International 300 N .Z««b Road Ann Arbor, Ml 48106 8229486 Amarakooo, Dhannaratne EXPERIMENTAL STUDY OF THE FLUID DRAG ON A TORUS AT LOW REYNOLDS NUMBER The Louisiana Slate University and Agricultural and Mechanical PH.D. CoL 1982 University Microfilms International 300 N. Zeeb Roul, Ann Arbor, M I 46106 EXPERIMENTAL STUDY OF THE FLUID DRAG ON A TORUS AT LOW REYNOLDS NUMBER A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College In partial fu lfillm e n t of the requirements for the degree of Doctor of Philosophy In The Department of Physics and Astronomy by Dharmaratne Amarakoon B.S., University of Sr1 Lanka, 1970 M.S., Louisiana State University, 1977 August 1982 ACKNOWLEDGEMENTS The author Is deeply Indebted to his research advisor, Dr. Robert G. Hussey* and wishes to express his most sincere gratitude not only for much helpful advice and encouragement, but also for the hours of revealing conversation and discussion which were so unselfishly given. The author wishes to express his heart-felt appreciation to Drs. Bill J. Good and Edward G. Grlmsal for th e ir helpful advice, suggestions and for th eir many contributions that made this project a successful one. Warmest and sincere thanks are extended to Drs. Burke Huner and Robert P. Roger for their valuable advice and constructive comments offered at various stages of this project. The author 1s indebted to Mr. Bud Schuler for his s k ill 1n the fabrication of the closed torus. 11 TABLE OF CONTENTS Page TITLE PAGE.................................................................................................... 1 ACKNOWLEDGEMENTS............................................................................................. 11 TABLE OF CONTENTS....................................................................................... 111 LIST OF TABLES........................................................................................ v LIST OF FIGURES........................................................ v1 ABSTRACT...........................................................................................................v11 CHAPTER 1: INTRODUCTION........................................................................ 1 A. Statement of the Problem ........................................................ 1 B. Theoretical Background ............................................................ 1 1. The equations of fluid flo w ......................................... 1 2. The Reynolds number and the Stokes approximation. 2 3 The drag fo^ce and solutions to equations of motion.................................................................................... 4 4. The Inertial corrections to the equations of motion......................................................................................... 15 5. Boundary effects 1n slow viscous f lo w.......................... 21 C. The Present Work .............................................................................. 26 CHAPTER 2: THE EXPERIMENT.......................................................................... 29 CHAPTER 3: RESULTS...................................................................................... 50 A. Presentation of Data ..................................................................... 50 B. Boundary Effects ............................................................................. 50 111 TABLE OF CONTENTS (continued) Paae 1. Influences of the free surface and bottom surface of the tank ............................................................................ 50 2. Relation between circular and square boundary e ffe c ts .................................................................................... 56 3. The weak boundary re g io n ................................................ 58 4. The boundary dominant region ........................................ 65 C. Inertial Effects ........................................................................ 70 D. Comparison with Theory ........................................................... 79 CHAPTER 4: SUMMARY AND CONCLUSIONS................................................... 85 REFERENCES.................................................................................................... 87 APPENDIX I: COMMENTS ON, AND DERIVATION OF THE ASYMPTOTIC FORMULA * nn F * 2to /3(n 0 + Jtn 4 + i ).................................. 91 APPENDIX I I : DELAY CIRCUIT.................................................................... 96 APPENDIX I I I : CORRELATION OF THE RESULTS IN THE STRONG BOUNDARY REGION IN TERMS OF G AND E ...................... 97 APPENDIX IV: VALUES OF THE KINEMATIC VISCOSITY OF FLUID B . 98 VITA ................................................................................................................ 99 1 v L IS T OF TABLES Page I . Drag Values for a Disk, Hemispherical Cap and a Cardioid of Revolution. ................................................ 6 I I . Values of the Dlmensionless Drag F* ......................................13 I I I . The Dimensions and Effective Weights of Rings (T o ri).............................................................................................31 IV. Data for the Top and Bottom E f f e c t ......................................52 V. Velocities of Rings in Different Boundaries and the Corresponding Viscosities .................................................... 59 VI. Values of the Dimensionless Drag F* Obtained 1n D ifferent Boundaries ............................................................ 61 V II. Data fo r Rings in the Boundary Dominant Case. 66 V III. Data for Simultaneous Boundary and In e rtia l Effects........................................................... 75 IX. Results (Values of K) of Sutterby for Spheres. 78 X. Valuesco of F1 ................................................................. 81 V LIST OF FIGURES Page I 1. The variation of dlmenslonless drag F^w lth geometrical parameter s0 .................................................................................. 14 2(a ). Geometrical Illu s tra tio n of the problem solved by Frazer ................................................................................................ 27 2(b). The problem of a torus close to a cylindrical boundary. 27 3. A perspective drawing of the big mechanical dropper used to release the torus 1n the liq u id ............................ 38 4. Sketch of the experimental setup ............................................. 43 5. The variation of the product of velocity U and viscosity u with depth 1n the liq u id .................................... 55 6. The variation of
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