# 臺灣博碩士論文加值系統

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 本文探討流體以微小均勻速度流經一具有滑移表面的球體和圓柱體的系統，試圖求出比Basset所得到之緩流運動的流場與拖曳係數解更高次之近似展開式。對於微小的雷諾數值，吾人利用奇異微擾法解析求解，在靠近和遠離物體(流體或圓柱體)的區域，各自選取局部適用之流線函數的展開式，它們分別稱為'Stokes'和'Oseen'展開式。此二展開式皆必須滿足Navier-Stokes方程式，而每一個展開式僅對應到各自的邊界條界(Stokes展開式需符合物體表面上滑移的條件，而Oseen展開式則要符合遠離物體處均勻流速分佈的情況)。另外，此二展開式亦需滿足互相對應配合的額外條件，才能決定出唯一的解。最後，吾人使用所求出的流線函數展開式解析解推導出物體在流體中拖曳係數的展開式，並將物體表面滑移摩擦係數和雷諾數對拖曳係數的影響效應一併討論。結果顯示球體與圓柱體的拖曳係數皆會隨著滑移摩擦係數和雷諾數的增加而增加，而其值主要是由雷諾數的大小來決定。
 The problem of obtaining higher approximations to the fluid flow past a slip sphere and a slip circular cylinder than those represented by the creeping-flow solutions of Basset is considered. The singular perturbation technique is employed to expand the solutions for small non-zero Reynolds numbers. Separate, locally valid expansions of the stream function are developed for the regions close to and far from the sphere or cylinder, and they are respectively called 'Stokes' and 'Oseen' expansions. Both expansions satisfy the Navier-Stokes equation, but only one set of physical boundary conditions is applicable to each expansion (the slip-surface conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion). Also, a 'matching' procedure is required to yield further boundary conditions for each expansion to ultimately determine the unique solutions. Finally, the stream function results are used to derive asymptotic expansions for the drag coefficient. The effects of the coefficient of sliding friction and the Reynolds number on the drag coefficient are discussed. For both cases of a sphere and a circular cylinder, the drag coefficient increases with the increase of the coefficient of sliding friction and the Reynolds number, and the latter plays a more important role than the former in determining the drag coefficient.
 第一章 緒論.......................................................................................1 1-1 球形粒子之緩流運動...........................................................1 1-2 慣性力的效應.......................................................................3 第二章 Stokes和Oseen的近似法....................................................9 2-1 流體流經一球體...................................................................9 2-2 流體流經一圓柱...................................................................12 第三章 流體流經具滑移表面之球體..............................................16 3-1 Stokes和Oseen展開式.......................................................16 3-2 展開式的首項.......................................................................20 3-3 Oseen展開式的第二項........................................................23 3-4 Stokes展開式的第二項.......................................................25 3-5 展開式的更高次項...............................................................29 3-6 結果與討論...........................................................................31 3-6.1 滑移參數的效應................................................... 32 3-6.2 雷諾數R的影響.........................................................35 第四章 流體流經具滑移表面之圓柱體..........................................37 4-1 Stokes和Oseen展開式.......................................................37 4-2 展開式的首項.......................................................................40 4-3 Oseen展開式的第二項........................................................42 4-4 展開式的更高次項...............................................................44 4-5 結果與討論...........................................................................48 4-5.1 滑移參數的效應... ............................................... 48 4-5.2 雷諾數R的影響.........................................................48 第五章 結論.......................................................................................53 符號說明.............................................................................................55 參考文獻.............................................................................................58
 Albano, A. M., D. Bedeaux, and P. Mazur, "On the motion of a sphere arbitrary slip in a viscous incompressible fluid." Physica A 80, 89 (1975).Basset, A. B., A Treatise on Hydrodynamics. Vol. 2, Dover, New York (1961).Davis, M. H., "Collisions of small cloud droplets: gas kinetic effects." J. Atmos. Sci. 29, 911 (1972).Faxen, H., Nova Acta Societatis Scientianum Upsaliensis. Volumen extra ordinem (1927).Felderhof, B. U., "Hydrodynamic interaction between two spheres." Physica A 88, 373 (1977).Goldstein, S., "The steady flow of viscous fluid past a fixed spherical obstacle at small Reynolds numbers." Proc. Roy. Soc. A. 123, 225 (1929).Hadamard, J. S., "Mouvement permanent lent d'Une sphere liquide et visqueuse dans un liquid visqueux." Compt. Rend. Acad. Sci. (Paris) 152, 1735 (1911).Hancock, G. J., "The self-propulsion of microscopic organisms through liquids." Proc. Roy. Soc. A. 217, 96. (1953).Happle, J., and H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, the Netherlands (1983).Kaplun, S., "Low Reynolds number flow past a circular cylinder." J. Math. Mech. 6, 595 (1957).Kaplun, S., and P. A. Lagerstrom, "Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers." J. Math. Mech. 6, 585 (1957).Kennard, E. H., Kinetic Theory of Gases, McGraw-Hill, New York (1938).Lamb, H., "On the uniform motion of a sphere through a viscous fluid." Phil. Mag. 21, 112 (1911).Loyalka, S. K., "Slip and jump coefficients for rarefied gas flows: variational results for Lennard-Jones and n(r)-6 potentials." Physica A. 163, 813 (1990).Loyalka, S. K., and J. L. Griffin, "Rotation of non-spherical axi-symmetric particles in the slip regime." J. Aerosol Sci. 25, 509 (1994).Oseen, C. W., "Ueber die Stokes'sche Formel, und uber eien verwandte Aufgabe in der Hydrodynamik." Ark. f. Mat. Astr. og Fys. 6, No. 29 (1910).Perry, R. H., Perry's Chemical Engineers' Handbook, 6th ed., McGraw -Hill, New York (1984).Proudman, I., and J. R. A. Pearson, "Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder." J. Fluid Mech. 2, 237 (1957).Rybczynski, W., "Uber die fortschreitende bewegung einer flussigen kugel in einem zahen medium." Bull. Acad. Sci. Cracovie A., 40 (1911).Stokes, G. G., "On the effect of the internal friction of fluid on pendulums." Trans. Camb. Phil. Soc. 9, 8 (1851).Talbot, L., R. K. Cheng, R. W. Schefer, and D. R.Willis, "Thermophoresis of particles in heated boundary layer." J. Fluid Mech. 101, 737 (1980).Tomotika, S., and T. Aoi, "The steady flow of viscous fluid past a sphere and circular cylinder at small Reynolds numbers." Quart. J. Mech. Appl. Math. 3, 140 (1950).Van Dyke, M., Perturbation Methods in Fluid Mechanics, the Parabolic Press (1975).Whitehead, A. N., "Second approximations to viscous fluid motion." Quart. J. Math. 23, 143 (1889).
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