A general solution for Stokes flow and its application to the problem of a rigid plate translating in a fluid

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• 作者：Xiang-Yu Li (1)
  Si-Cong Ren (2)
  Qi-Chang He (2) (3)
  
  1. School of Mechanics and Engineering ; Southwest Jiaotong University ; 610031 ; Chengdu ; China
  2. School of Mechanical Engineering ; Southwest Jiaotong University ; 610031 ; Chengdu ; China
  3. Laboratoire Mod茅lisation et Simulation Multi Echelle UMR 8208 CNRS ; Universit茅 Paris-Est ; 5 Boulevard Descartes ; 77454 ; Marne-la-Vall茅e ; France
  
• 关键词：Stokes flow ; 3D general solutions ; Potential theory method ; Circular plate ; Elliptic plate
• 刊名：Acta Mechanica Solida Sinica
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：31
• 期：1
• 页码：32-44
• 全文大小：1,126 KB
• 参考文献：1. Selvadurai, APS (2000) Partial Differential Equations in Mechanics 2. CrossRef
  2. Batchelor, GK (1970) Slender-body theory for particles of arbitrary cross-section in Stokes flow. Journal of Fluid Mechanics 44: pp. 419-440 CrossRef
  3. Crowdy, D G (2013) Exact solutions for cylindrical 鈥渟lip-stick鈥?Janus swimmers in Stokes flow. Journal of Fluid Mechanics 719: pp. R2 (1-9) CrossRef
  4. Youngren, GK, Acrivos, A (1975) Stokes flow past a particle of arbitrary shape: a numerical method of solution. Journal of Fluid Mechanics 69: pp. 377-403 CrossRef
  5. Zick, AA, Homsy, GM (1982) Stokes flows through periodic arrays of spheres. Journal of Fluid Mechanic 115: pp. 13-26 CrossRef
  6. Bird, RB, Stewart, WE, Lightfoot, EN (1960) Transport Phenomena. Wiley, New York
  7. Happel, J, Brenner, H (1983) Low Reynolds Number Hydrodynamics. Martinus Nijhoff, Kluwer Academic Publishers, Dordrecht
  8. Kim, S, Karrila, S J (1991) Microhydrodynamics: Principles and Selected Applications. Butterworth, Heinemann, Boston
  9. Leal, L G (1992) Laminar Flow and Convective Transport Processes. Scaling Principles and Asymptotic Analysis. Butterworth, Heinemann, Boston
  10. Ho, C M, Tai, Y C (1998) Micro-electro-mechanical-systems (MEMS) and fluid flows. Annual Review of Fluid Mechanics 32: pp. 579-612 CrossRef
  11. Stokes, G G (1851) On the effect of the internal friction of fluids on the motion of pendulums. Transactions of the Cambridge Philosophical Society 9: pp. 8-106
  12. Charalambopoulos, A, Dassios, G (2002) Complete decomposition of axisymmetric Stokes flow. International Journal of Engineering Science 40: pp. 1099-1111 CrossRef
  13. Boussinesq, J (1888) 脡quilibre d鈥櫭﹍asticit茅 d鈥檜n solide sans pesanteur, homog`ene et isotrope, dont les parties profondes sont maintenues fixes, pendant que sa surface 茅prouve des pressions ou des d茅placements connus, s鈥檃nnullant hors d鈥檜ne r茅gion restreinte o霉 ils sont arbitraires. Comptes Rendus de l鈥橝cad茅mie des Sciences Paris 106: pp. 1043-1048
  14. Naghdi, PM, Hsu, C S (1961) On a representation of displacements in linear elasticity in terms of three stress functions. Journal of Mathematics and Mechanics 10: pp. 233-245
  15. Xu, XS, Wang, MZ (1991) General complete solutions of the equations of spatial and axisymmetric Stokes鈥?flow. The Quarterly Journal of Mechanics and Applied Mathematics 44: pp. 537-548 CrossRef
  16. Amaranath, T (2007) General solutions of the Stokes equations and their applications. Proceedings in Applied Mathematics and Mechanics 7: pp. 1100701-1100702 CrossRef
  17. Kong, DL, Cui, Z, Pan, YX (2012) On the Papkovich-Neuber formulation for Stokes flows driven by a translating/rotating prolate spheroid at arbitrary angles. International Journal of Pure and Applied Mathematics 75: pp. 455-483
  18. Padmavati, BS, Amaranath, T (2002) A note on decomposition of solenoidal fields. Applied Mathematics Letters 15: pp. 803-805 CrossRef
  19. Padmavathi, BS, Rajasekhar, GP, Amaranath, T (1998) A note on complete general solutions of Stokes equations. The Quarterly Journal of Mechanics and Applied Mathematics 51: pp. 383-388 CrossRef
  20. Tran-Cong, T, Blake, JR (1982) General solutions of the Stokes flow equations. Journal of Mathematical Analysis and Applications 90: pp. 72-84 CrossRef
  21. Kellogg, OD (1929) Foundations of Potential Theory. F. Ungar Publishing, New York CrossRef
  22. Fabrikant, VI (1989) Applications of Potential Theory in Mechanics: A Selection of New Results. Kluwer Academic Publishers, Dordrecht
  23. Fabrikant, VI (1991) Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering. Kluwer Academic Publishers, Dordrecht
  24. Chen, WQ, Ding, HJ (2004) Potential theory method for 3D crack and contact problems of multi-field coupled media: A survey. Journal of Zhejiang University Science 5: pp. 1009-1021 CrossRef
  25. Chen, WQ, Pan, EN, Wang, HM (2010) Theory of indentation on multiferroic composite materials. Journal of the Mechanics and Physics of Solids 58: pp. 1524-1551 CrossRef
  26. Li, XY (2012) Fundamental electro-elastic field in an infinite transversely isotropic piezoelectric medium with a permeable external circular crack. Smart Materials and Structures 21: pp. 065019 CrossRef
  27. Li, XY (2013) Fundamental solutions of penny-shaped and half-infinite plane cracks embedded in an infinite space of 1D hexagonal quasi-crystal under thermal loading. Proceedings of the Royal Society A 469: pp. 20130023 CrossRef
  28. Li, XY, Chen, WQ, Wang, HY (2010) General steady-state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application. European Journal of Mechanics A/Solids 29: pp. 317-326 CrossRef
  29. Oberbeck, A (1876) Ueber station盲re fl眉ssigkeitsbewegungen mit berucksichtigung der inneren. Journal f眉r die reine und angewandte Mathematik 81: pp. 62-80
  30. Lamb, H (1932) Hydrodymamics. University Press, Combridge
  31. Jeffery, GB (1922) The motion of ellipsoidal particles immersed in a viscous fluid. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 102: pp. 161-179 CrossRef
  32. Boridy, E (1990) Potential flow past an open spherical cavity. Journal of Applied Physics 67: pp. 6687-6693 CrossRef
  33. Blaser, S (2002) Forces on the surface of small ellipsoidal particles immersed in a linear flow field. Chemical Engineering Science 57: pp. 515-526 CrossRef
  34. Srivastava, DK, Yadav, RR, Yadav, S (2012) Steady stokes flow around deformed sphere. Class of oblate axisymmetric bodies. International Journal of Applied Mathematics and Mechanics 8: pp. 17-53
  35. Fabrikant, VI (2005) A new symbolism for solving the Hertz contact problem. The Quarterly Journal of Mechanics and Applied Mathematics 53: pp. 368-381
  36. Fabrikant, VI (2007) Utilization of divergent integrals and a new symbolism in contact and crack analysis. IMA Journal of Applied Mathematics 72: pp. 180-190 CrossRef
  37. Lauga, E, Powers, TR (2009) The hydrodynamics of swimming microorganisms. Reports on Progress in Physics 72: pp. 096601 CrossRef
  38. Wang, CY (1991) Exact solutions of the steady-state Navier-Stokes equations. Annual Review of Fluid Mechanics 23: pp. 159-177 CrossRef
  39. Ding, HJ, Chen, B, Liang, J (1996) General solutions for coupled equations for piezoelectric media. International Journal of Solids and Structures 33: pp. 2283-2298 CrossRef
  40. Gradsbteyn, IS, Ryzbik, LM (2004) Table of Integrals, Series, and Products. Elsevier, Amsterdam
  
• 刊物类别：Engineering
• 刊物主题：Theoretical and Applied Mechanics
  Mechanics, Fluids and Thermodynamics
  Engineering Fluid Dynamics
  Numerical and Computational Methods in Engineering
  Chinese Library of Science
  
• 出版者：The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of
• ISSN：1614-3116

文摘

A general solution for 3D Stokes flow is given which is different from, and more compact than the existing ones and more compact than them in that it involves only two scalar harmonic functions. The general solution deduced is combined with the potential theory method to study the Stokes flow induced by a rigid plate of arbitrary shape translating along the direction normal to it in an unbounded fluid. The boundary integral equation governing this problem is derived. When the plate is elliptic, exact analytical results are obtained not only for the drag force but also for the velocity distributions. These results include and complete the ones available for a circular plate. Numerical examples are provided to illustrate the main results for circular and elliptic plates. In particular, the elliptic eccentricity of a plate is shown to exhibit significant influences.