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Small amplitude oscillations of a thin beam immersed in a viscous fluid near a solid surface

Phys. Fluids 17, 073102 (2005); doi:10.1063/1.1995467

Published 20 July 2005

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Christopher P. Green and John E. Sader
Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia
The hydrodynamic loading on a solid body moving in a viscous fluid can be strongly affected by its proximity to a surface. In this article, we calculate the hydrodynamic load on an infinitely long rigid beam of zero thickness that is undergoing small amplitude oscillations. The presence of a solid surface an arbitrary distance from the beam is rigorously accounted for using a boundary integral formulation. ©2005 American Institute of Physics
History: Received 30 November 2004; accepted 17 June 2005; published 20 July 2005
Permalink: http://link.aip.org/link/?PHFLE6/17/073102/1
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KEYWORDS and PACS

Keywords
PACS
  • 47.50.+d
    Non-Newtonian fluid flows
  • 47.35.+i
    Hydrodynamic waves
  • 02.60.Nm
    Integral and integrodifferential equations
  • YEAR: 2005

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PUBLICATION DATA

ISSN:
1070-6631 (print)   1089-7666 (online)
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REFERENCES (43)

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  1. G. G. Stokes, "On the effect of internal friction of fluids on the motion of pendulums," Trans. Cambridge Philos. Soc. 9, 8 (1851).
  2. L. Rosenhead, Laminar Boundary Layers (Clarendon, Oxford, 1963).
  3. M. Ray, "The vibration of an infinite elliptic cylinder in a viscous fluid," Z. Angew. Math. Mech. 16, 99 (1936).
  4. R. P. Kanwal, "Vibrations of an elliptic cylinder and of a flat plate in a viscous fluid," Z. Angew. Math. Mech. 35, 17 (1955).
  5. E. O. Tuck, "Calculation of unsteady flows due to small motions of cylinders in a viscous fluid," J. Eng. Math. 3, 29 (1969).
  6. R. E. Williams and R. G. Hussey, "Oscillating cylinders and the Stokes paradox," Phys. Fluids 15, 2083 (1972).
  7. B. Huner and R. G. Hussey, "Cylinder drag at low Reynolds number," Phys. Fluids 20, 1211 (1977).
  8. C. J. Lawrence and S. Weinbaum, "The unsteady force on a body at low Reynolds number—the axisymmetric motion of a spheroid," J. Fluid Mech. 189, 463 (1988).
  9. C. Pozrikidis, "A study of linearized oscillatory flow past particles by the boundary integral method," J. Fluid Mech. 202, 17 (1989).
  10. C. Pozrikidis, "A singularity method for unsteady linearized flow," Phys. Fluids A 1, 1508 (1989).
  11. M. Loewenberg, "Asymmetric, oscillatory motion of a finite-length cylinder: The macroscopic effect of particle edges," Phys. Fluids 6, 1095 (1994).
  12. M. Lowenberg, "Axisymmetric unsteady Stokes flow past an oscillating finite-length cylinder," J. Fluid Mech. 265, 265 (1994).
  13. J. E. Sader, "Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope," J. Appl. Phys. 84, 64 (1998).
  14. J. E. Sader, J. W. M. Chon, and P. Mulvaney, "Calibration of rectangular atomic force microscope cantilevers," Rev. Sci. Instrum. 70, 3967 (1999).
  15. C. Bergaud, N. Nicu, and A. Martinez, "Multi-mode air damping analysis of composite cantilever beams," Jpn. J. Appl. Phys., Part 1 38, 6521 (1999).
  16. M. P. Scherer, G. Frank, and A. W. Gummer, "Experimental determination of the mechanical impedance of atomic force microscopy cantilevers in fluids up to 70  kHz," J. Appl. Phys. 88, 2912 (2000).
  17. C. P. Green and J. E. Sader, "Torsional frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope," J. Appl. Phys. 92, 6262 (2002).
  18. C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, "Normal and torsional spring constants of atomic force microscope cantilevers," Rev. Sci. Instrum. 75, 1988 (2004).
  19. D. O. Clubb, O. V. L. Buu, R. M. Bowley, R. Nyman, and J. R. Owers-Bradley, "Quartz tuning fork viscometers for helium liquids," J. Low Temp. Phys. 136, 1 (2004).
  20. D. Anselmetti, R. Lüthi, E. Meyer, T. Richmond, M. Dreier, J. E. Frommer, and H. J. Güntherodt, "Attractive mode imaging of biological materials with dynamic force microscopy," Nanotechnology 5, 87 (1994).
  21. Q. Zong, D. Innis, K. Kjoller, and V. B. Elings, "Fractured polymer/silica fiber surface studied by tapping mode atomic force microscopy," Surf. Sci. Lett. 290, L688 (1993).
  22. H. G. Hansma, R. L. Sinsheimer, J. Groppe, T. C. Bruice, V. Elings, G. Gurley, M. Bezanilla, I. A. Mastrangelo, P. V. C. Hough, and P. K. Hansma, "Recent advances in atomic force microscopy of DNA," Scanning 15, 296 (1993).
  23. P. K. Hansma, J. P. Cleveland, M. Radmacher, D. A. Walters, P. E. Hillner, M. Bezanilla, M. Fritz, D. Vie, H. G. Hansma, C. B. Prater, J. Massie, L. Fukunaga, J. Gurley, and V. Elings, "Tapping mode atomic force microscopy in liquids," Appl. Phys. Lett. 64, 1738 (1994).
  24. Y. Huang and C. C. Williams, "Capacitance voltage measurement and modeling on a nanometer scale by scanning C-V microscopy," J. Vac. Sci. Technol. B 12, 369 (1994).
  25. Y. Huang, C. C. Williams, and J. Slinkman, "Quantitative 2-dimensional dopant profile measurement and inverse modeling by scanning capacitance microscopy," Appl. Phys. Lett. 66, 344 (1995).
  26. F. Pan, J. Kubby, E. Peeters, A. T. Tran, and S. Mukherjee, "Squeeze film damping effect on the dynamic response of a MEMS torsion mirror," J. Micromech. Microeng. 8, 200 (1998).
  27. A. Pavlov, Y. Pavlova, and R. Laiho, "Proposal of scanning probe microscope with MEMS cantilever for study of conductive and non-conductive materials," Rev. Adv. Mater. Sci. 5, 324 (2003).
  28. O. Reynolds, "On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments including an experimental determination of the viscosity of olive oil," Philos. Trans. R. Soc. London 177, 157 (1886).
  29. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, 1965).
  30. W. E. Langlois, "Isothermal squeeze films," Q. Appl. Math. 20, 131 (1962).
  31. D. F. Hays, "Squeeze films for rectangular plates," J. Basic Eng. 85, 243 (1963).
  32. W. S. Griffin, H. H. Richardson, and S. Yamanami, "A study of squeeze film damping," J. Basic Eng. 88, 451 (1966).
  33. M. H. Sadd and A. K. Stiffler, "Squeeze film dampers: Amplitude effects at low squeeze numbers," J. Eng. Ind. 97, 1366 (1975).
  34. R. Hsu and P. Ganatos, "The motion of a rigid body in viscous fluid bounded by a plane wall," J. Fluid Mech. 207, 29 (1989).
  35. S. Kim and S.J. Karrila, Microhydrodynamics (Butterworth-Heinemann, Boston, 1991).
  36. L.D. Landau and E.M. Lifshitz, Theory of Elasticity (Pergamon, London, 1959).
  37. G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1974).
  38. P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  39. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  40. The convention adopted for the Reynolds number conforms with Ref. 37. We note that the Reynolds number is often associated with the nonlinear convective inertial term in the Navier–Stokes equation. This latter convention has not been adopted here.
  41. This can be verified by plotting the pressure and vorticity obtained in this manner.
  42. A.P. Prudnikov, Y.A. Brychkov, and O.I. Marichev, Integrals and Series (Gordon and Breach, New York, 1990), Vol. 3.
  43. J.S. Marshall, Inviscid Incompressible Flow (Wiley, New York, 2001).

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