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Stokes flow for a stokeslet between two parallel flat plates


Velocity and pressure fields for Stokes flow due to a force singularity of arbitrary orientation and arbitrary distance between two parallel plates are found, using the image technique and a Fourier transform. Two alternative expressions for the solution, one in terms of infinite integrals and the other in terms of infinite series, are given. The infinite series solution is especially suitable for computation purposes being an exponentially decreasing series. From the series the “far field” behaviour is extracted. It is found that a force singularity parallel to the two planes has a far field behaviour of source and image for the parallel components (a two dimensional source doublet of height-dependent strength) whereas the normal component, and all fields due to a force singularity normal to the planes, die out exponentially. Velocity fields are compared with those of the one plane case. An estimate of the influence of the second wall and when its effect can be disregarded is obtained.

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  1. J. R. Blake, A note on the image system for a stokeslet in a no-slip boundary,Proc. Camb. Phil. Soc., 70 (1971) 303–310.

    Google Scholar 

  2. J. R. Blake, A spherical envelope approach to ciliary propulsion,J. Fluid Mech., 46 (1971) 199–208.

    Google Scholar 

  3. J. R. Blake, Infinite models for ciliary propulsion,J. Fluid Mech., 49 (1971) 209–222.

    Google Scholar 

  4. J. R. Blake, Self-propulsion due to oscillations on the surface of a cylinder at low Reynolds number,Bull. Aust. Math. Soc., 5 (1971) 255–264.

    Google Scholar 

  5. J. R. Blake, A model for the micro-structure in ciliated organisms,J. Fluid Mech., 55 (1972) 1–23.

    Google Scholar 

  6. J. R. Blake and A. T. Chwang, Fundamental singularities of viscous flow, part I,J. Engineering Math., 8 (1974) 23–29.

    Google Scholar 

  7. T. J. Lardner and W. J. Shack, Cilia transport,Bull. Math. Biophys., 34 (1972) 325–335.

    Google Scholar 

  8. H. A. Lorentz,Zittingsverlag. Akad. v. Wet., 5 (1896) 168–182.

    Google Scholar 

  9. N. J. De Mestre, Low-Reynolds-number fall of slender cylinders near boundaries,J. Fluid Mech., 58 (1973) 641–656.

    Google Scholar 

  10. S. M. Ross and S. Corrsin, Results of an analytical model of mucociliary pumping,J. of Applied Physiology, 37 (1974) 333–340.

    Google Scholar 

  11. I. N. Sneddon,Fourier transforms, McGraw-Hill Co., (1951) 62.

  12. G. N. Watson,A treatise on the theory of Bessel Functions, Cambridge Univ. Press, 1948, 2d ed.

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Liron, N., Mochon, S. Stokes flow for a stokeslet between two parallel flat plates. J Eng Math 10, 287–303 (1976).

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