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Integral equation methods for Stokes flow in doubly-periodic domains

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Abstract

A fast integral-equation technique is presented for the calculation of Stokes flow in doubly-periodic domains. While existing integral formulations typically rely on a Fourier series to compute the governing Greens' function, here a method of images is developed which is faster, more flexible, and easily incorporated into the fast multipole method. Accurate solutions can be obtained with obstacles of arbitrary shape at a cost roughly proportional to the number of points needed to resolve the interface. The performance of the method is illustrated with several numerical examples.

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References

  1. L. Greengard, M.C. Kropinski and A. Mayo, Integral equation methods for Stokes flow and isotropic elasticity in the plane. J. Comp. Phys. 125 (1996) 403–414.

    Google Scholar 

  2. M.C.A. Kropinski, Integral equation methods for particle simulations in creeping flows. Comp. Math. Appl. 38, (1999) 67–87.

    Google Scholar 

  3. L. Greengard and J. Helsing, On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites. J. Mech. Phys. Solids 46 (1998) 1441–1462.

    Google Scholar 

  4. R.E. Larson and J.J.L. Higdon, Microscopic flow near the surface of two-dimensional porous media. Part 1. Axial flow. J. Fluid Mech. 166 (1986) 449–472.

    Google Scholar 

  5. R.E. Larson and J.J.L. Higdon, Microscopic flow near the surface of two-dimensional porous media. Part 2. Transverse flow. J. Fluid Mech. 178 (1987) 119–136.

    Google Scholar 

  6. A.A. Zick and G.M. Homsy, Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115 (1982) 13–26.

    Google Scholar 

  7. T. Tran-Cong, N. Phan-Thien and A.L. Graham, Stokes problems of multiparticle systems: periodic arrays. Phys. Fluids A 2 (1990) 666–673.

    Google Scholar 

  8. J.E. Drummond and M.I. Tahir, Laminar viscous flow through regular arrays of parallel solid cylinders. Int. J. Multiphase Flow 10 (1984) 515–540.

    Google Scholar 

  9. A. van de Vorst, Numerical simulation of viscous sintering by a periodic lattice of a representative unit cell. J. Am. Ceram. Soc. 81 (1998) 2147–2156.

    Google Scholar 

  10. G.A.L. van de Vorst, Integral formulation to simulate the viscous sintering of a two-dimensional lattice of periodic unit cells. J. Engng. Math. 30 (1996) 97–118.

    Google Scholar 

  11. R. Charles and C. Pozrikidis, Significance of the dispersed-phase viscosity on the simple shear flow of suspensions of two-dimensional liquid drops. J. Fluid Mech. 365 (1998) 205–234.

    Google Scholar 

  12. X. Li, R. Charles and C. Pozrikidis, Simple shear flow of suspensions of liquid drops. J. Fluid Mech. 320 (1996) 395–416.

    Google Scholar 

  13. C. Pozrikidis, Computation of periodic Green's functions of Stokes flow. J. Engng. Math. 30 (1996) 79–96.

    Google Scholar 

  14. C. Pozrikidis, Computation of the pressure inside bubbles and pores in Stokes flow. J. Fluid. Mech. 474 (2003) 319–337.

    Google Scholar 

  15. X. J. Fan, N. Phan-Thien and R. Zheng, Completed double layer boundary element method for periodic suspensions. Z. Ang. Math. Phys. 49 (1998) 167–193.

    Google Scholar 

  16. J.F. Brady, R.J. Phillips, J.C. Lester and G. Bossis, Dynamic simulation of hydrodynamical interacting suspensions. J. Fluid Mech. 195 (1998) 257–280.

    Google Scholar 

  17. A. Sierou and J.F. Brady, Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448 (2001) 115–146.

    Google Scholar 

  18. I.L. Claeys and J.F. Brady, Suspensions of prolate spheroids in Stokes flow. Part 2. Statistically homogeneous dispersions. J. Fluid Mech. 251 (1993) 443–477.

    Google Scholar 

  19. A.S. Sangani and G. Mo, An O(N) algorithm for Stokes and Laplace interactions of particles. Phys. Fluids 8 (1996) 1990–2010.

    Google Scholar 

  20. D.A. Edwards, M. Shapiro, P. Bar-Yoseph and M. Shapira, The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders. Phys. Fluids A 2 (1990) 45–55.

    Google Scholar 

  21. C.Y. Wang, Stokes flow through an array of rectangular fibers. Int. J. Multiphase Flow 22 (1996) 185–194.

    Google Scholar 

  22. H.P.A. Souto and C. Moyne, Dispersion in two-dimensional periodic porous media. Part 1. Hydrodynamics. Phys. Fluids 9 (1997) 2243–2252.

    Google Scholar 

  23. H.P.A. Souto and C. Moyne, Dispersion in two-dimensional periodic porous media. Part 2. Dispersion tensor. Phys. Fluids 9 (1997) 2253–2263.

    Google Scholar 

  24. J. Carrier, L. Greengard and V. Rokhlin A fast adaptive multipole algorithm for particle simulations. SIAM J. Sci. Statist. Comput. 9 (1998) 669–686.

    Google Scholar 

  25. L. Greengard and V. Rokhlin, A fast algorithm for particle simulations. J. Comp. Phys. 73 (1987) 325–348.

    Google Scholar 

  26. L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems. Cambridge, Mass.: MIT Press (1988) 90 pp.

    Google Scholar 

  27. V. Rokhlin, Rapid solution of integral equations of classical potential theory. J. Comp. Phys. 60 (1985) 187–207.

    Google Scholar 

  28. S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications. Boston, Mass.: Butterworth-Heinemann (1991) 355 pp.

    Google Scholar 

  29. C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge, Mass.: Cambridge University Press (1992) 259 pp.

    Google Scholar 

  30. J.E. Gomez and H. Power, A parallel multipolar indirect boundary element method for the Neumann interior Stokes flow problem. Int. J. Numer. Methods Engng. 48 (2000) 523–543.

    Google Scholar 

  31. S.G. Mikhlin, Integral Equations. London: Pergammon Press (1957) 341 pp.

    Google Scholar 

  32. S.G. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Groningen: P. Noordhoff Ltd (1953) 704 pp.

    Google Scholar 

  33. V.Z. Parton and P.I. Perlin, Integral Equation Methods in Elasticity. Moscow: MIR (1982) 303 pp.

    Google Scholar 

  34. H. Hasimoto, On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (1959) 317–328.

    Google Scholar 

  35. P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 64 (1921) 253–287.

    Google Scholar 

  36. Lord Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of a medium. Phil. Mag. 34 (1892) 481–502.

    Google Scholar 

  37. W.T. Perrins, D.R. McKenzie and R.C. McPhedran, Transport properties of regular arrays of spheres. Proc. R. Soc. London A369 (1979) 207–225.

    Google Scholar 

  38. R.W. O'Brien, A method for the calculation of the effective transport properties of suspensions of interacting particles. J. Fluid Mech. 91 (1979) 17–39.

    Google Scholar 

  39. Y. Saad and M.H. Schultz, GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comp. 7 (1986) 856–869.

    Google Scholar 

  40. C.L. Berman and L. Greengard, A renormalization method for the evaluation of lattice sums. J. Math. Phys. 35 (1994) 6036–6048.

    Google Scholar 

  41. A.S. Sangani and A. Acrivos, Slow flow past periodic arrays of cylinders with application to heat transfer. Int. J. Multiphase Flow 8 (1982) 193–206.

    Google Scholar 

  42. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach (1969) 224 pp.

    Google Scholar 

  43. L. Greengard and V. Rokhlin, A new version of the fast multipole method for the laplace equation in three dimensions. Acta Numerica 6 (1997) 229–269.

    Google Scholar 

  44. L. Greengard and V. Rokhlin, Rapid evaluation of potential fields in three dimensions. In: C. Anderson and C. Greengard (eds.), Vortex Methods. Lecture Notes in Mathematics 1360. Berlin: Springer-Verlag (1988) pp. 121–141.

    Google Scholar 

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, New York, 10012, U.S.A

    Leslie Greengard

  2. Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada, V5A 1S6

    Mary Catherine Kropinski

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  1. Leslie Greengard
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  2. Mary Catherine Kropinski
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Greengard, L., Kropinski, M.C. Integral equation methods for Stokes flow in doubly-periodic domains. Journal of Engineering Mathematics 48, 157–170 (2004). https://doi.org/10.1023/B:ENGI.0000011923.59797.92

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  • DOI: https://doi.org/10.1023/B:ENGI.0000011923.59797.92

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