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Simple and efficient representations for the fundamental solutions of Stokes flow in a half-space

  • Z. Gimbutas (a1), L. Greengard (a2) (a3) and S. Veerapaneni (a4)

We derive new formulae for the fundamental solutions of slow viscous flow, governed by the Stokes equations, in a half-space. They are simpler than the classical representations obtained by Blake and collaborators, and can be efficiently implemented using existing fast solver libraries. We show, for example, that the velocity field induced by a Stokeslet can be annihilated on the boundary (to establish a zero-slip condition) using a single reflected Stokeslet combined with a single Papkovich–Neuber potential that involves only a scalar harmonic function. The new representation has a physically intuitive interpretation.

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Aderogba, K. & Blake, J. R. 1978a Action of a force near the planar surface between semi-infinite immiscible liquids at very low Reynolds numbers: addendum. Bull. Austral. Math. Soc. 19 (2), 309318.
Aderogba, K. & Blake, J. R. 1978b Action of a force near the planar surface between two semi-infinite immiscible liquids at very low Reynolds numbers. Bull. Austral. Math. Soc. 18 (3), 345356.
Bhattacharya, S. & Bławzdziewicz, J. 2002 Image system for Stokes-flow singularity between two parallel planar walls. J. Math. Phys. 43 (11), 57205731.
Blake, J. R. 1971 A note on the image system for a Stokeslet in a no-slip boundary. Math. Proc. Camb. Phil. Soc. 70 (2), 303310.
Blake, J. R. 1974 Singularities of viscous flow. J. Engng Maths 8 (2), 113124.
Blake, J. R. & Chwang, A. T. 1974 Fundamental singularities of viscous flow. J. Engng Maths 8 (1), 2329.
Cichocki, B. & Jones, R. B. 1998 Image representation of a spherical particle near a hard wall. Physica A 258 (3), 273302.
Fong, W. & Darve, E. 2009 The black-box fast multipole method. J. Comput. Phys. 228 (23), 87128725.
Frangi, A. 2005 A fast multipole implementation of the qualocation mixed-velocity–traction approach for exterior Stokes flows. Engng Anal. Bound. Elem. 29 (11), 10391046.
Fu, Y., Klimkowski, K. J., Rodin, G. J., Berger, E., Browne, J. C., Singer, J. K., Van De Geijn, R. A. & Vemaganti, S. K. 1998 A fast solution method for three-dimensional many-particle problems of linear elasticity. Intl J. Numer. Meth. Engng 42 (7), 12151229.
Fu, Y. & Rodin, G. J. 2000 Fast solution method for three-dimensional Stokesian many-particle problems. Commun. Numer. Meth. Engng 16 (2), 145149.
Gimbutas, Z. & Greengard, L.2012, STFMMLIB3 – fast multipole method (FMM) library for the evaluation of potential fields governed by the Stokes equations in $R^{3}$ ,
Gumerov, N. A. & Duraiswami, R. 2006 Fast multipole method for the biharmonic equation in three dimensions. J. Comput. Phys. 215 (1), 363383.
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media. Springer.
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lorentz, H. A. 1896 Eene algemeene stelling omtrent de beweging eener vloeistof met wrijving en eenige daaruit afgeleide gevolgen. Versl. K. Akad. W. Amsterdam 5, 168175.
Lorentz, H. A. 1907 Ein allgemeiner Satz, die Bewegung einer reibenden Flüssigkeit betreffend, nebst einigen Anwendungen desselben. Abhand. Theor. Phys. 1, 2342.
Mindlin, R. D. 1936 Force at a point in the interior of a semi-infinite solid. J. Appl. Phys. 7 (5), 195202.
Neuber, H. 1934 Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. Der Hohlkegel unter Einzellast als Beispiel. Z. Angew. Math. Mech. 14 (4), 203212.
Papkovich, P. F. 1932 Solution générale des équations differentielles fondamentales d’élasticité exprimée par trois fonctions harmoniques. C. R. Acad. Sci. Paris 195, 513515.
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Spagnolie, S. E. & Lauga, E. 2012 Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J. Fluid Mech. 700, 105147.
Tornberg, A. K. & Greengard, L. 2008 A fast multipole method for the three-dimensional Stokes equations. J. Comput. Phys. 227 (3), 16131619.
Veerapaneni, S. K., Rahimian, A., Biros, G. & Zorin, D. 2011 A fast algorithm for simulating vesicle flows in three dimensions. J. Comput. Phys. 230 (14), 56105634.
Wang, H., Lei, T., Li, J., Huang, J. & Yao, Z. 2007 A parallel fast multipole accelerated integral equation scheme for 3D Stokes equations. Intl J. Numer. Meth. Engng 70 (7), 812839.
Wang, H. T. & Yao, Z. H. 2005 A new fast multipole boundary element method for large scale analysis of mechanical properties in 3d particle-reinforced composites. Comput. Model. Engng Sci. 7 (1), 8595.
Wang, X., Kanapka, J., Ye, W., Aluru, N. R. & White, J. 2006 Algorithms in FastStokes and its application to micromachined device simulation. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 25 (2), 248257.
Ying, L., Biros, G. & Zorin, D. 2004 A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys. 196 (2), 591626.
Yoshida, K., Nishimura, N. & Kobayashi, S. 2001 Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3d. Intl J. Numer. Meth. Engng 50 (3), 525547.
Yu, H. Y. 2003 Fundamental singularities in a two-fluid Stokes flow with a plane interface. J. Mech. 19 (1), 263270.
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Journal of Fluid Mechanics
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  • EISSN: 1469-7645
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