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Nearly Singular Integrals in 3D Stokes Flow

  • Svetlana Tlupova (a1) and J. Thomas Beale (a2)
Abstract

A straightforward method is presented for computing three-dimensional Stokes flow, due to forces on a surface, with high accuracy at points near the surface. The flow quantities are written as boundary integrals using the free-space Green’s function. To evaluate the integrals near the boundary, the singular kernels are regularized and a simple quadrature is applied in coordinate charts. High order accuracy is obtained by adding special corrections for the regularization and discretization errors, derived here using local asymptotic analysis. Numerical tests demonstrate the uniform convergence rates of the method.

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Corresponding author.Email:stlupova@umich.edu
References
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[1]Aris, R., Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Dover, New York, 1962.
[2]Beale, J. T. and Lai, M.-C., A method for computing nearly singular integrals, SIAM J. Numer. Anal., 38 (2001), 19021925.
[3]Beale, J. T., A grid-based boundary integral method for elliptic problems in three dimensions, SIAM J. Numer. Anal., 42 (2004), 599620.
[4]Bruno, O. P. and Kunyansky, L. A., A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests and applications, J. Comput. Phys., 169 (2001), 80110.
[5]Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.
[6]Cortez, R., The method of regularized Stokeslets, SIAM J. Sci. Comput., 23(4) (2001), 12041225.
[7]Cortez, R., Fauci, L. and Medovikov, A., The method of regularized Stokeslets in three dimensions: analysis, validation, and application to helical swimming, Phys. Fluids, 17 (2005), 114.
[8]Flores, H., Lobaton, E., Meéndez-Diez, S., Tlupova, S. and Cortez, R., A study of bacterial flagellar bundling, Bull. Math. Biol., 67 (2005), 137168.
[9]Greengard, L., Kropinski, M. C. and Mayo, A., Integral equation methods for Stokes flow and isotropic elasticity in the plane, J. Comput. Phys., 125 (1996), 403414.
[10]Helsing, J. and Ojala, R., On the evaluation of layer potentials close to their sources, J. Comput. Phys., 227 (2008), 28992921.
[11]Hsiao, G. C. and Wendland, W. L., Boundary Integral Equations, Springer-Verlag, Berlin, 2008.
[12]Johnston, P. R., Application of sigmoidal transformations to weakly singular and near-singular boundary element integrals, Int. J.Numer. Meth. Eng., 45(10) (1999), 13331348.
[13]Kress, R., Linear Integral Equations, Springer-Verlag, New York, 2nd edition, 1999.
[14]Layton, A. T. and Beale, J. T., A partially implicit hybrid method for computing interface motion in Stokes flow, Discrete Contin. Dyn. Syst. Ser. B, 27 (2012), 11391153.
[15]Lighthill, J., Helical distributions of Stokeslets, J. Eng. Math., 30 (1996), 3578.
[16]Nicholas, M. J., A higher order numerical method for 3-d doubly periodic electromagnetic scattering problems, Commun. Math. Sci., 6(3) (2008), 669694.
[17]Pozrikidis, C., Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, Cambridge, 1992.
[18]Pozrikidis, C., Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, Oxford, 1997.
[19]Pozrikidis, C., Interfacial dynamics for Stokes flow, J. Comput. Phys., 169 (2001), 250301.
[20]Sierou, A. and Brady, J. F., Accelerated Stokesian dynamics simulations, J. Fluid Mech., 448 (2001), 115146.
[21]Tlupova, S. and Cortez, R., Boundary integral solutions of coupled Stokes and Darcy flows, J. Comput. Phys., 228(1) (2009), 158179.
[22]Veerapaneni, S. K., Rahimian, A., Biros, G., and Zorin, D., A fast algorithm for simulating vesicle flows in three dimensions, J. Comput. Phys., 230(14) (2011), 56105634.
[23]Ying, L., Biros, G., and Zorin, D., A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains, J. Comput. Phys., 219 (2006), 247275.
[24]Ying, W. and Beale, J. T., A fast accurate boundary integral method for potentials on closely packed cells, submitted to Commun. Comput. Phys., 2012.
[25]Zinchenko, A. Z., Rother, M. A., and Davis, R. H., Cusping, capture, and breakup of interacting drops by a curvatureless boundary-integral algorithm, J. Fluid Mech., 391 (1999), 249292.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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