Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T17:58:56.905Z Has data issue: false hasContentIssue false
  • English
  • Français

Nearly Singular Integrals in 3D Stokes Flow

Published online by Cambridge University Press:  03 June 2015

Svetlana Tlupova  and
J. Thomas Beale
Svetlana Tlupova*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA
J. Thomas Beale*
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708-0320, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

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.

Keywords

35J4765N8076D07Stokes equationsboundary integral equation methodnearly singular integrals
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 5 , November 2013 , pp. 1207 - 1227
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aris, R., Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Dover, New York, 1962.Google Scholar
[2]Beale, J. T. and Lai, M.-C., A method for computing nearly singular integrals, SIAM J. Numer. Anal., 38 (2001), 1902–1925.Google Scholar
[3]Beale, J. T., A grid-based boundary integral method for elliptic problems in three dimensions, SIAM J. Numer. Anal., 42 (2004), 599–620.CrossRefGoogle Scholar
[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), 80–110.CrossRefGoogle Scholar
[5]Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.Google Scholar
[6]Cortez, R., The method of regularized Stokeslets, SIAM J. Sci. Comput., 23(4) (2001), 1204–1225.Google Scholar
[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), 1–14.Google Scholar
[8]Flores, H., Lobaton, E., Meéndez-Diez, S., Tlupova, S. and Cortez, R., A study of bacterial flagellar bundling, Bull. Math. Biol., 67 (2005), 137–168.Google Scholar
[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), 403–414.Google Scholar
[10]Helsing, J. and Ojala, R., On the evaluation of layer potentials close to their sources, J. Comput. Phys., 227 (2008), 2899–2921.CrossRefGoogle Scholar
[11]Hsiao, G. C. and Wendland, W. L., Boundary Integral Equations, Springer-Verlag, Berlin, 2008.Google Scholar
[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), 1333–1348.3.0.CO;2-Q>CrossRefGoogle Scholar
[13]Kress, R., Linear Integral Equations, Springer-Verlag, New York, 2nd edition, 1999.CrossRefGoogle Scholar
[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), 1139–1153.Google Scholar
[15]Lighthill, J., Helical distributions of Stokeslets, J. Eng. Math., 30 (1996), 35–78.Google Scholar
[16]Nicholas, M. J., A higher order numerical method for 3-d doubly periodic electromagnetic scattering problems, Commun. Math. Sci., 6(3) (2008), 669–694.Google Scholar
[17]Pozrikidis, C., Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, Cambridge, 1992.Google Scholar
[18]Pozrikidis, C., Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, Oxford, 1997.Google Scholar
[19]Pozrikidis, C., Interfacial dynamics for Stokes flow, J. Comput. Phys., 169 (2001), 250–301.Google Scholar
[20]Sierou, A. and Brady, J. F., Accelerated Stokesian dynamics simulations, J. Fluid Mech., 448 (2001), 115–146.Google Scholar
[21]Tlupova, S. and Cortez, R., Boundary integral solutions of coupled Stokes and Darcy flows, J. Comput. Phys., 228(1) (2009), 158–179.Google Scholar
[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), 5610–5634.CrossRefGoogle Scholar
[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), 247–275.Google Scholar
[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.Google Scholar
[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), 249–292.CrossRefGoogle Scholar