Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-15T09:20:16.550Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduction of the Regularization Error of the Method of Regularized Stokeslets for a Rigid Object Immersed in a Three-Dimensional Stokes Flow

Published online by Cambridge University Press:  03 June 2015

Hoang-Ngan Nguyen  and
Ricardo Cortez
Hoang-Ngan Nguyen*
Affiliation:
Center for Computational Science, Tulane University, 6823 St. Charles Ave, New Orleans, Louisiana, 70118, USA
Ricardo Cortez
Affiliation:
Center for Computational Science, Tulane University, 6823 St. Charles Ave, New Orleans, Louisiana, 70118, USA
*
Corresponding author.Email:[email protected]
Get access

Abstract

We focus on the problem of evaluating the velocity field outside a solid object moving in an incompressible Stokes flow using the boundary integral formulation. For points near the boundary, the integral is nearly singular, and accurate computation of the velocity is not routine. One way to overcome this problem is to regularize the integral kernel. The method of regularized Stokeslet (MRS) is a systematic way to regularize the kernel in this situation. For a specific blob function which is widely used, the error of the MRS is only of first order with respect to the blob parameter. We prove that this is the case for radial blob functions with decay property ϕ(r)=O(r−3−α) when r→∞ for some constant α>1. We then find a class of blob functions for which the leading local error term can be removed to get second and third order errors with respect to blob parameter. Since the addition of these terms might give a flow field that is not divergence free, we introduce a modification of these terms to make the divergence of the corrected flow field close to zero while keeping the desired accuracy. Furthermore, these dominant terms are explicitly expressed in terms of blob function and so the computation time is negligible.

Keywords

76D0765N99Stokes flowregularized Stokesletboundary integral equationnearly singular integral
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 1 , January 2014 , pp. 126 - 152
Copyright
Copyright © Global Science Press Limited 2014

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] Ainley, J., Durkin, S., Embid, R., Boindala, P., and Cortez, R. The method of images for regularized stokeslets. Journal of Computational Physics, 227(9):4600–4616, 2008.Google Scholar
[2] Beale, J. A grid-based boundary integral method for elliptic problems in three dimensions. SIAM Journal on Numerical Analysis, 42(2):599–620, 2004.Google Scholar
[3] Beale, J. T. and Lai, M.-C. A method for computing nearly singular integrals. SIAM Journal on Numerical Analysis, 38(6):1902–1925, 2001.Google Scholar
[4] Chwang, A. T. and Wu, T. Y.-T. Hydromechanics of low-reynolds-number flow. Part 2. Singularity method for stokes flows. Journal of Fluid Mechanics, 67(04):787–815, 1975.Google Scholar
[5] Cortez, R. The method of regularized stokeslets. SIAM Journal on Scientific Computing, 23(4):1204–1225, 2001.Google Scholar
[6] Cortez, R., Fauci, L., and Medovikov, A. The method of regularized stokeslets in three dimensions: Analysis, validation, and application to helical swimming. Physics of Fluids, 17(3):031504, 2005.CrossRefGoogle Scholar
[7] Gyrya, V., Lipnikov, K., Aranson, I., and Berlyand, L. Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate concentrations. Journal of Mathematical Biology, 62:707–740, 2011. 10.1007/s00285-010-0351-y.Google Scholar
[8] Haines, B. M., Aranson, I. S., Berlyand, L., and Karpeev, D. A. Effective viscosity of bacterial suspensions: a three-dimensional PDE model with stochastic torque. Communications on Pure and Applied Analysis, 11:19–46, 2012.Google Scholar
[9] Hernández-Ortiz, J. P., Pablo, J. J. de, and Graham, M. D. Fast computation of many-particle hydrodynamic and electrostatic interactions in a confined geometry. Phys. Rev. Lett., 98:140602, 2007.Google Scholar
[10] Kumar, A. and Graham, M. D. Accelerated boundary integral method for multiphase flow in non-periodic geometries. Journal of Computational Physics, 231(20):6682–6713, 2012.CrossRefGoogle Scholar
[11] Lauga, E. and Powers, T. R. The hydrodynamics of swimming microorganisms. Reports on Progress in Physics, 72(9):096601, 2009.CrossRefGoogle Scholar
[12] O’Malley, S. and Bees, M. The orientation of swimming biflagellates in shear flows. Bulletin of Mathematical Biology, 74:232–255, 2012. 10.1007/s11538-011-9673-1.CrossRefGoogle ScholarPubMed
[13] Pozrikidis, C. Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press, 1992.Google Scholar
[14] Smith, D. J. A boundary element regularized stokeslet method applied to cilia- and flagella-driven flow. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 465(2112):3605–3626, 2009.Google Scholar