Commun. Comput. Phys., 15 (2014), pp. 126-152. |
Reduction of the Regularization Error of the Method of Regularized Stokeslets for a Rigid Object Immersed in a Three-Dimensional Stokes Flow Hoang-Ngan Nguyen ^{1*}, Ricardo Cortez ^{1} 1 Center for Computational Science, Tulane University, 6823 St. Charles Ave, New Orleans, Louisiana, 70118, USA.Received 2 November 2012; Accepted (in revised version) 29 April 2013 Available online 26 July 2013 doi:10.4208/cicp.021112.290413a 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 \phi(r) = O(r^{-3-\alpha}) when r \to \infty for some constant \alpha > 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. AMS subject classifications: 76D07, 65N99Notice: Undefined variable: pac in /var/www/html/readabs.php on line 165 Key words: Stokes flow, regularized Stokeslet, boundary integral equation, nearly singular integral. *Corresponding author. Email: hnguyen4@tulane.edu (H.-N. Nguyen), rcortez@tulane.edu (R. Cortez) |