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Using theory and experiments of spheres moving near boundaries to optimize the method of images for regularized Stokeslets

Hoa Nguyen1,*, Amelia Gibbs1,†, Frank Healy2,‡, Orrin Shindell3,§, Ricardo Cortez4,∥, Kathleen M. Brown5,¶, Jonathan McCoy5,#, and Bruce Rodenborn5,**

  • *Contact author: hnguyen5@trinity.edu
  • †Contact author: tgibbs@trinity.edu
  • ‡Contact author: fhealy@trinity.edu
  • §Contact author: oshindel@trinity.edu
  • ∥Contact author: rcortez@tulane.edu
  • Contact author: kate.brown@centre.edu
  • #Contact author: jonathan.mccoy@centre.edu
  • **Contact author: bruce.rodenborn@centre.edu
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Phys. Rev. Fluids 10, 033101 – Published 12 March, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.033101

Abstract

The general system of images for regularized Stokeslets (GSIRS) developed by Cortez and Varela [J. Comput. Phys. 285, 41 (2015)] is used extensively to model Stokes flow phenomena such as microorganisms swimming near a boundary. Our collaborative team uses dynamically similar scaled macroscopic experiments to test theories for forces and torques on spheres moving near a boundary and uses these data and the method of regularized Stokeslets (MRS) created by Cortez et al. [Phys. Fluids 17, 031504 (2005)] to calibrate the GSIRS. We find excellent agreement between theory and experiments, which provides experimental validation of exact series solutions for spheres moving near an infinite plane boundary. We test two surface discretization methods commonly used in the literature: the 6-patch method and the spherical centroidal Voronoi tessellation (SCVT) method. Our data show that a discretization method, such as SCVT, that uniformly distributes points provides the most accurate results when the motional symmetry is broken by the presence of a boundary. We use theory and the MRS to find optimal values for the regularization parameter in free space for a given surface discretization and show that the optimal regularization parameter values can be fit with simple formulas when using the SCVT method. We also present a regularization function with higher-order accuracy when compared with the regularization function previously introduced by Cortez et al. [Phys. Fluids 17, 031504 (2005)]. The simulated force and torque values compare very well with experiments and theory for a wide range of boundary distances. However, we find that for a fixed discretization of the sphere, the simulations lose accuracy when the gap between the edge of the sphere and the wall is smaller than the average distance between discretization points in the SCVT method. We also show an alternative method to calibrate the GSIRS to simulate sphere motion arbitrarily close to the boundary. Our computational parameters and methods along with our matlab and python implementations of the series solution of Lee and Leal [J. Fluid Mech. 98, 193 (1980)], MRS, and GSIRS provide researchers with important resources to optimize the GSIRS and other numerical methods, so that they can efficiently and accurately simulate spheres moving near a boundary.

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Phys. Rev. Fluids 10, 033101– Published 12 March, 2025
  • Received 10 July 2024
  • Accepted 19 February 2025
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