Vol. 11, No. 2, 2016

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
Hydrodynamics of suspensions of passive and active rigid particles: a rigid multiblob approach

Florencio Balboa Usabiaga, Bakytzhan Kallemov, Blaise Delmotte, Amneet Pal Singh Bhalla, Boyce E. Griffith and Aleksandar Donev

Vol. 11 (2016), No. 2, 217–296
Abstract

We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a standard Krylov solver converges in a modest number of iterations that is essentially independent of the number of particles. Key to the efficiency of the method is a technique for fast computation of the product of the blob-blob mobility matrix and a vector. For unbounded suspensions, we rely on existing analytical expressions for the Rotne–Prager–Yamakawa tensor combined with a fast multipole method (FMM) to obtain linear scaling in the number of particles. For suspensions sedimented against a single no-slip boundary, we use a direct summation on a graphical processing unit (GPU), which gives quadratic asymptotic scaling with the number of particles. For fully confined domains, such as periodic suspensions or suspensions confined in slit and square channels, we extend a recently developed rigid-body immersed boundary method by B. Kallemov, A. P. S. Bhalla, B. E. Griffith, and A. Donev (Commun. Appl. Math. Comput. Sci. 11 (2016), no. 1, 79–141) to suspensions of freely moving passive or active rigid particles at zero Reynolds number. We demonstrate that the iterative solver for the coupled fluid and rigid-body equations converges in a bounded number of iterations regardless of the system size. In our approach, each iteration only requires a few cycles of a geometric multigrid solver for the Poisson equation, and an application of the block-diagonal preconditioner, leading to linear scaling with the number of particles. We optimize a number of parameters in the iterative solvers and apply our method to a variety of benchmark problems to carefully assess the accuracy of the rigid multiblob approach as a function of the resolution. We also model the dynamics of colloidal particles studied in recent experiments, such as passive boomerangs in a slit channel, as well as a pair of non-Brownian active nanorods sedimented against a wall.

Keywords
Stokes flow, colloidal suspensions, Stokesian dynamics, immersed boundary method
Mathematical Subject Classification 2010
Primary: 76M25
Milestones
Received: 7 June 2016
Revised: 20 November 2016
Accepted: 6 December 2016
Published: 12 January 2017
Authors
Florencio Balboa Usabiaga
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012
United States
Bakytzhan Kallemov
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012
United States
Energy Geosciences Division
Lawrence Berkeley National Laboratory
Berkeley, CA 94720
United States
Blaise Delmotte
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012
United States
Amneet Pal Singh Bhalla
Department of Mathematics
University of North Carolina
Chapel Hill, NC 27599
United States
Boyce E. Griffith
Department of Mathematics
University of North Carolina
Chapel Hill, NC 27599
United States
Department of Biomedical Engineering
University of North Carolina
Chapel Hill, NC 27599
United States
Aleksandar Donev
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012
United States