Hydrodynamics of suspensions of passive and active rigid particles: a rigid multiblob approach

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 ["An immersed boundary method for rigid

Brownian dynamics of confined suspensions of active microrollers

“…When deciding which method should be used one has to bear in mind that often, a simple GPU implementation can be faster than a method with a better theoretical scaling unless the number of particles is above about a hundred thousand for present-day hardware and FMM implementations [14]. heights.…”

“…A simple block diagonal preconditioner, which neglects the hydrodynamic interactions between distinct particles but does include the hydrodynamic interactions of each particle with the wall [14], gives only a very small improvement in the number of iterations. A preconditioner based on a incomplete Cholesky factorization of a truncated mobility matrix helps convergence greatly but also significantly increases the memory requirements and the complexity of the code, and also increases the cost per iteration.…”

“…However, they did not include thermal fluctuations nor considered many-body hydrodynamic interactions. In a previous work we consider many body simulations of arbitrary shaped passive or active colloids near a planar wall using a rigid multiblob method [14], however, we did not discuss how to include Brownian motion. Recently, Singh and Adhikari [11] have used a Galerkin boundary integral representation for active spheres [15] to study the crystallization of a two-dimensional rotating suspension of spheres near a boundary; similar crystal behavior has been observed in experiments with bacteria and synthetic active colloids [6,16], as well in simulations of unconfined rotating spheres [17,18].…”

“…However, the midpoint scheme of Fixman not only involves the square root of the mobility but also the square root of its inverse, the resistance matrix. Since there is no pairwise approximations to the resistance matrix even considering only far field hydrodynamic interactions [35], solving resistance problems is generally notably harder than solving mobility problems [14,36]. This makes Fixman's midpoint scheme expensive in practice [37], especially in unbounded systems where the mobility matrix is ill-conditioned.…”

“…In this work we only include hydrodynamic interactions in a far field approximation and do not use a preconditioner [40]. However, it is important to note that both the RFD and the Krylov method are general techniques that can effectively be used with other methods such as the boundary integral method of Adhikari et al [11,15] or our rigid multiblob method [14], with an appropriate preconditioner.…”