Stokes flow past three spheres

In this paper we present a numerical method to calculate the dynamics of three spheres in a quiescent viscous fluid. The method is based on Lamb’s solution to Stokes flow and the Method of Reflections, and is arbitrarily accurate given sufficient computer memory and time. It is more accurate than multipole methods, but much less efficient. Although it is too numerically intensive to be suitable for more than three spheres, it can easily handle spheres of different sizes. We find no convergence difficulties provided we study mobility problems, rather than resistance problems. After validating against the existing literature, we make a direct comparison with Stokesian Dynamics (SD), and find that the largest errors in SD occur at a sphere separation around 0.1 radius. Finally, we present results for an example system having different-sized spheres

Many-body effects and matrix inversion in low-Reynolds-number hydrodynamics

It is shown that the method of reflections in resistance form (with truncated multipoles) is one of many possible iterative methods to obtain the inverse of the mobility matrix (with the same truncation) in low-Reynolds-number hydrodynamics. Although the method of reflections in the mobility form is guaranteed to converge, it is found that in the resistance form the method may fail to converge. This breakdown is overcome by conjugate-gradient-type iterative methods, and the implications of the iterative method for low-Reynolds-number hydrodynamics are discussed

Few-body hydrodynamic interactions probed by optical trap pulling experiment

We study the hydrodynamic coupling of neighboring micro-beads placed in a dual optical trap setup allowing us to precisely control the degree of coupling and directly measure time-dependent trajectories of the entrained beads. Average experimental trajectories of a probe bead entrained by the motion of a neighboring scan bead are compared with theoretical computation, illustrating the role of viscous coupling and setting timescales for probe bead relaxation. The findings provide direct experimental corroborations of hydrodynamic coupling at larger, micron spatial scales and millisecond timescales, of relevance to hydrodynamic-assisted colloidal assembly as well as improving the resolution of optical tweezers. We repeat the experiments for three bead setups

Stokesian Dynamics in Python

This is a Python 3 implementation of the Stokesian Dynamics method for polydisperse spheres suspended in a 3D Newtonian background fluid. The fluid-filled domain may be unbounded or a periodic cube. Physical setups and custom forces on the particles are easily implemented using Python; meanwhile, the computational speed is handled by Numba.This software is aimed at researchers in low Reynolds number fluid dynamics who are looking for an easy-to-use yet flexible implementation of this popular method. The startup cost of writing a Stokesian Dynamics code is high, given the need for pre-computed resistance scalars.Several minus sign errors in the literature also need resolving first. These have been resolved and validated before being implemented in this code. The hope is that many months of future researchers’ time (often PhD students’ time) will be saved by no longer reinventing the wheel

Generating, from scratch, the near-field asymptotic forms of scalar resistance functions for two unequal rigid spheres in low Reynolds number flow

The motion of rigid spherical particles suspended in a low Reynolds number fluid can be related to the forces, torques and stresslets acting upon them by 22 scalar resistance functions, commonly notated $X^A_{11}$, $X^A_{12}$, $Y^A_{11}$, etc. Near-field asymptotic forms of these resistance functions were derived in Jeffrey and Onishi (J. Fluid Mech., 1984) and Jeffrey (Phys. Fluids A, 1992); these forms are now used in several numerical methods for suspension mechanics. However, the first of these important papers contains a number of small errors which make it difficult for the reader to correctly evaluate the functions for parameters not explicitly tabulated. This short article comprehensively corrects these errors, and adds formulae that were originally omitted from both papers, so that the reader can verify and implement the equations independently. The corrected expressions, rationalised and using contemporary nondimensionalisation, are shown to match mid-field values of these scalars which are calculated through an alternative method. A Python script to generate and evaluate these functions is provided

A fast integral equation method for solid particles in viscous flow using quadrature by expansion

Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models

Fast Multipole Method for the Biharmonic Equation

The evaluation of sums (matrix-vector products) of the solutions of the three-dimensional biharmonic equation can be accelerated using the fast multipole method, while memory requirements can also be significantly reduced. We develop a complete translation theory for these equations. It is shown that translations of elementary solutions of the biharmonic equation can be achieved by considering the translation of a pair of elementary solutions of the Laplace equations. Compared to previous methods that required the translation of five Laplace elementary solutions for the biharmonic Green's function, and much larger numbers for higher order multipoles, our method is significantly more efficient. The theory is implemented and numerical tests presented that demonstrate the performance of the method for varying problem sizes and accuracy requirements. In our implementation the FMM\ is faster than direct solution for a matrix size of $550$ for an accuracy of $10^{-3},$ 950 for an accuracy of $10^{-6} and$N=3550$for an accuracy of$10^{-9}$

Stokes flow past three sphere

Abstract In this paper we present a numerical method to calculate the dynamics of three spheres in a quiescent viscous fluid. The method is based on Lamb's solution to Stokes flow and the method of reflections, and is arbitrarily accurate given sufficient computer memory and time. It is more accurate than multipole methods, but much less efficient. Although it is too numerically intensive to be suitable for more than three spheres, it can easily handle spheres of different sizes. We find no convergence difficulties provided we study mobility problems, rather than resistance problems. After validating against the existing literature, we make a direct comparison with Stokesian Dynamics (SD), and find that the largest errors in SD occur at a sphere separation around 0.1 radius. Finally, we present results for an example system having different-sized spheres

Spectral Ewald Acceleration of Stokesian Dynamics for polydisperse suspensions

In this work we develop the Spectral Ewald Accelerated Stokesian Dynamics (SEASD), a novel computational method for dynamic simulations of polydisperse colloidal suspensions with full hydrodynamic interactions. SEASD is based on the framework of Stokesian Dynamics (SD) with extension to compressible solvents, and uses the Spectral Ewald (SE) method [Lindbo & Tornberg, J. Comput. Phys. 229 (2010) 8994] for the wave-space mobility computation. To meet the performance requirement of dynamic simulations, we use Graphic Processing Units (GPU) to evaluate the suspension mobility, and achieve an order of magnitude speedup compared to a CPU implementation. For further speedup, we develop a novel far-field block-diagonal preconditioner to reduce the far-field evaluations in the iterative solver, and SEASD-nf, a polydisperse extension of the mean-field Brownian approximation of Banchio & Brady [J. Chem. Phys. 118 (2003) 10323]. We extensively discuss implementation and parameter selection strategies in SEASD, and demonstrate the spectral accuracy in the mobility evaluation and the overall $\mathcal{O}(N\log N)$ computation scaling. We present three computational examples to further validate SEASD and SEASD-nf in monodisperse and bidisperse suspensions: the short-time transport properties, the equilibrium osmotic pressure and viscoelastic moduli, and the steady shear Brownian rheology. Our validation results show that the agreement between SEASD and SEASD-nf is satisfactory over a wide range of parameters, and also provide significant insight into the dynamics of polydisperse colloidal suspensions.Comment: 39 pages, 21 figure