Boundary integral methods are attractive for simulating Stokes flow with particles or droplets due to the reduction in dimensionality and natural handling of the geometry. In many problems walls are present, and it becomes necessary to evaluate singular or nearly singular layer potentials over the wall. In this paper we show how this can be done using quadrature by expansion (QBX), a relatively new method based on local expansions of the layer potential. We present results for the Laplace single layer potential and the Stokes double layer potential. QBX can be used to evaluate the potentials to high accuracy arbitrarily close to the wall and on the wall. We also discuss how some quantities can be precomputed and how geometric symmetries can be used to reduce precomputation and storage.

Boundary integral methods are highly suited for problems with complicated geometries, but require special quadrature methods to accurately compute the singular and nearly singular layer potentials that appear in them. This article presents a boundary integral method that can be used to study the motion of rigid particles in three-dimensional periodic Stokes flow with confining walls. A centerpiece of our method is the highly accurate special quadrature method, which is based on a combination of upsampled quadrature and quadrature by expansion, accelerated using a precomputation scheme. The method is demonstrated for rodlike and spheroidal particles, with the confining geometry given by a pipe or a pair of flat walls. A parameter selection strategy for the special quadrature method is presented and tested. Periodic interactions are computed using the spectral Ewald fast summation method, which allows our method to run in O(n log n) time for n grid points in the primary cell, assuming the number of geometrical objects grows while the grid point concentration is kept fixed.

Understanding particle drift in suspension flows is of the highest importance in numerous engineering applications where particles need to be separated and filtered out from the suspending fluid. Commonly known drift mechanisms such as the Magnus force, Saffman force and Segre-Silberberg effect all arise only due to inertia of the fluid, with similar effects on all non-spherical particle shapes. In this work, we present a new shape-selective lateral drift mechanism, arising from particle inertia rather than fluid inertia, for ellipsoidal particles in a parabolic velocity profile. We show that the new drift is caused by an intermittent tumbling rotational motion in the local shear flow together with translational inertia of the particle, while rotational inertia is negligible. We find that the drift is maximal when particle inertial forces are of approximately the same order of magnitude as viscous forces, and that both extremely light and extremely heavy particles have negligible drift. Furthermore, since tumbling motion is not a stable rotational state for inertial oblate spheroids (nor for spheres), this new drift only applies to prolate spheroids or tri-axial ellipsoids. Finally, the drift is compared with the effect of gravity acting in the directions parallel and normal to the flow. The new drift mechanism is stronger than gravitational effects as long as gravity is less than a critical value. The critical gravity is highest (i.e. the new drift mechanism dominates over gravitationally induced drift mechanisms) when gravity acts parallel to the flow and the particles are small.

A unified treatment for the fast and spectrally accurate evaluation of electrostatic potentials with periodic boundary conditions in any or none of the three spatial dimensions is presented. Ewald decomposition is used to split the problem into real-space and Fourier-space parts, and the Fast Fourier Transform (FFT)-based Spectral Ewald (SE) method is used to accelerate computation of the latter, yielding the total runtime O(N log N) for N sources. A key component is a new FFT-based solution technique for the free-space Poisson problem. The computational cost is further reduced by a new adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling factors. The SE method is most efficient in the triply periodic case where the cost of computing FFTs is the lowest, whereas the rest of the algorithm is essentially independent of periodicity. We show that removing periodic boundary conditions from one or two directions out of three will only moderately increase the total runtime, and in the free-space case, the runtime is around four times that of the triply periodic case. The Gaussian window function previously used in the SE method is compared with a new piecewise polynomial approximation of the Kaiser–Bessel window, which further reduces the runtime. We present error estimates and a parameter selection scheme for all parameters of the method, including a new estimate for the shape parameter of the Kaiser–Bessel window. Finally, we consider methods for force computation and compare the runtime of the SE method with that of the fast multipole method.

A fast and spectrally accurate Ewald summation method for the evaluation of stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is presented. This work extends the previously developed Spectral Ewald method for Stokes flow to periodic boundary conditions in any number (three, two, one, or none) of the spatial directions, in a unified framework. The periodic potential is split into a short-range and a long-range part, where the latter is treated in Fourier space using the fast Fourier transform. A crucial component of the method is the modified kernels used to treat singular integration. We derive new modified kernels, and new improved truncation error estimates for the stokeslet and stresslet. An automated procedure for selecting parameters based on a given error tolerance is designed and tested. Analytical formulas for validation in the doubly and singly periodic cases are presented. We show that the computational time of the method scales like O(N log N) for N sources and targets, and investigate how the time depends on the error tolerance and window function, i.e. the function used to smoothly spread irregular point data to a uniform grid. The method is fastest in the fully periodic case, while the run time in the free-space case is around three times as large. Furthermore, the highest efficiency is reached when applying the method to a uniform source distribution in a primary cell with low aspect ratio. The work presented in this paper enables efficient and accurate simulations of three-dimensional Stokes flow with arbitrary periodicity using e.g. boundary integral and potential methods.

In boundary integral methods, special quadrature methods are needed to approximate layer potentials, integrals where the integrand is singular or sharply peaked for evaluation points on or close to the boundaries. In this paper, we study a method based on extrapolation or interpolation along a line, sometimes called the Hedgehog method. In this method, the layer potential is evaluated with a regular quadrature method for evaluation points along a line, and an approximant is constructed and evaluated in an area of interest where the original layer potential is difficult to evaluate due to it being singular or sharply peaked.

We analyze the errors in the Hedgehog method with polynomial approximation, and use this to construct optimal distributions of sample points. Furthermore, rational approximation is introduced in the Hedgehog method, and compared with polynomial approximation. It is found that rational approximation can typically achieve a lower error than polynomial approximation, and does not increase the computational cost of the method significantly. Strategies for avoiding and dealing with spurious poles in rational approximation are discussed.

We compare extrapolation (no sample point on the boundary) with interpolation (sample point present) in the Hedgehog method, and find that the error in our example is lower in the interpolation case by around one order of magnitude, compared to the extrapolation case.

We consider a specific test case, consisting of two rigid rodlike particles in Stokes flow. Parameter selection and error estimation for the Hedgehog method is discussed for this test case. The accuracy and computational cost of the Hedgehog method is examined, and compared with another special quadrature method, namely quadrature by expansion (QBX). We find that the Hedgehog method should be able to compete with QBX in this context, but further investigation is needed for strict tolerances.