A unified treatment for fast and spectrally accurate evaluation of electrostatic potentials subject to periodic boundary conditions in any or none of the three space dimensions is presented. Ewald decomposition is used to split the problem into a real space and a Fourier space part, and the FFT based Spectral Ewald (SE) method is used to accelerate the computation of the latter. A key component in the unified treatment is an FFT based solution technique for the free-space Poisson problem in three, two or one dimensions, depending on the number of non-periodic directions. The cost of calculations is furthermore reduced by employing an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling rates. The SE method will always be most efficient for the triply periodic case as the cost for computing FFTs will be the smallest, whereas the computational cost for the rest of the algorithm is essentially independent of the periodicity. We show that the cost of removing periodic boundary conditions from one or two directions out of three will only marginally increase the total run time. Our comparisons also show that the computational cost of the SE method for the free-space case is typically about four times more expensive as compared to the triply periodic case.The Gaussian window function previously used in the SE method, is here compared to an approximation of the Kaiser-Bessel window function, recently introduced in [2]. With a carefully tuned shape parameter that is selected based on an error estimate for this new window function, runtimes for the SE method can be further reduced.
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. K E Y W O R D Sboundary integral equations, fast Ewald summation, quadrature by expansion, rigid particle suspensions, Stokes flow, streamline computation INTRODUCTIONMicrohydrodynamics is the study of fluid flow at low Reynolds numbers, also known as Stokes flow or creeping flow. Applications are found in biology, for example in the swimming of microorganisms 1 and in blood flow, 2 as well as in the field of microfluidics, which concerns the design and construction of miniaturized fluid devices. 3 Suspensions of rigid particles in Stokes flow are important both in various applications and in fundamental fluid mechanics. [4][5][6][7] In this article, we describe a boundary integral method that can be used to study the motion of rigid particles of different shapes in Stokes flow. The particle suspension may also be confined in a container geometry, such as a pipe or a pair of flat walls. The flow in the fluid domain (i.e., within the container but outside the particles) is governed by the Stokes equations, which for an incompressible Newtonian fluid take the formThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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 Segré–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.
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