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Journal articles on the topic 'Stokeslet'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Aboelkassem, Yasser. "Chaotic mixing by oscillating a Stokeslet in a circular Hele-Shaw microfluidic device." Mathematics of Quantum Technologies 5, no. 1 (January 1, 2016): 1–8. http://dx.doi.org/10.1515/nsmmt-2016-0001.

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AbstractChaotic mixing by oscillating a Stokeslet in a circular Hele-Shaw microffluidic device is presented in this article. Mathematical modeling for the induced flow motions by moving a Stokeslet along the x-axis is derived using Fourier expansion method. The solution is formulated in terms of the velocity stream function. The model is then used to explore different stirring dynamics as function of the Stokeslet parameters. For instance, the effects of using various oscillation amplitudes and force strengths are investigated. Mixing patterns using Poincaré maps are obtained numerically and have been used to characterize the mixing efficiency. Results have shown that, for a given Stokeslet’s strength, efficient mixing can be obtained when small oscillation amplitudes are used. The present mixing platform is expected to be useful for many of biomicrofluidic applications.

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2

Cortez, Ricardo. "Regularized Stokeslet segments." Journal of Computational Physics 375 (December 2018): 783–96. http://dx.doi.org/10.1016/j.jcp.2018.08.055.

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3

Chan, A. T., and A. T. Chwang. "The unsteady stokeslet and oseenlet." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 214, no. 1 (January 1, 2000): 175–79. http://dx.doi.org/10.1243/0954406001522895.

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The unsteady low Reynolds number flow of an incompressible viscous fluid past a singular forcelet is investigated analytically. New fundamental three-dimensional solutions for a concentrated impulsive force are derived for the Stokes and the Oseen equations. These elementary solutions can be used as fundamental Green's functions to obtain solutions for flows over singularities with any time-dependent nature. The fundamental singularities are employed to construct some well-known solutions to demonstrate their validity and usefulness in solving unsteady problems governed by the Stokes and the Oseen equations. A new solution is presented for an unsteady Oseen flow with a constant acceleration.

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4

Gallagher, M. T., and D. J. Smith. "The art of coarse Stokes: Richardson extrapolation improves the accuracy and efficiency of the method of regularized stokeslets." Royal Society Open Science 8, no. 5 (May 2021): 210108. http://dx.doi.org/10.1098/rsos.210108.

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The method of regularized stokeslets is widely used in microscale biological fluid dynamics due to its ease of implementation, natural treatment of complex moving geometries, and removal of singular functions to integrate. The standard implementation of the method is subject to high computational cost due to the coupling of the linear system size to the numerical resolution required to resolve the rapidly varying regularized stokeslet kernel. Here, we show how Richardson extrapolation with coarse values of the regularization parameter is ideally suited to reduce the quadrature error, hence dramatically reducing the storage and solution costs without loss of accuracy. Numerical experiments on the resistance and mobility problems in Stokes flow support the analysis, confirming several orders of magnitude improvement in accuracy and/or efficiency.

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5

Popov, I. Yu. "On operator treatment of a Stokeslet." Siberian Mathematical Journal 35, no. 5 (September 1994): 1022–26. http://dx.doi.org/10.1007/bf02104580.

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6

Walker, B. J., K. Ishimoto, H. Gadêlha, and E. A. Gaffney. "Filament mechanics in a half-space via regularised Stokeslet segments." Journal of Fluid Mechanics 879 (October 1, 2019): 808–33. http://dx.doi.org/10.1017/jfm.2019.723.

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We present a generalisation of efficient numerical frameworks for modelling fluid–filament interactions via the discretisation of a recently developed, non-local integral equation formulation to incorporate regularised Stokeslets with half-space boundary conditions, as motivated by the importance of confining geometries in many applications. We proceed to utilise this framework to examine the drag on slender inextensible filaments moving near a boundary, firstly with a relatively simple example, evaluating the accuracy of resistive force theories near boundaries using regularised Stokeslet segments. This highlights that resistive force theories do not accurately quantify filament dynamics in a range of circumstances, even with analytical corrections for the boundary. However, there is the notable and important exception of movement in a plane parallel to the boundary, where accuracy is maintained. In particular, this justifies the judicious use of resistive force theories in examining the mechanics of filaments and monoflagellate microswimmers with planar flagellar patterns moving parallel to boundaries. We proceed to apply the numerical framework developed here to consider how filament elastohydrodynamics can impact drag near a boundary, analysing in detail the complex responses of a passive cantilevered filament to an oscillatory flow. In particular, we document the emergence of an asymmetric periodic beating in passive filaments in particular parameter regimes, which are remarkably similar to the power and reverse strokes exhibited by motile$9+2$cilia. Furthermore, these changes in the morphology of the filament beating, arising from the fluid–structure interactions, also induce a significant increase in the hydrodynamic drag of the filament.

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7

Blake, J. R., and S. R. Otto. "Ciliary propulsion, chaotic filtration and a ?blinking? stokeslet." Journal of Engineering Mathematics 30, no. 1-2 (March 1996): 151–68. http://dx.doi.org/10.1007/bf00118828.

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8

Blake, J. R., S. R. Otto, and D. A. Blake. "Filter Feeding, Chaotic Filtration, and a Blinking Stokeslet." Theoretical and Computational Fluid Dynamics 10, no. 1-4 (January 1, 1998): 23–36. http://dx.doi.org/10.1007/s001620050049.

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9

Falade, A., and H. Brenner. "Stokes wall effects for particles moving near cylindrical boundaries." Journal of Fluid Mechanics 154 (May 1985): 145–62. http://dx.doi.org/10.1017/s002211208500146x.

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An asymptotic scheme is derived for calculating values of the ‘reflected’ Stokeslet-field velocity dyadic V and its gradient ${\boldmath \nabla V}$ back at the Stokeslet location for situations in which this singular point lies in close proximity to the wall of an infinitely long circular cylinder. The asymptotic formulas furnished by this scheme permit calculations of first- and second-order wall effects in the non-dimensional parameter κ = a/R0 (a ≡ characteristic particle radius, R0 ≡ cylinder radius) upon the Stokes resistance of a particle of arbitrary shape, location and orientation when translating and/or rotating near the wall of an otherwise quiescent fluid-filled or fluid-surrounded circular cylinder. This reflection-type calculation is applicable for circumstances in which the inequalities κ [Lt ] 1 and 1 − β [Lt ] 1 are each separately satisfied, while simultaneously K/(1 − β) [Lt ] 1. (Here β = b/R0 is the fractionally eccentric Stokeslet location, or equivalently the centre of reaction of the particle, with b its distance from the tube axis.) The main result of this paper is the development of the pair of asymptotic wall-correction formulas \[ W_{jk}\sim {}_0C_{jk}(1-\beta)^{-1}+{}_1C_{jk}+{}_2C_{jk}(1-\beta)+O(1-\beta)^2 \] and \[ W_{jk,\,l}\sim {}_0D_{jkl}(1-\beta)^{-2}+{}_1D_{jkl}(1-\beta)^{-1}+{}_2D_{jkl}+O(1-\beta) \] to particle resistance, with W ≡ Wjk and ${\boldmath \nabla W}\equiv W_{kl,j}$ respectively the non-dimensional normalized wall-effect dyadic and its gradient at the Stokeslet location β. The numerical, rational fraction, β-independent, nCjk and nDjkl coefficients (n = 0, 1, 2) appearing above are evaluated by solving a recursive sequence of Stokes-flow boundary-value problems in the semi-infinite fluid domain bounded by a plane wall. These simple asymptotic formulas are shown to agree excellently in the range near β = 1 with existing values derived from the exact solution of the original circular-cylinder boundary-value problem, involving tedious infinite-series summations of complicated Bessel-function integrands extended over infinite integration domains. Generalizations of the scheme to particle motion in the space external to a circular cylinder is briefly sketched, as too is the case of cylinders of non-circular cross-section.

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10

Usha, R., and S. D. Nigam. "Flow in a spherical cavity due to a stokeslet." Fluid Dynamics Research 11, no. 1-2 (January 1993): 75–78. http://dx.doi.org/10.1016/0169-5983(93)90006-v.

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11

Blinova, Irina V., Ksenia N. Kyz’yurova, and Igor Yu Popov. "Stokes flow driven by a Stokeslet in a cone." Acta Mechanica 225, no. 11 (March 22, 2014): 3115–21. http://dx.doi.org/10.1007/s00707-014-1117-1.

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12

Gallagher, Meurig T., and David J. Smith. "Passively parallel regularized stokeslets." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2179 (August 3, 2020): 20190528. http://dx.doi.org/10.1098/rsta.2019.0528.

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Stokes flow, discussed by G.G. Stokes in 1851, describes many microscopic biological flow phenomena, including cilia-driven transport and flagellar motility; the need to quantify and understand these flows has motivated decades of mathematical and computational research. Regularized stokeslet methods, which have been used and refined over the past 20 years, offer significant advantages in simplicity of implementation, with a recent modification based on nearest-neighbour interpolation providing significant improvements in efficiency and accuracy. Moreover this method can be implemented with the majority of the computation taking place through built-in linear algebra, entailing that state-of-the-art hardware and software developments in the latter, in particular multicore and GPU computing, can be exploited through minimal modifications (‘passive parallelism’) to existing Matlab computer code. Hence, and with widely available GPU hardware, significant improvements in the efficiency of the regularized stokeslet method can be obtained. The approach is demonstrated through computational experiments on three model biological flows: undulatory propulsion of multiple Caenorhabditis elegans , simulation of progression and transport by multiple sperm in a geometrically confined region, and left–right symmetry breaking particle transport in the ventral node of the mouse embryo. In general an order-of-magnitude improvement in efficiency is observed. This development further widens the complexity of biological flow systems that are accessible without the need for extensive code development or specialist facilities. This article is part of the theme issue ‘Stokes at 200 (part 2)’.

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13

Spagnolie, Saverio E., and Eric Lauga. "Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations." Journal of Fluid Mechanics 700 (April 16, 2012): 105–47. http://dx.doi.org/10.1017/jfm.2012.101.

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AbstractThe swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. A general axisymmetric swimmer is described as a linear combination of fundamental solutions to the Stokes equations: a Stokeslet dipole, a source dipole, a Stokeslet quadrupole, and a rotlet dipole. The effects of nearby walls or stress-free surfaces on swimming trajectories are described through the contribution of each singularity, and we address the question of how accurately this multipole approach captures the wall effects observed in full numerical solutions of the Stokes equations. The reduced model is used to provide simple but accurate predictions of the wall-induced attraction and pitching dynamics for model Janus particles, ciliated organisms, and bacteria-like polar swimmers. Transitions in attraction and pitching behaviour as functions of body geometry and propulsive mechanism are described. The reduced model may help to explain a number of recent experimental results.

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14

Ohm, Laurel. "Remarks on Regularized Stokeslets in Slender Body Theory." Fluids 6, no. 8 (August 14, 2021): 283. http://dx.doi.org/10.3390/fluids6080283.

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We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius ϵ>0. Denoting the regularization parameter by δ, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive L∞ bounds for the difference between regularized SBT and its classical counterpart in terms of δ, ϵ, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to log(δ/ϵ); in particular, δ=ϵ is necessary to avoid an O(1) discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to log(1+δ2/ϵ2), and any choice of δ∝ϵ will result in an O(1) discrepancy as ϵ→0. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed slender body PDE which classical SBT is known to approximate. Numerics verify this O(1) discrepancy but also indicate that the difference may have little impact in practice.

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15

Lu, D. Q. "Generation of free-surface gravity waves by an unsteady Stokeslet." Archive of Applied Mechanics 79, no. 4 (April 29, 2008): 311–22. http://dx.doi.org/10.1007/s00419-008-0233-6.

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16

Maul, Christine, and Sangtae Kim. "Image systems for a Stokeslet inside a rigid spherical container." Physics of Fluids 6, no. 6 (June 1994): 2221–23. http://dx.doi.org/10.1063/1.868223.

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17

Jánosi, Imre, Tamás Tél, Dietrich Wolf, and Jason Gallas. "Chaotic particle dynamics in viscous flows: The three-particle Stokeslet problem." Physical Review E 56, no. 3 (September 1997): 2858–68. http://dx.doi.org/10.1103/physreve.56.2858.

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18

Thygesen, Uffe Høgsbro, and Thomas Kiørboe. "Diffusive transport in Stokeslet flow and its application to plankton ecology." Journal of Mathematical Biology 53, no. 1 (March 6, 2006): 1–14. http://dx.doi.org/10.1007/s00285-006-0374-6.

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19

Smith, David J. "A nearest-neighbour discretisation of the regularized stokeslet boundary integral equation." Journal of Computational Physics 358 (April 2018): 88–102. http://dx.doi.org/10.1016/j.jcp.2017.12.008.

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20

Yu, H. Y. "Fundamental Singularities in a Two-Fluid Stokes Flow with a Plane Interface." Journal of Mechanics 19, no. 1 (March 2003): 263–70. http://dx.doi.org/10.1017/s1727719100004305.

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ABSTRACTA modified image method is presented for obtaining the solutions of the fundamental singularities in the neighborhood of a plane interface between two semi-infinite, immiscible, and incompressible viscous fluids. The fundamental singularities considered are the stokeslet, rotlet, stresslet, stokes-doublet, source, and source-doublet. The Galerkin vector function representation introduced reduces the complexity of the expressions for the solutions. Moreover, the physical meaning of each solution is clearly identified by these new expressions.

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21

Daddi-Moussa-Ider, Abdallah, Badr Kaoui, and Hartmut Löwen. "Axisymmetric Flow due to a Stokeslet Near a Finite-Sized Elastic Membrane." Journal of the Physical Society of Japan 88, no. 5 (May 15, 2019): 054401. http://dx.doi.org/10.7566/jpsj.88.054401.

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22

Mužík, Juraj. "The use of dual reciprocity method for 2D laminar viscous flow." MATEC Web of Conferences 310 (2020): 00044. http://dx.doi.org/10.1051/matecconf/202031000044.

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The paper presents the use of the dual reciprocity multidomain singular boundary method (SBMDR) for the solution of the laminar viscous flow problem described by Navier-Stokes equations. A homogeneous part of the solution is solved using a singular boundary method with the 2D Stokes fundamental solution - Stokeslet. The dual reciprocity approach has been chosen because it is ideal for the treatment of the nonhomogeneous and nonlinear terms of Navier-Stokes equations. The presented SBMDR approach to the solution of the 2D flow problem is demonstrated on a standard benchmark problem - lid-driven cavity.

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23

Mužík, Juraj, and Roman Bulko. "Lid-driven cavity flow using dual reciprocity." MATEC Web of Conferences 313 (2020): 00043. http://dx.doi.org/10.1051/matecconf/202031300043.

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The paper presents the use of the multi-domain dual reciprocity method of fundamental solutions (MD-MFSDR) for the analysis of the laminar viscous flow problem described by Navier-Stokes equations. A homogeneous part of the solution is solved using the method of fundamental solutions with the 2D Stokes fundamental solution Stokeslet. The dual reciprocity approach has been chosen because it is ideal for the treatment of the non-homogeneous and nonlinear terms of Navier-Stokes equations. The presented DR-MFS approach to the solution of the 2D flow problem is demonstrated on a standard benchmark problem - lid-driven cavity.

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24

Lim, C. C., and I. H. Mccomb. "Stability of Normal Modes and Subharmonic Bifurcations in the 3-Body Stokeslet Problem." Journal of Differential Equations 121, no. 2 (September 1995): 384–405. http://dx.doi.org/10.1006/jdeq.1995.1133.

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25

Smith, D. J. "A boundary element regularized Stokeslet method applied to cilia- and flagella-driven flow." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2112 (September 25, 2009): 3605–26. http://dx.doi.org/10.1098/rspa.2009.0295.

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A boundary element implementation of the regularized Stokeslet method of Cortez is applied to cilia and flagella-driven flows in biology. Previously published approaches implicitly combine the force discretization and the numerical quadrature used to evaluate boundary integrals. By contrast, a boundary element method can be implemented by discretizing the force using basis functions, and calculating integrals using accurate numerical or analytic integration. This substantially weakens the coupling of the mesh size for the force and the regularization parameter, and greatly reduces the number of degrees of freedom required. When modelling a cilium or flagellum as a one-dimensional filament, the regularization parameter can be considered a proxy for the body radius, as opposed to being a parameter used to minimize numerical errors. Modelling a patch of cilia, it is found that: (i) for a fixed number of cilia, reducing cilia spacing reduces transport, (ii) for fixed patch dimension, increasing cilia number increases the transport, up to a plateau at 9×9 cilia. Modelling a choanoflagellate cell, it is found that the presence of a lorica structure significantly affects transport and flow outside the lorica, but does not significantly alter the force experienced by the flagellum.

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26

Lu, Dong-qiang. "Analytical solutions for the interfacial viscous capillary-gravity waves due to an oscillating Stokeslet." Journal of Hydrodynamics 31, no. 6 (December 2019): 1139–47. http://dx.doi.org/10.1007/s42241-019-0087-0.

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27

OTTO, S. R., A. N. YANNACOPOULOS, and J. R. BLAKE. "Transport and mixing in Stokes flow: the effect of chaotic dynamics on the blinking stokeslet." Journal of Fluid Mechanics 430 (March 10, 2001): 1–26. http://dx.doi.org/10.1017/s0022112000002494.

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28

Gallagher, Meurig T., Debajyoti Choudhuri, and David J. Smith. "Sharp Quadrature Error Bounds for the Nearest-Neighbor Discretization of the Regularized Stokeslet Boundary Integral Equation." SIAM Journal on Scientific Computing 41, no. 1 (January 2019): B139—B152. http://dx.doi.org/10.1137/18m1191816.

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29

Rorai, C., M. Zaitsev, and S. Karabasov. "On the limitations of some popular numerical models of flagellated microswimmers: importance of long-range forces and flagellum waveform." Royal Society Open Science 6, no. 1 (January 2019): 180745. http://dx.doi.org/10.1098/rsos.180745.

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For a sperm-cell-like flagellated swimmer in an unbounded domain, several numerical models of different fidelity are considered based on the Stokes flow approximation. The models include a regularized Stokeslet method and a three-dimensional finite-element method, which serve as the benchmark solutions for several approximate models considered. The latter include the resistive force theory versions of Lighthill, and Gray and Hancock, as well as a simplified approximation based on computing the hydrodynamic forces exerted on the head and the flagellum separately. It is shown how none of the simplified models is robust enough with regards to predicting the effect of the swimmer head shape change on the swimmer dynamics. For a range of swimmer motions considered, the resulting solutions for the swimmer force and velocities are analysed and the applicability of the Stokes model for the swimmers in question is probed.

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30

ZHANG, LU, R. SHANKAR SUBRAMANIAN, and R. BALASUBRAMANIAM. "Motion of a drop in a vertical temperature gradient at small Marangoni number – the critical role of inertia." Journal of Fluid Mechanics 448 (November 26, 2001): 197–211. http://dx.doi.org/10.1017/s0022112001005997.

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When a drop moves in a uniform vertical temperature gradient under the combined action of gravity and thermocapillarity at small values of the thermal Péclet number, it is shown that inclusion of inertia is crucial in the development of an asymptotic solution for the temperature field. If inertia is completely ignored, use of the method of matched asymptotic expansions, employing the Péclet number (known as the Marangoni number) as the small parameter, leads to singular behaviour of the outer temperature field. The origin of this behaviour can be traced to the interaction of the slowly decaying Stokeslet, arising from the gravitational contribution to the motion of the drop, with the temperature gradient field far from the drop. When inertia is included, and the method of matched asymptotic expansions is used, employing the Reynolds number as a small parameter, the singular behaviour of the temperature field is eliminated. A result is obtained for the migration velocity of the drop that is correct to O(Re2 log Re).

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31

Hsiao, C.-H., and D. L. Young. "The Derivation and Application of Fundamental Solutions for Unsteady Stokes Equations." Journal of Mechanics 31, no. 6 (September 18, 2015): 683–91. http://dx.doi.org/10.1017/jmech.2015.70.

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AbstractIn this paper, two formulations in explicit form to derive the fundamental solutions for two and three dimensional unsteady unbounded Stokes flows due to a mass source and a point force are presented, based on the vector calculus method and also the Hörmander’s method. The mathematical derivation process for the fundamental solutions is detailed. The steady fundamental solutions of Stokes equations can be obtained from the unsteady fundamental solutions by the integral process. As an application, we adopt fundamental solutions: an unsteady Stokeslet and an unsteady potential dipole to validate a simple case that a sphere translates in Stokes or low-Reynolds-number flow by using the singularity method instead by the traditional method which in general limits to the assumption of oscillating flow. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions by making use of the fundamental solutions of the linearly unsteady Stokes equations.

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32

Hill, Reghan J. "Corona charge regulation in nanoparticle electrophoresis." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2183 (November 2015): 20150522. http://dx.doi.org/10.1098/rspa.2015.0522.

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Nanoparticle (NP) size and charge play key roles in bioconjugation chemistry, imaging and drug delivery. Although the electrophoretic mobility and hydrodynamic size are routinely measured, interpreting these data can be extremely difficult. Here, the challenge is addressed via an electrokinetic model for spheres bearing a soft amphoteric corona, the charge of which is regulated by a multi-component electrolyte. The model is applied to NPs with a metallic core to which are grafted poly(ethylene glycol) chains with either weak acid or amphiprotic end groups. The results elucidate the separate roles of electrolyte pH and ionic strength on the electrophoretic mobility and diffusion coefficient. In this study, the forces were evaluated directly, rather than from the Stokeslet velocity disturbances. While the second-order convergence was demonstrated by both methods, the direct approach, which uses only the inner part of the global solution, furnished superior accuracy and robustness. This may benefit future attempts to model the dielectric and electroacoustic properties of these complex nanoparticulates.

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33

Kos, Žiga, and Miha Ravnik. "Elementary Flow Field Profiles of Micro-Swimmers in Weakly Anisotropic Nematic Fluids: Stokeslet, Stresslet, Rotlet and Source Flows." Fluids 3, no. 1 (February 8, 2018): 15. http://dx.doi.org/10.3390/fluids3010015.

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34

Lockerby, D. A., and B. Collyer. "Fundamental solutions to moment equations for the simulation of microscale gas flows." Journal of Fluid Mechanics 806 (October 3, 2016): 413–36. http://dx.doi.org/10.1017/jfm.2016.606.

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Fundamental solutions (Green’s functions) to Grad’s steady-state linearised 13-moment equations for non-equilibrium gas flows are derived. The creeping microscale gas flows, to which they pertain, are important to understanding the behaviour of atmospheric particulate and the performance of many potential micro/nano technologies. Fundamental solutions are also derived for the regularised form of the steady-state linearised 13-moment equations, due to Struchtrup & Torrilhon (Phys. Fluids, vol. 15 (9), 2003, pp. 2668–2680). The solutions are compared to their classical and ubiquitous counterpart: the Stokeslet. For an illustration of their utility, the fundamental solutions to Grad’s equations are implemented in a linear superposition approach to modelling external flows. Such schemes are mesh free, and benefit from not having to truncate and discretise an infinite three-dimensional domain. The high accuracy of the technique is demonstrated for creeping non-equilibrium gas flow around a sphere, for which an analytical solution exists for comparison. Finally, to demonstrate the method’s geometrical flexibility, the flow generated between adjacent spheres held at a fixed uniform temperature difference is explored.

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35

Pepper, Rachel E., Marcus Roper, Sangjin Ryu, Paul Matsudaira, and Howard A. Stone. "Nearby boundaries create eddies near microscopic filter feeders." Journal of The Royal Society Interface 7, no. 46 (November 26, 2009): 851–62. http://dx.doi.org/10.1098/rsif.2009.0419.

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We show through calculations, simulations and experiments that the eddies often observed near sessile filter feeders are frequently due to the presence of nearby boundaries. We model the common filter feeder Vorticella , which is approximately 50 µm across and which feeds by removing bacteria from ocean or pond water that it draws towards itself. We use both an analytical stokeslet model and a Brinkman flow approximation that exploits the narrow-gap geometry to predict the size of the eddy caused by two parallel no-slip boundaries that represent the slides between which experimental observations are often made. We also use three-dimensional finite-element simulations to fully solve for the flow around a model Vorticella and analyse the influence of multiple nearby boundaries. Additionally, we track particles around live feeding Vorticella in order to determine the experimental flow field. Our models are in good agreement both with each other and with experiments. We also provide approximate equations to predict the experimental eddy sizes owing to boundaries both for the case of a filter feeder between two slides and for the case of a filter feeder attached to a perpendicular surface between two slides.

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36

Nguyen, Hoang-Ngan, and Ricardo Cortez. "Reduction of the Regularization Error of the Method of Regularized Stokeslets for a Rigid Object Immersed in a Three-Dimensional Stokes Flow." Communications in Computational Physics 15, no. 1 (January 2014): 126–52. http://dx.doi.org/10.4208/cicp.021112.290413a.

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AbstractWe focus on the problem of evaluating the velocity field outside a solid object moving in an incompressible Stokes flow using the boundary integral formulation. For points near the boundary, the integral is nearly singular, and accurate computation of the velocity is not routine. One way to overcome this problem is to regularize the integral kernel. The method of regularized Stokeslet (MRS) is a systematic way to regularize the kernel in this situation. For a specific blob function which is widely used, the error of the MRS is only of first order with respect to the blob parameter. We prove that this is the case for radial blob functions with decay property ϕ(r)=O(r−3−α) when r→∞ for some constant α>1. We then find a class of blob functions for which the leading local error term can be removed to get second and third order errors with respect to blob parameter. Since the addition of these terms might give a flow field that is not divergence free, we introduce a modification of these terms to make the divergence of the corrected flow field close to zero while keeping the desired accuracy. Furthermore, these dominant terms are explicitly expressed in terms of blob function and so the computation time is negligible.

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37

Solomentsev, Yuri, and John L. Anderson. "Electrophoresis of slender particles." Journal of Fluid Mechanics 279 (November 25, 1994): 197–215. http://dx.doi.org/10.1017/s0022112094003885.

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The hydrodynamic theory of slender bodies is used to model electrophoretic motion of a slender particle having a charge (zeta potential) that varies with position along its length. The theory is limited to systems where the Debye screening length of the solution is much less than the typical cross-sectional dimension of the particle. A stokeslet representation of the hydrodynamic force is combined with the Lorentz reciprocal theorem for Stokes flow to develop a set of linear equations which must be solved for the components of the translational and angular velocities of the particle. Sample calculations are presented for the electrophoretic motion of straight spheroids and cylinders and a torus in a uniform electric field. The theory is also applied to a straight uniformly charged particle in a spatially varying electric field. The uniformly charged particle rotates into alignment with the principal axes of ∇E∞; we suggest that such alignment can lead to electrophoretic transport of particles through a small aperture in an otherwise impermeable wall. The theory developed here is more general than just for electrophoresis, since the final result is expressed in terms of a general 'slip velocity’ at the surface of the particle. Thus, the results are applicable to diffusiophoresis of slender particles if the proper slip-velocity coefficient is used.

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38

Jiang, Houshuo, and Thomas Kiørboe. "The fluid dynamics of swimming by jumping in copepods." Journal of The Royal Society Interface 8, no. 61 (January 12, 2011): 1090–103. http://dx.doi.org/10.1098/rsif.2010.0481.

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Copepods swim either continuously by vibrating their feeding appendages or erratically by repeatedly beating their swimming legs, resulting in a series of small jumps. The two swimming modes generate different hydrodynamic disturbances and therefore expose the swimmers differently to rheotactic predators. We developed an impulsive stresslet model to quantify the jump-imposed flow disturbance. The predicted flow consists of two counter-rotating viscous vortex rings of similar intensity, one in the wake and one around the body of the copepod. We showed that the entire jumping flow is spatially limited and temporally ephemeral owing to jump-impulsiveness and viscous decay. In contrast, continuous steady swimming generates two well-extended long-lasting momentum jets both in front of and behind the swimmer, as suggested by the well-known steady stresslet model. Based on the observed jump-swimming kinematics of a small copepod Oithona davisae , we further showed that jump-swimming produces a hydrodynamic disturbance with much smaller spatial extension and shorter temporal duration than that produced by a same-size copepod cruising steadily at the same average translating velocity. Hence, small copepods in jump-swimming are in general much less detectable by rheotactic predators. The present impulsive stresslet model improves a previously published impulsive Stokeslet model that applies only to the wake vortex.

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Mathijssen, A. J. T. M., A. Doostmohammadi, J. M. Yeomans, and T. N. Shendruk. "Hydrodynamics of micro-swimmers in films." Journal of Fluid Mechanics 806 (September 29, 2016): 35–70. http://dx.doi.org/10.1017/jfm.2016.479.

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One of the principal mechanisms by which surfaces and interfaces affect microbial life is by perturbing the hydrodynamic flows generated by swimming. By summing a recursive series of image systems, we derive a numerically tractable approximation to the three-dimensional flow fields of a stokeslet (point force) within a viscous film between a parallel no-slip surface and a no-shear interface and, from this Green’s function, we compute the flows produced by a force- and torque-free micro-swimmer. We also extend the exact solution of Liron & Mochon (J. Engng Maths, vol. 10 (4), 1976, pp. 287–303) to the film geometry, which demonstrates that the image series gives a satisfactory approximation to the swimmer flow fields if the film is sufficiently thick compared to the swimmer size, and we derive the swimmer flows in the thin-film limit. Concentrating on the thick-film case, we find that the dipole moment induces a bias towards swimmer accumulation at the no-slip wall rather than the water–air interface, but that higher-order multipole moments can oppose this. Based on the analytic predictions, we propose an experimental method to find the multipole coefficient that induces circular swimming trajectories, allowing one to analytically determine the swimmer’s three-dimensional position under a microscope.

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40

Kim, Sangtae, and William B. Russel. "Modelling of porous media by renormalization of the Stokes equations." Journal of Fluid Mechanics 154 (May 1985): 269–86. http://dx.doi.org/10.1017/s0022112085001525.

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The permeability of a random array of fixed spheres has been calculated over the range of volume fractions from dilute to almost closest packing, by assuming pairwise-additive (low-Reynolds-number) hydrodynamic interactions within an effective medium. Non-convergent pair interactions arising from the long-range decay of the Stokeslet were removed by renormalizing the Stokes equation to determine the permeability of the effective medium, i.e. to include the mean screening effect of the other spheres. Pair interactions in this Brinkman medium were calculated by the method of reflections in the far field and boundary collocation in the near field.The permeability predicted by the theory asymptotes correctly to established results for dilute arrays, and compares favourably (within 15%) with the Carman correlation for volume fractions between 0.3 and 0.5. The magnitude also falls within the range of exact results for periodic arrays at the higher concentrations, but our model does not reproduce the dependence on structure.Use of the Brinkman equation with an effective viscosity leads to an apparent slip velocity at the boundary of a porous medium. Our calculation of the bulk stress via volume averaging determines the effective viscosity and hence the slip coefficient unambiguously for dilute porous media. However, at concentrations corresponding to the available experimental results the lengthscale characterizing pressure or velocity gradients becomes comparable to the interparticle spacing, and the averaging technique fails. Indeed the Brinkman equation itself is no longer valid.

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41

Schnitzer, Ory, and Ehud Yariv. "Small-solid-fraction approximations for the slip-length tensor of micropillared superhydrophobic surfaces." Journal of Fluid Mechanics 843 (March 26, 2018): 637–52. http://dx.doi.org/10.1017/jfm.2018.187.

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Fakir-like superhydrophobic surfaces, formed by doubly periodic arrays of thin pillars that sustain a lubricating gas layer, exhibit giant liquid-slip lengths that scale as $\unicode[STIX]{x1D719}^{-1/2}$ relative to the periodicity, $\unicode[STIX]{x1D719}$ being the solid fraction (Ybert et al., Phys. Fluids, vol. 19, 2007, 123601). Considering arbitrarily shaped pillars distributed over an arbitrary Bravais lattice, we employ matched asymptotic expansions to calculate the slip-length tensor in the limit $\unicode[STIX]{x1D719}\rightarrow 0$. The leading $O(\unicode[STIX]{x1D719}^{-1/2})$ slip length is determined from a local analysis of an ‘inner’ region close to a single pillar, in conjunction with a global force balance. This leading term, which is independent of the lattice geometry, is related to the drag due to pure translation of a flattened disk shaped like the pillar cross-section; its calculation is illustrated for the case of elliptical pillars. The $O(1)$ slip length is associated with the excess velocity induced about a given pillar by all the others. Since the field induced by each pillar corresponds on the ‘outer’ lattice scale to a Stokeslet whose strength is fixed by the shear rate, the $O(1)$ slip length depends upon the lattice geometry but is independent of the cross-sectional shape. Its calculation entails asymptotic evaluation of singular lattice sums. Our approximations are in excellent agreement with existing numerical computations for both circular and square pillars.

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42

Smith, Andrew A., Thomas D. Johnson, David J. Smith, and John R. Blake. "Symmetry breaking cilia-driven flow in the zebrafish embryo." Journal of Fluid Mechanics 705 (April 13, 2012): 26–45. http://dx.doi.org/10.1017/jfm.2012.117.

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AbstractFluid mechanics plays a vital role in early vertebrate embryo development, an example being the establishment of left–right asymmetry. Following the dorsal–ventral and anterior–posterior axes, the left–right axis is the last to be established; in several species it has been shown that an important process involved with this is the production of a left–right asymmetric flow driven by ‘whirling’ cilia. It has previously been established in experimental and mathematical models of the mouse ventral node that the combination of a consistent rotational direction and posterior tilt creates left–right asymmetric flow. The zebrafish organizing structure, Kupffer’s vesicle, has a more complex internal arrangement of cilia than the mouse ventral node; experimental studies show that the flow exhibits an anticlockwise rotational motion when viewing the embryo from the dorsal roof, looking in the ventral direction. Reports of the arrangement and configuration of cilia suggest two possible mechanisms for the generation of this flow from existing axis information: (a) posterior tilt combined with increased cilia density on the dorsal roof; and (b) dorsal tilt of ‘equatorial’ cilia. We develop a mathematical model of symmetry breaking cilia-driven flow in Kupffer’s vesicle using the regularized Stokeslet boundary element method. Computations of the flow produced by tilted whirling cilia in an enclosed domain suggest that a possible mechanism capable of producing the flow field with qualitative and quantitative features closest to those observed experimentally is a combination of posteriorly tilted roof and floor cilia, and dorsally tilted equatorial cilia.

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43

Choudhary, A., T. Renganathan, and S. Pushpavanam. "Inertial migration of an electrophoretic rigid sphere in a two-dimensional Poiseuille flow." Journal of Fluid Mechanics 874 (July 12, 2019): 856–90. http://dx.doi.org/10.1017/jfm.2019.479.

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There has been a recent interest in integrating external fields with inertial microfluidic devices to tune particle focusing. In this work, we analyse the inertial migration of an electrophoretic particle in a two-dimensional Poiseuille flow with an electric field applied parallel to the walls. For a thin electrical double layer, the particle exhibits a slip-driven electrokinetic motion along the direction of the applied electric field, which causes the particle to lead or lag the flow (depending on its surface charge). The fluid disturbance caused by this slip-driven motion is characterized by a rapidly decaying source-dipole field which alters the inertial lift on the particle. We determine this inertial lift using the reciprocal theorem. Assuming no wall effects, we derive an analytical expression for a ‘phoretic lift’ which captures the modification to the inertial lift due to electrophoresis. We also take wall effects into account, at the leading order, using the method of reflections. We find that for a leading particle, the phoretic lift acts towards the regions of high shear (i.e. walls), while the reverse is true for a lagging particle. Using an order-of-magnitude analysis, we obtain different components of the inertial force and classify them on the basis of the interactions from which they emerge. We show that the dominant contribution to the phoretic lift originates from the interaction of the source-dipole field (generated by the electrokinetic slip at the particle surface) with the stresslet field (generated due to particle’s resistance to strain in the background flow). Furthermore, to contrast the slip-driven phenomenon (electrophoresis) from the force-driven phenomenon (buoyancy) in terms of their influence on the inertial migration, we also study a non-neutrally buoyant particle. We show that the gravitational effects alter the inertial lift primarily through the interaction of the background shear with the buoyancy-induced Stokeslet field.

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44

Lighthill, James. "Helical distributions of stokeslets." Journal of Engineering Mathematics 30, no. 1-2 (March 1996): 35–78. http://dx.doi.org/10.1007/bf00118823.

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45

Cortez, Ricardo. "The Method of Regularized Stokeslets." SIAM Journal on Scientific Computing 23, no. 4 (January 2001): 1204–25. http://dx.doi.org/10.1137/s106482750038146x.

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46

Zhao, Shunliu, and Alex Povitsky. "Method of Submerged Stokeslets for Slip Flow About Ensembles of Particles." Journal of Nanoscience and Nanotechnology 8, no. 7 (July 1, 2008): 3790–801. http://dx.doi.org/10.1166/jnn.2008.18343.

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A boundary singularity method with submerged Stokeslets is applied to the low Reynolds number flows about a set of spheres. Newtonian fluid is considered with no slip or partial slip boundary conditions at the wall. The validity of the method for Stokes flows about representative sets of spheres is investigated. The considered cases include (i) a uniform flow about a stationary set of particles typical for filtration and chemical vapor deposition, (ii) a flow induced by particles moving toward each other typical for self-assembly processes and (iii) a flow induced by spinning particles typical for micro-pump applications. The dependence of the flowfield on the number of Stokeslets is investigated in order to establish the needed number of Stokeslets. Comparison of flow field for the no-slip (Kn = 0) and partial-slip boundary conditions (Kn = 0.1) shows that the partial slip at the particles' surface significantly affect the velocity field and pressure distribution.

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47

Sayas, Francisco-Javier, and Virginia Selgas. "Variational views of stokeslets and stresslets." SeMA Journal 63, no. 1 (February 12, 2014): 65–90. http://dx.doi.org/10.1007/s40324-014-0013-x.

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48

Nitsche, Ludwig C., and Uwe Schaflinger. "A swarm of Stokeslets with interfacial tension." Physics of Fluids 13, no. 6 (June 2001): 1549–53. http://dx.doi.org/10.1063/1.1369124.

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49

Bouzarth, Elizabeth L., and Michael L. Minion. "A multirate time integrator for regularized Stokeslets." Journal of Computational Physics 229, no. 11 (June 2010): 4208–24. http://dx.doi.org/10.1016/j.jcp.2010.02.006.

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50

Ainley, Josephine, Sandra Durkin, Rafael Embid, Priya Boindala, and Ricardo Cortez. "The method of images for regularized Stokeslets." Journal of Computational Physics 227, no. 9 (April 2008): 4600–4616. http://dx.doi.org/10.1016/j.jcp.2008.01.032.

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