Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-09T15:23:32.778Z Has data issue: false hasContentIssue false
  • English
  • Français

Unsteady Stokes flow near boundaries: the point-particle approximation and the method of reflections

Published online by Cambridge University Press:  01 March 2018

A. Simha Open the ORCID record for A. Simha [Opens in a new window] ,
J. Mo  and
P. J. Morrison
A. Simha*
Affiliation:
Department of Physics and Institute for Fusion Studies, University of Texas at Austin, Austin, TX 78712, USA
J. Mo
Affiliation:
Department of Physics and Center for Nonlinear Dynamics, University of Texas at Austin, Austin, TX 78712, USA
P. J. Morrison
Affiliation:
Department of Physics and Institute for Fusion Studies, University of Texas at Austin, Austin, TX 78712, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Problems of particle dynamics involving unsteady Stokes flows in confined geometries are typically harder to solve than their steady counterparts. Approximation techniques are often the only resort. Felderhof (see e.g. J. Phys. Chem. B, vol. 109 (45), 2005, pp. 21406–21412; J. Fluid Mech., vol. 637, 2009, pp. 285–303) has developed a point-particle approximation framework to solve such problems, especially in the context of Brownian motion. Despite excellent agreement with past experiments, this framework produces unsteady drag coefficients that depend on particle density. This is inconsistent, since the problem can be formulated mathematically without any reference to the particle’s density. We address this inconsistency in our work. Upon implementing our modifications, the framework passes consistency checks that it previously failed. Further, it is not obvious that such an approximation should work for short-time-scale motion. We investigate its validity by deriving it from a general formalism based on integral equations through a series of systematic approximations. We also compare results from the point-particle framework against a calculation performed using the method of reflections, for the specific case of a sphere near a full-slip plane boundary. We find from our analysis that the reasons for the success of the point-particle approximation are subtle and have to do with the nature of the unsteady Oseen tensor. Finally, we provide numerical predictions for Brownian motion near a full-slip and a no-slip plane wall based on the point-particle approximation as used by Felderhof, our modified point-particle approximation and the method of reflections. We show that our modifications to Felderhof’s framework would become significant for systems of metallic nanoparticles in liquids.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 841 , 25 April 2018 , pp. 883 - 924
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alam, M. S., Ishii, K. & Hasimoto, H. 1980 Slow motion of a small sphere outside of a circular cylinder. J. Phys. Soc. Japan 49 (1), 405–408.Google Scholar
Albano, A. M., Bedeaux, D. & Mazur, P. 1975 On the motion of a sphere with arbitrary slip in a viscous incompressible fluid. Physica A 80 (1), 89–97.Google Scholar
Ardekani, A. M. & Rangel, R. H. 2006 Unsteady motion of two solid spheres in Stokes flow. Phys. Fluids 18 (10), 103306.CrossRefGoogle Scholar
Balakrishnan, V. 1979 Fluctuation–dissipation theorems from the generalised Langevin equation. Pramana 12 (4), 301–315.Google Scholar
Bedeaux, D. & Mazur, P. 1974 A generalization of Faxén’s theorem to nonsteady motion of a sphere through a compressible fluid in arbitrary flow. Physica 78 (3), 505–515.CrossRefGoogle Scholar
Berg-Sørensen, K. & Flyvbjerg, H. 2004 Power spectrum analysis for optical tweezers. Rev. Sci. Instrum. 75 (3), 594–612.Google Scholar
Brennen, C. E.1982 A review of added mass and fluid inertial forces. Tech. Rep. CR 82.010. Contract no. N62583-81-MR-554. Naval Civil Engineering Laboratory.Google Scholar
Choi, C.-H. & Kim, C.-J. 2006 Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys. Rev. Lett. 96 (6), 066001.Google Scholar
Clarke, R. J., Jensen, O. E., Billingham, J. & Williams, P. M. 2006 Three-dimensional flow due to a microcantilever oscillating near a wall: an unsteady slender-body analysis. Proc. R. Soc. A 462 (2067), 913–933.CrossRefGoogle Scholar
Clercx, H. J. H. & Schram, P. P. J. M. 1992 Brownian particles in shear flow and harmonic potentials: a study of long-time tails. Phys. Rev. A 46 (4), 1942–1950.Google Scholar
Faxén, H.1921 Einwirkung der gefässwände auf den widerstand gegen die bewegung einer kleinen kugel in einer zähen flüssigkeit. PhD thesis, Uppsala University.Google Scholar
Felderhof, B. U. 1976 Force density induced on a sphere in linear hydrodynamics: II. Moving sphere, mixed boundary conditions. Physica A 84 (3), 569–576.Google Scholar
Felderhof, B. U. 2005 Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion. J. Phys. Chem. B 109 (45), 21406–21412.Google Scholar
Felderhof, B. U. 2006a Diffusion and velocity relaxation of a Brownian particle immersed in a viscous compressible fluid confined between two parallel plane walls. J. Chem. Phys. 124 (5), 054111.Google Scholar
Felderhof, B. U. 2006b Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion. J. Phys. Chem. B 110 (26), 13304.Google Scholar
Felderhof, B. U. 2009a Estimating the viscoelastic moduli of a complex fluid from observation of Brownian motion. J. Chem. Phys. 131 (16), 164904.CrossRefGoogle ScholarPubMed
Felderhof, B. U. 2009b Transient flow of a viscous incompressible fluid in a circular tube after a sudden point impulse. J. Fluid Mech. 637, 285–303.CrossRefGoogle Scholar
Felderhof, B. U. 2012 Hydrodynamic force on a particle oscillating in a viscous fluid near a wall with dynamic partial-slip boundary condition. Phys. Rev. E 85 (4), 046303.Google Scholar
Franosch, T., Grimm, M., Belushkin, M., Mor, F. M., Foffi, G., Forró, L. & Jeney, S. 2011 Resonances arising from hydrodynamic memory in Brownian motion. Nature 478 (7367), 85–88.Google Scholar
Franosch, T. & Jeney, S. 2009 Persistent correlation of constrained colloidal motion. Phys. Rev. E 79 (3), 031402.Google Scholar
Frydel, D. & Rice, S. A. 2006 Lattice–Boltzmann study of the transition from quasi-two-dimensional to three-dimensional one particle hydrodynamics. Mol. Phys. 104 (08), 1283–1297.Google Scholar
Grimm, M., Franosch, T. & Jeney, S. 2012 High-resolution detection of Brownian motion for quantitative optical tweezers experiments. Phys. Rev. E 86 (2), 021912.Google Scholar
Hajizadeh, F. & Reihani, S. N. S. 2010 Optimized optical trapping of gold nanoparticles. Opt. Express 18 (2), 551–559.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Prentice-Hall.Google Scholar
Jackson, J. D. 1999 Classical Electrodynamics. Wiley.Google Scholar
Jeney, S., Lukić, B., Kraus, J. A., Franosch, T. & Forró, L. 2008 Anisotropic memory effects in confined colloidal diffusion. Phys. Rev. Lett. 100 (24), 240604.CrossRefGoogle ScholarPubMed
Jones, R. B. 2004 Spherical particle in Poiseuille flow between planar walls. J. Chem. Phys. 121 (1), 483–500.Google Scholar
Kheifets, S., Simha, A., Melin, K., Li, T. & Raizen, M. G. 2014 Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss. Science 343 (6178), 1493–1496.Google Scholar
Kim, S. & Karrila, S. J. 2013 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon Press.Google Scholar
Lauga, E. & Squires, T. M. 2005 Brownian motion near a partial-slip boundary: a local probe of the no-slip condition. Phys. Fluids 17 (10), 103102.Google Scholar
Li, T. & Raizen, M. G. 2013 Brownian motion at short time scales. Ann. Phys. 525 (4), 281–295.Google Scholar
Lorentz, H. A. 1907 Ein allgemeiner satz, die bewegung einer reibenden flüssigkeit betreffend, nebst einigen anwendungen desselben. In Abhandlungen über theoretische Physik, pp. 23–42. B. G. Teubner.Google Scholar
Luke, J. H. C. 1989 Convergence of a multiple reflection method for calculating stokes flow in a suspension. SIAM J. Appl. Maths 49 (6), 1635–1651.Google Scholar
Maul, C. & Kim, S. 1996 Image of a point force in a spherical container and its connection to the Lorentz reflection formula. In The Centenary of a Paper on Slow Viscous Flow by the Physicist H. A. Lorentz, pp. 119–130. Springer.Google Scholar
Mazur, P. & Bedeaux, D. 1974 A generalization of Faxén’s theorem to nonsteady motion of a sphere through an incompressible fluid in arbitrary flow. Physica 76 (2), 235–246.Google Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics. Courier Corporation.Google Scholar
Mo, J., Simha, A., Kheifets, S. & Raizen, M. G. 2015a Testing the Maxwell–Boltzmann distribution using Brownian particles. Opt. Express 23 (2), 1888–1893.Google Scholar
Mo, J., Simha, A. & Raizen, M. G. 2015b Broadband boundary effects on Brownian motion. Phys. Rev. E 92 (6), 062106.Google ScholarPubMed
Mo, J., Simha, A. & Raizen, M. G. 2017 Brownian motion as a new probe of wettability. J. Chem. Phys. 146 (13), 134707.Google ScholarPubMed
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. (International Series in Pure and Applied Physics) , vol. 1. McGraw-Hill.Google Scholar
Neto, C., Evans, D. R., Bonaccurso, E., Butt, H.-J. & Craig, V. S. J. 2005 Boundary slip in Newtonian liquids: a review of experimental studies. Rep. Prog. Phys. 68 (12), 2859–2897.Google Scholar
O’Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11 (01), 67–74.Google Scholar
O’Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27 (04), 705–724.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.Google Scholar
Sardar, R., Funston, A. M., Mulvaney, P. & Murray, R. W. 2009 Gold nanoparticles: past, present, and future. Langmuir 25 (24), 13840–13851.Google Scholar
Schäffer, E., Nørrelykke, S. F. & Howard, J. 2007 Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers. Langmuir 23 (7), 3654–3665.Google Scholar
Shibuichi, S., Onda, T., Satoh, N. & Tsujii, K. 1996 Super water-repellent surfaces resulting from fractal structure. J. Phys. Chem. 100 (50), 19512–19517.Google Scholar
Shinohara, M. & Hashimoto, H. 1979 The lateral force on a small sphere sedimenting in a viscous fluid bounded by a cylindrical wall. J. Phys. Soc. Japan 46 (1), 320–327.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8.Google Scholar
Svoboda, K. & Block, S. M. 1994 Optical trapping of metallic Rayleigh particles. Opt. Lett. 19 (13), 930–932.Google Scholar
Tuck, E. O. 1967 A simple ‘Filon–trapezoidal’ rule. Maths Comput. 21 (98), 239–241.Google Scholar
Vinogradova, O. I. 1999 Slippage of water over hydrophobic surfaces. Intl J. Miner. Process. 56 (1), 31–60.CrossRefGoogle Scholar
Yang, F.-L. 2010 A formula for the wall-amplified added mass coefficient for a solid sphere in normal approach to a wall and its application for such motion at low Reynolds number. Phys. Fluids 22 (12), 123303.Google Scholar
Zwanzig, R. & Bixon, M. 1970 Hydrodynamic theory of the velocity correlation function. Phys. Rev. A 2 (5), 2005–2012.Google Scholar
Zwanzig, R. & Bixon, M. 1975 Compressibility effects in the hydrodynamic theory of Brownian motion. J. Fluid Mech. 69 (01), 21–25.Google Scholar