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The hydrodynamics of confined dispersions

Published online by Cambridge University Press:  17 October 2011

James W. Swan  and
John F. Brady
James W. Swan*
Affiliation:
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
John F. Brady
Affiliation:
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: [email protected]
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Abstract

A method is proposed for computing the low-Reynolds-number hydrodynamic forces on particles comprising a suspension confined by two parallel, no-slip walls. This is constructed via the two-dimensional analogue of Hasimoto’s solution (J. Fluid Mech., vol. 5, 1959, pp. 317–328) for a periodic array of point forces in a viscous, incompressible fluid, and, like Hasimoto, the summation of interactions is accelerated by substitution and superposition of ‘Ewald-like’ forcing. This method is akin to the accelerated Stokesian dynamics technique (J. Fluid Mech., vol. 448, 2001, pp. 115–146) and models the suspension dynamics with log–linear computational scaling. The effectiveness of this approach is demonstrated with a calculation of the high-frequency dynamic viscosity of a colloidal dispersion as function of volume fraction and channel width. Similarly, the short-time self-diffusivity for and the sedimentation rate of spherical particles in a confined suspension are determined. The results demonstrate the influence of confining geometry on the transport of small particles, which is becoming increasingly important for micro- and biofluidics.

JFM classification

Complex Fluids: Colloids Mathematical Foundations: Computational methods Low-Reynolds-number flows: Stokesian dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 687 , 25 November 2011 , pp. 254 - 299
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

‡

Present address: Department of Chemical Engineering and Center for Molecular and Engineering Thermodynamics, University of Delaware, 150 Academy St., Newark, DE 19716, USA.

References

1. Ali, I., Marenduzzo, D. & Yeomans, J. M. 2004 Dynamics of polymer packaging. J. Chem. Phys. 121 (17), 8635–8642.Google Scholar
2. Allen, M. P. & Tildesley, D. J. 1989 Computer Simulation of Liquids. Oxford University Press.Google Scholar
3. Ashton, R., Padala, C. & Kane, R. S. 2003 Microfluidic separation of DNA. Current Opinions in Biotechnology 14 (5), 497–504.Google Scholar
4. Beenakker, C. W. J. 1986 Ewald sum of the Rotne–Prager tensor. J. Chem. Phys. 85 (3), 1581–1582.Google Scholar
5. Bhattacharya, S. 2008 Cooperative motion of spheres arranged in periodic grids between two walls. J. Chem. Phys. 128, 074709.CrossRefGoogle Scholar
6. Bhattacharya, S. & Blawzdziewicz, J. 2002 Image system for Stokes-flow singularity between two parallel planar walls. J. Math. Phys. 43 (11), 5720–5731.Google Scholar
7. Bhattacharya, S., Blawzdziewicz, J. & Wajnryb, E. 2005 Hydrodynamic interactions of spherical particles in suspensions confined between two planar walls. J. Fluid Mech. 541, 263–292.Google Scholar
8. Blake, J. R. 1971 A note on the image system for a Stokeslet in a no-slip boundary. Proc. Camb. Phil. Soc. 70, 303–310.Google Scholar
9. Bossis, G., Meunier, A. & Sherwood, J. D. 1991 Stokesian dynamics simulations of particle trajectoriers near a plane. Phys. Fluids A 3 (8), 1853–1858.Google Scholar
10. Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111–157.Google Scholar
11. Brady, J. F, Phillips, R. J., Lester, J. C. & Bossis, G. 1988 Dynamic simulation of hydrodynamically interacting suspensions. J. Fluid Mech. 195, 257–280.CrossRefGoogle Scholar
12. Butler, J. E. & Bonnecaze, R. T. 1999 Imaging of particle shear migration with electrical impedance tomography. Phys. Fluids 11 (8), 1982.Google Scholar
13. Cichocki, B., Jones, R. B., Kutteh, R. & Wajnryb, E. 2000 Friction and mobility for colloidal spheres in Stokes flow near a boundary: the multipole method and applications. J. Chem. Phys. 112 (5), 2548–2561.Google Scholar
14. Cox, R. G. & Hsu, S. K. 1977 The lateral migration of solid particles in a laminar flow near a plane. Intl J. Multiphase Flow 3 (3), 201–222.Google Scholar
15. Cox, R. G. & Mason, S. G. 1971 Suspended particles in fluid flow through tubes. Annu. Rev. Fluid Mech. 3, 291–316.CrossRefGoogle Scholar
16. Daniels, B. R., Masi, B. C. & Wirtz, D 2006 Probing single-cell micromechanics in vivo: the microrheology of C. Elegans developing embryos. Biophys. J. 90 (12), 4712–4719.CrossRefGoogle Scholar
17. Darden, T., York, D. & Pedersen, L. 1993 Particle mesh Ewald: an method for Ewald sums in large systems. J. Chem. Phys. 98 (12), 10089–10093.Google Scholar
18. Donev, A., Stillinger, F. H. & Torquato, S. 2005 Neighbor list collision-driven molecular dynamics simulation for non-spherical particles. I. Algorithmic details II. Applications to ellipses and ellipsoids. J. Comput. Phys. 202 (2), 737–793.Google Scholar
19. Dufresne, E. R., Altman, D. & Grier, D. G. 2001 Brownian dynamics of a sphere between parallel walls. Europhys. Lett. 53 (2), 264–270.Google Scholar
20. Durlofsky, L. J. & Brady, J. F. 1989 Dynamic simulation of bounded suspensions of hydrodynamically interacting particles. J. Fluid Mech. 200, 39–67.CrossRefGoogle Scholar
21. Faxén, H. 1921 Dissertations, Uppsala University. Ark. Mat. Astron. Fys. 17, 27.Google Scholar
22. Frank, M., Anderson, D., Weeks, E. R. & Morris, J. F. 2003 Particle migration in pressure-driven flow of a Brownian suspension. J. Fluid Mech. 493, 363–378.Google Scholar
23. Frenkel, D. & Smit, B. 2001 Understanding Molecular Simulations, 2nd edn. Academic Press.Google Scholar
24. de Gennes, P. G. 1979 Conjectures on the transition from Poiseuille to plug flow in suspensions. J. Phys. 40 (8), 783–787.Google Scholar
25. Glendinning, A. B. & Russel, W. B. 1982 A pairwise additive description of sedimentation and diffusion in concentrated suspensions of hard spheres. J. Colloid Interface Sci. 89 (1), 123–143.Google Scholar
26. Goldsmith, H. L. & Mason, S. G. 1962 The of suspensions through tubes. I. Single spheres, rods and discs. J. Colloid Sci. 17 (5), 448–476.Google Scholar
27. Gregory, J. 1981 Flocculation in laminar tube flow. Chem. Engng Sci. 36 (11), 1789–1794.Google Scholar
28. Guth, E. & Simha, R. 1936 Untersuchungen über die viskosität von suspensionen und lösungen. 3. Über die viskosität von kugelsuspensionen. Colloid Polym. Sci. 74 (3), 266.Google Scholar
29. Happel, J. & Brenner, H. 1986 Low Reynolds Number Hydrodynamics, 2nd edn. Prentice-Hall.Google Scholar
30. Hasimoto, H. 1959 On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (2), 317–328.Google Scholar
31. Hernández-Ortiz, J. P., de Pablo, J. J. & Graham, M. D. 2007 Fast computation of many-particle hydrodynamic and electrostatic interactions in a confined geometry. Phys. Rev. Lett. 98 (14), 140602.Google Scholar
32. Hockney, R. W. & Eastwood, J. W. 1989 Computer Simulation Using Particles. Taylor and Francis.Google Scholar
33. Jones, R. B. 2004 Spherical particle in Poiseuille flow between planar walls. J. Chem. Phys. 121, 483–500.Google Scholar
34. Karnis, A., Goldsmith, H. L. & Mason, S. G. 1966 The kinetics of flowing dispersions: I. Concentrated suspensions of rigid particles. J. Colloid Interface Sci. 22 (6), 521–553.Google Scholar
35. Kim, S. & Karrila, S. J. 2005 Microhydrodynamics, 2nd edn. Dover Publications.Google Scholar
36. Koh, C. J., Hookham, P. & Leal, L. G. 1994 An experimental investigation of concentrated suspension flows in a rectangular channel. J. Fluid Mech. 266, 1–32.CrossRefGoogle Scholar
37. Ladyzhenskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.Google Scholar
38. Lee, S. H., Chadwick, R. S. & Leal, L. G. 1979 Motion of a sphere in the presence of a plane interface. Part 1: an approximate solution by generalization of the method of Lorentz. J. Fluid Mech. 93 (4), 705–726.CrossRefGoogle Scholar
39. Leighton, D. T. & Acrivos, A. 1987 Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. Fluid Mech. 177, 109–131.Google Scholar
40. Liron, N. 1984 Stokeslet arrays in a pipe and their application to ciliary transport. J. Fluid Mech. 143, 173–195.Google Scholar
41. Liron, N. & Mochon, S. 1976 Stokes flow for a Stokeslet between two parallel flat plates. J. Engng Maths 10, 287–303.Google Scholar
42. Lyon, M. K. & Leal, L. G. 1998 An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. J. Fluid Mech. 363, 25–56.Google Scholar
43. Marcus, A. H., Lin, B. & Rice, S. A. 1996 Self-diffusion in quasi-two-dimension hard sphere suspensions. Phys. Rev. E 53 (2), 1765–1776.Google Scholar
44. Norman, J. T., Nayak, H. V. & Bonnecaze, R. T. 2005 Migration of buoyant particles in low-Reynolds-number pressure driven flows. J. Fluid Mech. 523, 1–35.Google Scholar
45. Nott, P. R. & Brady, J. F. 1994 Pressure driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157–199.CrossRefGoogle Scholar
46. O’Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27 (4), 705–724.CrossRefGoogle Scholar
47. Oseen, C. W. 1928 Neuere Methoden und Ergebnisse in der Hydrodynamik. Dissertation, Akademische Verlagsgesellschaft, Leipzig.Google Scholar
48. Phung, T. N., Brady, J. F. & Bossis, G. 1996 Stokesian dynamics simulation of Brownian suspensions. J. Fluid Mech. 313, 181–207.Google Scholar
49. Ramachandran, A. & Leighton, D. T. 2007 Viscous resuspension in a tube: the impact of secondary flows resulting from second normal stress differences. Phys. Fluids 19 (5), 053301.Google Scholar
50. Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115–146.Google Scholar
51. Seshadri, V. & Sutera, S. P. 1970 Apparent viscosity of coarse concentrated suspensions in tube flow. J. Rheol. 14 (3), 351–373.Google Scholar
52. Staben, M. E., Zinchenko, A. Z. & Davis, R. H. 2003 Motion of a particle between two parallel plane walls in low-Reynolds-number Poiseuille flow. Phys. Fluids 15, 1711–1734.Google Scholar
53. Swan, J. W. & Brady, J. F. 2007 Simulation of hydrodynamically interacting particles near a no-slip boundary. Phys. Fluids 19, 113306.Google Scholar
54. Swan, J. W. & Brady, J. F. 2010 Particle motion between parallel walls: hydrodynamics and simulation. Phys. Fluids 22, 103301.Google Scholar
55. Zarraga, I. E. & Leighton, D. T. 2002 Measurement of an unexpectedly large shear-induced self-diffusivity in a dilute suspension of spheres. Phys. Fluids 14 (7), 2194.Google Scholar
56. Zurita-Gotor, M., Blawzdziewicz, J. & Wajnryb, E. 2007 Motion of a rod-like particle between parallel walls with application to suspension rheology. J. Rheol. 51 (1), 71–97.Google Scholar
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