Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-19T22:10:55.021Z Has data issue: false hasContentIssue false
  • English
  • Français

Tangential electroviscous drag on a sphere surrounded by a thin double layer near a wall for arbitrary particle–wall separations

Published online by Cambridge University Press:  27 May 2010

S. M. TABATABAEI  and
T. G. M. VAN DE VEN
S. M. TABATABAEI
Affiliation:
Department of Hydraulic Engineering, University of Zabol, 98615-538 Zabol, Iran
T. G. M. VAN DE VEN*
Affiliation:
Pulp and Paper Research Centre, Department of Chemistry, McGill University, Montreal, CanadaH3A 2A7
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

When a charged particle moves along a charged wall in a polar fluid, it experiences an electroviscous lift force normal to the surface and an electroviscous drag, superimposed on the viscous drag, parallel to the surface. Here a theoretical analysis is presented to determine the electroviscous drag on a charged spherical particle surrounded by a thin electrical double layer near a charged plane wall, when the particle translates parallel to the wall without rotation, in a symmetric electrolyte solution at rest. The electroviscous (electro-hydrodynamic) forces, arising from the coupling between the electrical and hydrodynamic equations, are determined as a solution of three partial differential equations, for electroviscous ion concentration (perturbed ion clouds), electroviscous potential (perturbed electric potential) and electroviscous or electro-hydrodynamic flow field (perturbed flow field). The problem was previously solved for small gap widths and low Peclet numbers in the inner region around the gap between the sphere and the wall, using lubrication theory. Here the restriction on the particle–wall distances is removed, and an analytical and numerical solution is obtained valid for the whole domain of interest. For large sphere–wall separations the solution approaches that for the electroviscous drag on an isolated sphere in an unbounded fluid. For small particle–wall distances it differs from that obtained by the use of lubrication theory, showing that lubrication theory is inadequate for electroviscous problems. The analytical results are in complete agreement with the full numerical calculations. For small particle–wall distances a model is given which provides both physical insight and an easy way to calculate the force with high precision.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 656 , 10 August 2010 , pp. 360 - 406
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Booth, F. 1950 Electroviscous effect for suspensions of solid spherical particles. Proc. R. Soc. A 203, 533–551.Google Scholar
Cooley, M. D. A. & O'Neill, M. E. 1968 On the slow rotation of a sphere about a diameter parallel to a nearby plane wall. J. Inst. Math. Appl. 4, 163–173.CrossRefGoogle Scholar
Cox, R. G. 1997 Electroviscous forces on a charged particle suspended in a flowing liquid. J. Fluid Mech. 338, 1–34.CrossRefGoogle Scholar
Dean, W. R. & O'Neill, M. E. 1963 A slow motion of viscous liquid caused by the rotation of a solid sphere. Mathematika 10, 13–24.CrossRefGoogle Scholar
Happel, J. O. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. SIAM.Google Scholar
Helmholtz, H. V. 1879 Über electrische Grenzschichten. Wied. Ann. 7, 337–382.CrossRefGoogle Scholar
Hinch, E. J. & Sherwood, J. D. 1983 The primary electroviscous effect in a suspension of spheres with thin double layers. J. Fluid Mech. 132, 337–347.CrossRefGoogle Scholar
Jeffery, G. B. 1912 On a form of the solution of Laplace's equation suitable for problems relating to two spheres. Proc. R. Soc. A 87, 109–120.Google Scholar
Keh, H. J. & Anderson, J. L. 1985 Boundary effects on electrophoretic motion of colloidal spheres. J. Fluid Mech. 153, 417–439.CrossRefGoogle Scholar
Krasny-Ergen, W. 1936 Untersuchungen über die Viskosität von Suspensionen und Lösungen. 2. Zur Theorie der Electroviskosität, Kolloidzschr 74, 172–178.Google Scholar
Lever, D. A. 1979 Large distortion of the electrical double layer around a charged particle by a shear flow. J. Fluid Mech. 92, 421–433.CrossRefGoogle Scholar
Macrobert, T. M. 1967 Spherical Harmonics. Pergamon.Google Scholar
O'Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11, 67–74.CrossRefGoogle Scholar
O'Neill, M. E. & Stewartson, K. 1967 On slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27, 705–724.CrossRefGoogle Scholar
Ohshima, H., Healy, T. W., White, L. R. & O'Brien, R. W. 1984 Sedimentation velocity and potential in a dilute suspension of charged spherical colloidal particles. J. Chem. Soc. Faraday Trans. 2 (80), 1299–1317.CrossRefGoogle Scholar
Russel, W. B. 1978 The rheology of suspensions of charged rigid particles. J. Fluid Mech. 85, 673–683.CrossRefGoogle Scholar
Sellier, A. 2001 On the boundary effects in electrophoresis. C.R. Acad. Sci. Paris, Série IIb, 565–570.Google Scholar
Sherwood, J. D. 1980 The primary electroviscous effect in a suspension of spheres. J. Fluid Mech. 101, 609–629.CrossRefGoogle Scholar
Smoluchowski, M. 1914 Elektrische Endosmose and Strömungsströme. In Handbuch der Elektrizität und des Magnetismus, Band II. pp. 366–512. Liefering 2, Leipzig.Google Scholar
Tabatabaei, S. M. 2003 Electroviscous particle–wall interactions. PhD thesis, Department of Chemical Engineering McGill University, Montreal, Canada.Google Scholar
Tabatabaei, S. M., van de Ven, T. G. M. & Rey, A. D. 2006 a Electroviscous cylinder-wall interactions. J. Colloid Interface Sci. 295 (2), 504–519.CrossRefGoogle ScholarPubMed
Tabatabaei, S. M., van de Ven, T. G. M. & Rey, A. D. 2006 b Electroviscous sphere–wall interactions. J. Colloid Interface Sci. 301 (1), 291–301.CrossRefGoogle ScholarPubMed
van de Ven, T. G. M. 1988 On the role of ion size in coagulation. J. Colloid Interface Sci. 124, 138–145.CrossRefGoogle Scholar
Warszynski, P. & van de Ven, T. G. M. 1991 Effects of electroviscous drag on the coagulation and deposition of electrically charged colloidal particles. Adv. Colloid Interface Sci. 36, 33–63.CrossRefGoogle Scholar
Warszynski, P. & van de Ven, T. G. M. 2000 Electroviscous forces on a charged cylinder moving near a charged wall. J. Colloid Interface Sci. 223 (1), 1–15.CrossRefGoogle Scholar
Wu, X., Warszynski, P. & van de Ven, T. G. M. 1996 Electrokinetic lift: observations and comparisons with theories. J. Colloid Interface Sci. 180, 61–69.CrossRefGoogle Scholar