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The viscous flow of charged particles through a charged cylindrical tube

Published online by Cambridge University Press:  26 April 2006

Paul Venema
Paul Venema
Affiliation:
Laboratory of Colloid Chemistry, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands Present address: Ingeny B.V., Einsteinweg 5, PO Box 685, 2300 AR Leiden. The Netherlands.
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Abstract

An analysis is given for the electro-kinetic transport properties in a system consisting of a line of identical spheres placed equidistantly with their centres on the axis of a cylindrical tube containing a viscous fluid. Both the spheres and the wall of the tube are charged and a two-species symmetrical electrolyte with valence Z is present in the system. As a result of the charges on the surface of the spheres and on the surface of the tube electrical double layers will develop. When an electrical field is applied to the system an electrokinetic motion is induced. We will use the thin double layer theory (Dukhin & Derjaguin 1974; O'Brien 1983), valid for sufficiently high electrolyte concentration and where the polarization of the electrical double layer is included. Using a multipole expansion an infinite set of linear equations for the multipoles will be derived from which the electro-kinetic transport coefficients may be determined. These coefficients depend on the system parameters, such as the radius of the tube R, the radius of the sphere a, the separation between the spheres d, the Debije radius κ-1, the zeta-potentials of the spheres ζp and of the wall of the tube ζw and the valency Z of the electrolyte. From these coefficients a relation is found between the pressure drop Δp per unit length and the drag force D on the spheres on one side and with the velocity U of the spheres, the total discharge Q and the applied electrical field E0 on the other side. For some values for the system parameters we have numerically solved the infinite set of linear equations by truncation and calculated the transport coefficients. We have also calculated the streamlines for some situations. The plots of these streamlines show that depending on the conditions on the system vortices may appear.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 282 , 10 January 1995 , pp. 45 - 73
Copyright
© 1995 Cambridge University Press

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References

Chen, S. B. & Keh, H. J. 1992 Axisymmetric electrophoresis of multiple colloidal spheres. J. Fluid Mech. 238, 251.Google Scholar
Dukhin, S. S. & Derjaguin, B. V. 1974 Electrokinetic phenomena. In Surface and Colloid Science (ed. E. Matijevic), vol. 7. Wiley.
Fåhraeus, R. 1928 Die strömungsverhältnisse und die Verteilung der Blutzellen in Gefässystem. Klinische Wochenschr. 7, 100.Google Scholar
Fåhraeus, R. & Lindqvist, T. 1931 The viscosity of the blood in narrow capillary tubes. Am. J. Physiol. 96, 562.Google Scholar
Goldsmith, H. L. & Mason, S. G. 1962 The flow of suspensions through tubes — I. Single spheres, rods, and discs. J. Colloid Interface Sci. 17, 448.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Hobson, E. W. 1955 The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press.
Keh, H. J. & Anderson, J. L. 1985 Boundary effects on the electrophoretic motion of colloidal spheres. J. Fluid Mech. 153, 417.Google Scholar
Keh, H. J. & Chen, S. B. 1988 Electrophoresis of a colloidal sphere parallel to a dielectrical plane. J. Fluid Mech. 194, 377.Google Scholar
Keh, H. J. & Chen, S. B. 1989a Particle interactions in electrophoresis — I. Motion of two spheres along their line of centers. J. Colloid Interface Sci. 130, 542.Google Scholar
Keh, H. J. & Chen, S. B. 1989b Particle interactions in electrophoresis — II. Motion of two spheres normal to their line of centers. J. Colloid Interface Sci. 130, 556.Google Scholar
MacRobert, T. M. 1948 Spherical Harmonics. Dover.
McKenzie, D. R., McPhedran, R. C. & Derrick, G. H. 1978 The conductivity of lattices of spheres. — II. The body centred and face centred cubic lattices. Proc. R. Soc. Lond. A 362, 211.Google Scholar
O'Brien, R. W. 1981 The electrical conductivity of a dilute suspension of charged particles. J. Colloid Interface Sci. 81, 234.Google Scholar
O'Brien, R. W. 1983 The solution of the electrokinetic equations for colloidal particles with thin double layers. J. Colloid Interface Sci. 92, 204.Google Scholar
O'Brien, R. W. 1986 Electro-osmosis in porous materials. J. Colloid Interface Sci. 110, 477.Google Scholar
O'Brien, R. W. & Hunter, R. J. 1981 The electrophoretic mobility of large colloidal particles. Can. J. Chem. 59, 1878.Google Scholar
O'Brien, R. W. & Perrins, W. T. 1984 The electrical conductivity of a porous plug. J. Colloid Interface Sci. 99, 20.Google Scholar
O'Brien, R. W. & Ward, D. N. 1988 The electrophoresis of a spheroid with a thin double layer. J. Colloid Interface Sci. 121, 402.Google Scholar
O'Brien, R. W. & White, L. R. 1978 Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc. Faraday Trans. II 74, 1607.Google Scholar
Rice, C. L. & Whitehead, R. 1965 Electrokinetic flow in a narrow cylindrical capillary. J. Phys. Chem. 69, 4017.Google Scholar
Saville, D. A. 1977 Electrokinetic effects with small particles. Ann. Rev. Fluid Mech. 9, 321.Google Scholar
Smoluchowski, M. von 1921 Elektrische Endoosmose und Strömungslehre. In Handbuch der Elektrizität und des Magnetismus, vol. 2 (ed. L. Graetz). Leipzig: Barth.
Tikhomolova, K. P. 1993 Electro-osmosis. Ellis Horwood.
Wang, H. & Skalak, R. 1969 Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38, 75 (referred to herein as W & S).Google Scholar
Watson, G. N. 1944 Theory of Bessel Functions. Cambridge University Press.