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The dynamic mobility of particles in a non-dilute suspension

Published online by Cambridge University Press:  26 April 2006

P. F. Rider  and
R. W. O'Brien
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Abstract

When an alternating electric field is applied to a colloid the particles oscillate at a velocity proportional to the applied field strength. The complex constant of proportionality is termed the dynamic mobility (O'Brien 1988). Although this quantity can now be determined from electroacoustic measurements in suspensions of arbitrary concentration (O'Brien 1990), the theory for interpreting these measurements in terms of the size and charge of the particles is limited to dilute suspensions.

In this paper we derive an expression for the O(ϕ) correction to the dynamic mobility in a random suspension of uniform spheres with volume fraction ϕ. It is assumed that the particle radius is much greater than the double layer thickness but much smaller than the sound wavelength. The mobility is calculated using O'Brien's 1979 macroscopic boundary integral technique. This method ensures a correct mathematical formulation of the problem, and yields an absolutely convergent expression for the average particle velocity. The evaluation of this expression to O(ϕ) involves the determination of the velocities of an isolated pair of particles at various separations and frequencies of oscillation. These velocities are computed using the collocation technique and the O(ϕ) correction to the dynamic mobility is then obtained by numerically integrating over all particle separations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 257 , December 1993 , pp. 607 - 636
Copyright
© 1993 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1970 The stress distribution in a suspension of force free particles. J. Fluid Mech. 41, 545.Google Scholar
Batchelor, G. K. 1974 Transport properties of two-phase materials with random structure. Ann. Rev. Fluid Mech. 6, 227.Google Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian Dynamics. Ann. Rev. Fluid Mech. 20, 111.Google Scholar
Chen, S. B. & Keh, H. J. 1988 Electrophoresis in a dilute suspension of colloidal spheres. AIChE J. 34, 1075.Google Scholar
Clercx, H. J. H. 1991 The dependence of transport coefficients of suspensions on quasistatic and retarded hydrodynamic interactions. PhD Thesis, Eindhoven University of Technology.
Gluckman, M. J., Pfeffer, R. & Weinbaum, S. 1971 A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids. J. Fluid Mech. 50, 705.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1966 The slow motion of two identical arbitrarily oriented spheres through a viscous fluid. Chem. Engng Sci. 21, 1151.Google Scholar
Hunter, R. J. 1987 Foundations of Colloid Science, vol. 1, Clarendon Press.
James, R. O., Texter, J. & Scales, P. J. 1991 Frequency dependence of electroacoustic (electrophoretic) mobilities. Langmuir, 7, 1993.Google 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. Lond. A 97, 109.Google Scholar
Jeffrey, D. J. 1976 Appendix to ‘Hydrodynamic interaction between gas bubbles in liquid’ by L. van Wijngaarden. J. Fluid Mech. 77, 27.Google Scholar
Kim, S. & Russel, W. B. 1985 The hydrodynamic interaction between two spheres in a Brinkman medium. J. Fluid Mech. 154, 34.Google Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.
Lawrence, C. J. & Weinbaum, S. 1986 The force on an axisymmetric body in linearized, time-dependent motion. J. Fluid Mech. 171, 209.Google Scholar
Loewenberg, M. & O'Brien, R. W. 1992 The dynamic mobility of non-spherical particles. J. Colloid Interface Sci. 150, 158.Google Scholar
Maxwell, J. C. 1873 Electricity and Magnetism, 1st edn. Clarendon Press.
Morrison, F. A. 1970 Electrophoresis of a particle of arbitrary shape. J. Colloid Interface Sci. 34, 210.Google Scholar
O'Brien, R. W. 1979 A method for calculating the effective transport properties of suspensions of interacting particles. J. Fluid Mech. 91, 17–39.Google Scholar
O'Brien, R. W. 1986 The high frequency dielectric dispersion of a colloid. J. Colloid Interface Sci. 113, 81.Google Scholar
O'Brien, R. W. 1988 Electroacoustic effects in a dilute suspension of spherical particles. J. Fluid Mech. 190, 71.Google Scholar
O'Brien, R. W. 1990 The electroacoustic equations for a colloidal suspension. J. Fluid Mech. 212, 81.Google Scholar
Oja, T., Petersen, G. L. & Cannon, D. W. 1985 A method for measuring the electrokinetic properties of a solution. United States Patent no. 449, 7, 207.
Pozrikidis, C. 1989 A singularity method for unsteady linearized flow. Phys. Fluids A 1 (9), 1508.Google Scholar
Reed, L. D. & Morrison, F. A. 1976 Hydrodynamic interactions in electrophoresis. J. Colloid Interface Sci. 54, 117.Google Scholar
Stimson, M. & Jeffery, G. B. 1926 The motion of two spheres in a viscous fluid. Proc. R. Soc. Lond. A 111, 110.Google Scholar
Williams, W. E. 1966 A note on slow vibrations in a viscous fluid. J. Fluid Mech. 25, 589.Google Scholar