Journal of Fluid Mechanics

Diffusion in a dilute polydisperse system of interacting spheres

G. K.  Batchelor a1
a1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge

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When m different species of small particles are dispersed in fluid the existence of a (small) spatial gradient of concentration of particles of type j is accompanied, as a consequence of Brownian motion of the particles, by a flux of particles of type i. The flux and the gradient are linearly related, and the tensor diffusivity Dij is the proportionality constant. When the total volume fraction of the particles is small, Dij is approximately a linear function of the volume fractions φ1, φ2, …s, φm, with coefficients which depend on the interactions between pairs of particles. The complete analytical expressions for these coefficients given here for the case of spherical particles are a linear combination of the second virial coefficient for the osmotic pressure of the dispersion (measuring the effective force acting on particles when there is a unit concentration gradient) and an analogous virial coefficient for the bulk mobility of the particles. Extensive calculations of the average velocities of the different species of spherical particles in a sedimenting polydisperse system have recently been published (Batchelor & Wen 1982) and some of the results given there (viz. those for small Péclet number of the relative motion of particles) refer in effect to the bulk mobilities wanted for the diffusion problem. It is thus possible to obtain numerical values of the coefficient of φk in the expression for Dij, as a function of the ratios of the radii of the spherical particles of types i, j and k. The numerical values for ‘hard’ spheres are found to be fitted closely by simple analytical expressions for the diffusivity; see (4.6) and (4.7). The dependence of the diffusivity on an interparticle force representing the combined action of van der Waals attraction and Coulomb repulsion in a simplified way is also investigated numerically for two species of particles of the same size. The diffusivity of a tracer particle in a dispersion of different particles is one of the many special cases for which numerical results are given; and the result for a tracer ‘hard’ sphere of the same size as the other particles is compared with that found by Jones & Burfield (1982) using a quite different approach.

(Published Online April 20 2006)
(Received August 30 1982)