256573 A Macrotransport Equation for the Flow of a Concentrated Suspension of Non-Colloidal Particles Through a Capillary Tube
By implementing a two time-scale perturbation expansion of the suspension balance model1, we formally derive a cross-section-averaged convection-dispersion equation for the flow of a concentrated suspension of neutrally buoyant, non-colloidal particles through a tube. The Taylor-dispersion coefficient Deff in this macrotransport equation scales as h(Φ)UR3/a2; here U is the velocity of the suspension in the tube, R is the tube radius, a is the particle radius and h is a monotonically increasing function of the particle volume fraction Φ. The linear dependence of Deff on U implies that changes in the cross-section averaged axial concentration profile are dependent only on the total axial strain experienced by the suspension. This stipulates that the spatial evolution of the mixing length scale for a given fluctuation in the inlet concentration of the suspension, or the spatial evolution of the width of the mixing zone between two suspensions of different concentrations in the tube will be independent of the suspension velocity in the tube. The result contrasts from the quadratic dependence on the velocity that is expected from the classical problem of Taylor dispersion of a passive tracer in the same geometry, which results in faster dispersion at higher average velocities. A second major point of difference between the two problems is that the effective velocity of the particulate phase is concentration-dependent, and this can lead to either sharpening or relaxation of concentration gradients. Concentration pulses relax asymmetrically, and positive concentration gradients along the flow direction can lead to time-independent distributions in an appropriately chosen frame of reference; both effects are not observed in solutal dispersion. The results in this paper are relevant for microchannel flows, and can be used to design geometries that minimize or enhance mixing in suspension flows through microchannels.
1. P. R. Nott and J. F. Brady, Pressure-driven flow of suspensions: simulation and theory, J. Fluid Mech, 275, 157-199 (1994).
2. D. Leighton and A. Acrivos, The shear-induced migration of particles in concentrated suspensions, J. Fluid Mech, 181, 415-439 (1987).