(132g) The Taylor Dispersivity of a Passive Tracer in the Pressure-Driven Flow of a Concentrated Suspension of Rigid, Non-Colloidal Spheres

It is well known that the axial spreading of a slug of passive tracer in a Newtonian fluid flowing in a conduit under laminar conditions is governed by a Taylor dispersivity¹, which scales as U²B²/D. Here U is the average velocity of the fluid in the conduit, B is the characteristic depth of the conduit, and D is the molecular diffusivity. The Taylor dispersivity is directly proportional to characteristic equilibration time, t_c, of the solute concentration inhomogeneities over the cross-section. The scaling of the Taylor dispersivity may be viewed as U (Ut_c), where t_c is the diffusion time scale over the characteristic depth of the channel, B²/ D. The shorter this time scale, the larger the Taylor dispersivity, and the faster the axial spreading of the solute.

In this work, the Taylor dispersivity of a solute being carried in the suspending fluid of a concentrated suspension of rigid, non-colloidal particles is analyzed. The model for solute transport employs the phase-averaging approach adopted by Zydney and Colton², but incorporates a tensorial self-diffusion tensor based on experimental measurements. Particle migration effects are included via the suspension balance model of Nott and Brady³ along with the constitutive equations of Zarraga and Leighton⁴.

For suspensions, in addition to molecular diffusion, there are two more mechanisms influencing the characteristic equilibration time, t_c: shear-induced self-diffusion⁵, and secondary currents driven by second normal stress differences. As the effects of shear-induced self diffusion and secondary currents become stronger, t_c, and therefore the Taylor dispersivity are decreased. Regimes where each of the three mechanisms can provide a dominant contribution to the Taylor dispersivity are identified in terms of three dimensionless parameters: B²/a²g(φ), kδ(φ) and Pe=UB/D. Here a is the particle radius, g(φ) represents the dependence of shear-induced migration rate on particle volume fraction φ, δ(φ) is the reduced second normal stress difference N₂/τ, Pe is the Peclet number, and k is a constant related to the shape of the cross-section. The scaling for Taylor dispersivity is established in each regime, and is verified with numerical computations.

The theoretical results developed in this work will allow us to interpret experimental measurements of Taylor dispersivity to deduce the magnitudes of shear-induced solute self- diffusivities at various volume fractions, and to measure the contribution of secondary currents in solute transport. This work is a step towards a more general understanding of how solute is transported in more complicated suspensions, such as blood.

References

1. G. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. Lond. A 219, 186-203 (1953).


2. A. L. Zydney and C. K. Colton, Augmented solute transport in the shear flow of a concentrated suspension, Physicochem. Hyd. 10, 77?96 (1988).


3. P. R. Nott and J. F. Brady, Pressure-driven flow of suspensions: simulation and theory, J. Fluid Mech 275, 157-199 (1994).


4. I. E. Zarraga, D. A. Hill and D. T. Leighton, The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids., J. Rheol 181, 415-439 (1987).


5. E. C. Eckstein, D. G. Bailey and A. H. Shapiro, Self diffusion of particles in shear flow of a suspension, J. Fluid Mech. 79 191?208 (1976).