(519f) A Macrotransport Equation for the Hele-Shaw Flow of Concentrated Suspensions of Non-Colloidal Spheres in Newtonian Fluids

A two time-scale perturbation expansion of the suspension balance model¹ coupled with the constitutive equations of Zarraga et al.² is employed to formally derive the depth-averaged convection-dispersion equation for the flow of a concentrated suspension of neutrally-buoyant, non-colloidal particles between two parallel plates.   The Taylor-dispersion coefficient D_eff in this macrotransport equation scales as h(f) U b³ / a²; here, U is the characteristic velocity scale, b is the half depth of the channel, a is the particle radius and h is a monotonically increasing function of the particle volume fraction f.  Analogous to the macrotransport equation for tube suspension flow³, the evolution of concentration distribution is dependent only on the total strain experienced by the suspension, and is independent of the suspension velocity.  However, unlike tube suspension flow, a positive concentration gradient along the flow direction does not reach an asymptotic distribution; rather, it is susceptible to miscible viscous fingering.  A linear stability analysis of the viscous fingering phenomenon based on the macrotransport equation is presented.

References

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

2. I. E. Zarraga, D. A. Hill and D. T. Leighton Jr., “The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids”, J. Rheol. 44, 185-220 (2000).

3.  A. Ramachandran, “A macrotransport equation for the particle distribution in the flow of a concentrated, non-colloidal suspension through a circular tube”, J. Fluid Mech. 734, 219-252 (2013).