Pacific Institute for the Mathematical Sciences - PIMS

Macrotransport processes in the pressure-driven flow of a concentrated, non-colloidal suspension

In this talk, I will discuss two examples of macrotransport analysis performed on the low Reynolds number, pressure-driven flow of a concentrated, non-Brownian suspension of rigid, spherical particles. The first part of the talk will consider the Taylor dispersivity of a solute being carried in the suspending fluid of a concentrated suspension. For suspensions, in addition to molecular diffusion, there are two other mechanisms influencing Taylor dispersion: shear-induced self diffusion, and secondary currents driven by second normal stress differences. As these two mechanisms become stronger, the Taylor dispersivity diminishes. A scaling analysis is used to demarcate parameter regimes where the three mechanisms can provide a dominant contribution to the Taylor dispersivity. In the regime where self-diffusion dominates, the Taylor dispersivity scales as UL3/a2, thus departing from the traditional U2L2/D scaling. Here, U is the characteristic velocity, L is a length scale in the conduit cross-section, a is the particle radius, and D is the solute diffusivity. The trends from scaling analysis are compared with numerical computations based on the suspension balance model of Nott and Brady (J. Fluid Mech. 1994) and the phase-averaged solute distribution equations of Zydney and Colton (Physicochem. Hydrodyn. 1988). Three different conduit cross-sections: the circle, the ellipse and the square, are examined to elucidate the effect of cross-sectional shape. The results will guide the design of conduit geometries and flow conditions that minimize or enhance axial solute dispersion and mixing in suspension flows through microfluidic geometries.

In the second part of the talk, I will present macrotransport equations that describe the evolution of the depth-averaged volume fraction phi of particles in tube and Hele-Shaw flow of suspensions. An interesting result is that the effective suspension velocity is a monotonically decreasing function of phi. For the tube geometry, this implies that positive volume fraction gradients in the flow direction will evolve to a steady state volume fraction distribution in an appropriate frame of reference. For the Hele-Shaw geometry, the calculation of the evolution of the volume fraction in time and space requires the solution of two coupled, 2-D PDEs, which can be implemented easily on any standard PDE solver. This represents a tremendous saving of computational cost in the determination of volume fraction distributions when compared to the full 3-D problem.

Location: ESB 2012