(142ba) Augmentation of Mass Transport of Solutes in Flowing Suspensions Due to Secondary Currents Driven by Second Normal Stress Differences

It has been long known that the diffusivity of solutes in suspensions of both rigid and deformable particles is enhanced by shear-induced self diffusion^1,2.  The augmentation of the solutal diffusivity, which is defined as D_eff/D₀-1, scales as  g(Φ)Ua²/BD₀, where D_eff is the observed diffusivity, D₀ is the molecular diffusivity, B is a characteristic dimension of the cross-section, a is the particle radius and gis a monotonically increasing function of the volume fraction Φ. 

For suspensions of rigid particles flowing through conduits of non-axisymmetric cross-sections, it was recently demonstrated³ that the flow is not unidirectional; the main flow along the axis is accompanied by a secondary flow within the cross-section driven by second normal stress differences (N₂).  The magnitude of the secondary currents scales as kδ(Φ)U, where k is a constant that depends on the degree of non-axisymmetry of the geometry, and δ is proportional to N₂, which is an increasing function of Φ.   Notably, it is independent of the particle radius (for non-Brownian suspensions only).

In this work, we demonstrate that N₂-driven secondary currents represent a new, convective mechanism for mass transport in flowing suspensions.  The characteristic Peclet number for the secondary current convection is kδ UB/(D₀+ gUa²/B).  It can be seen that when the particle size is small enough to render the enhancement of diffusivity by shear-induced self-diffusion weak, the Peclet number can still be large, since the magnitude of the secondary currents does not depend on particle size.  Therefore, in this limit, secondary convection, and not the traditionally-considered shear-induced self-diffusion, could be the dominant mechanism for enhancement of mass transfer.  For large flow velocities, when shear-induced self-diffusion is the dominant diffusive mechanism for mass transfer, the Peclet number reduces to kδB²/ ga², independent of the suspension velocity.   We find that for moderately concentrated suspensions, the Peclet number in this limit can be much larger than 1, because of the large value of the aspect ratio B/ ain most practical suspension flows.  Thus, secondary convection can provide additional enhancement of mass transfer over that due to shear-induced self-diffusion, and the relative augmentation can be by as much as a factor of 4 for some realistic examples.  However, it must be stated that since the streamlines associated with secondary currents are closed, the enhancement of the mass transfer rate by convection asymptotes to a constant for high Peclet numbers.

The practical relevance of this new mechanism of mass transfer is in the transport of large molecular weight solute molecules in blood flow.  Such molecules are associated with small diffusivities, and are associated with significant concentration gradients within blood vessels (higher mass transfer resistances) during their transport from blood into tissues.  Our model and calculations will permit improved modeling of the mass transfer of such solutes.

References

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

2.  W. Cha and R. L. Beissinger, Augmented solute transport to surfaces in sheared suspensions. J. Colloid Interface Sci. 178,  1–9 (1996).

3.  A. Ramachandran and D. T. Leighton, The influence of secondary flows induced by normal stress differences on the shear-induced migration of particles in concentrated suspensions, J. Fluid Mech. 603, 207-243 (2008).