We develop a coarse-grained theory to predict the concentration distribution of a suspension of vesicles or red blood cells in a wall-bound Couette flow. This model balances the wall-induced hydrodynamic lift on deformable particles with the flux due to binary collisions, which we represent via a second-order kinetic master equation. Our theory predicts a depletion of particles near the channel wall (i.e., the Fahraeus-Lindqvist effect), followed by a near-wall formation of particle layers. We quantify the effect of channel height, viscosity ratio, and shear-rate on the cell-free layer thickness (i.e., the Fahraeus-Lindqvist effect). The results agree with in vitro experiments as well as boundary integral simulations of suspension flows. Lastly, we examine a new type of collective particle motion for red blood cells induced by hydrodynamic interactions near the wall. These “swapping trajectories,” coined by Zurita-Gotor et al [J. Fluid Mech. 592, 447–469 (2007)], could explain the origin of particle layering near the wall. The theory we describe represents a significant improvement in terms of time savings and predictive power over current large-scale numerical simulations of suspension flows.
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See supplementary material at http://dx.doi.org/10.1063/1.4810808 for (1) “Vesicle_lift_and_collision.xls” which lists the raw data we obtained for the lift velocity and binary collision displacements from our DNS simulations. The lift velocities are for ν = 0.95 vesicles at CaB = 8, and the collision data are for ν = 0.95 vesicles at CaB = 1. We find that the collisional displacements differ by less than 1% between CaB = 8 and CaB = 1 (data not shown), indicating that the collisional processes are relatively insensitive to capillary number for CaB > 1; (2) “RBC_lift_and_collision.xls” contains the raw data we obtained for the lift and collisional processes of red blood cells. We examine shear capillary numbers in between 0.25 < Cas < 2, and viscosity ratios in between 0.25 < λ < 3; (3) “coll_wall_h15_dz_1.avi”—This movie illustrates a binary collision between two red blood cells near a wall, where the two particles are initially separated by Δz = 1 in the wall-normal direction, with the lower particle at a height h = 1.5 above the wall. This movie illustrates a standard binary collision, where the wall does not induce the “swapping” effect mentioned in our paper; and (4) “coll_wall_h15_dz_025_v2.avi” movie illustrates the “swapping” effect of two red blood cells near a wall. The two particles are initially separated by Δz = 0.25 in the wall-normal direction, with the lower particle at a height h = 1.5 above the wall. After a long period of time, the lower particle eventually overtakes and moves above the upper particle.
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2013
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