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Coarse-grained theory to predict the concentration distribution of red blood cells in wall-bounded Couette flow at zero Reynolds number
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10.1063/1.4810808
/content/aip/journal/pof2/25/6/10.1063/1.4810808
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/6/10.1063/1.4810808
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Example concentration profiles of deformable particles. (a) Volume fraction profile of red blood cells in Poiseuille flow (from Zhao ). (b) Density profile of red blood cells in Couette flow. We note that the concentration profiles exhibit a depletion layer at the wall (the Fahraeus-Lindqvist effect), followed by near-wall particle layering. To see examples of layering for other types of deformable particles, see Li and Pozrikidis for droplets, and the following authors for capsules: Pranay, Chia, and Tan. Figure 1(a) is reprinted with permission from H. Zhao , Phys. Fluids , 011902 (2012). Copyright 2012 American Institute of Physics.

Image of FIG. 2.
FIG. 2.

DNS simulation of lift velocity of deformable particles near a wall. (a) Simulation domain. (b) Lift velocity vs. height for nearly spherical vesicles (ν = 0.95) at capillary number and various viscosity ratios. (c) Comparison of the far-field (/ ≫ 1) lift velocities with those from vesicle experiments for a wide variety of vesicle shapes (i.e., reduced volumes). Reprinted with permission from H. Zhao , Phys. Fluids , 121901 (2011). Copyright 2011 American Institute of Physics.

Image of FIG. 3.
FIG. 3.

DNS simulation of binary collision process. (a) Simulation domain. Two particles start at the configuration shown above, and we track the final displacement of the top particle in the shear flow direction after collision. (b) Typical plot of δ vs. Δ for nearly spherical vesicles (ν = 0.95) in the same vorticity plane (i.e., Δ = 0). The capillary number is , and we examine several viscosity ratios. (c) Typical trajectory of a vesicle during a collisional process. Dots are experimental data from Kantsler , while the lines are from our boundary integral simulations. Reprinted with permission from H. Zhao and E. S. G. Shaqfeh, J. Fluid Mech. , 709–731 (2013). Copyright 2013 Cambridge University Press.

Image of FIG. 4.
FIG. 4.

DNS Simulation of suspensions in wall bounded shear flow. (a) Simulation domain. (b) Snapshots of ν = 0.95 vesicle suspensions at fixed capillary number ( = 8) and different viscosity ratios (λ = 1, 8, and 14).

Image of FIG. 5.
FIG. 5.

Concentration profiles of nearly spherical vesicles, as computed by DNS. The vesicles have a reduced volume ν = 0.95, the suspension is at 20% volume fraction, and the capillary number is = 8. (a) Height = 12 and (b) Height = 18. We examine several viscosity ratios for each simulation.

Image of FIG. 6.
FIG. 6.

Comparison between theory and simulation for vesicle suspensions The suspensions are at 20% volume fraction, and at matched viscosity ratio (λ = 1). We plot several curves for our theory, each one representing a different hydrodynamic screening radius (see Sec. II B for more details). (a) Height = 12 and (b) Height = 18.

Image of FIG. 7.
FIG. 7.

Comparison between theory and DNS simulations, red blood cell concentration profiles. We compare concentration profiles in wall-bounded Couette flow at two volume fractions (ϕ = 0.1 and 0.2), matched viscosity ratio (λ = 1), and unit capillary number ( = 1). (a) Height = 12 and (b) Height = 18.

Image of FIG. 8.
FIG. 8.

Lift velocity and binary collisions, red blood cells, and nearly spherical vesicles. We plot the lift velocity and binary collisions for ν = 0.95 vesicles and red blood cells at matched viscosity ratio (λ = 1). (a) Lift velocity vs. height. The capillary numbers are = 8 for the vesicles, and = 1 for the red blood cells. (b) Collisional displacement vs. δ. The collisions are in the same vorticity plane (i.e., Δ = 0). The capillary numbers are = 1 for the vesicles, and = 1 for the red blood cells.

Image of FIG. 9.
FIG. 9.

Definition of cell-free layer and inclination angle. (a) Schematic of cell-free layer, which is the minimum distance from the wall to the edge of the first layer of cells. (b) Schematic of inclination angle. (c) Plot of inclination angle versus viscosity ratio for a single red blood cell in free shear flow. The error bars represent the minimum and maximum of the inclination angle during the tank-treading motion.

Image of FIG. 10.
FIG. 10.

Comparison between theory and experiments: CFL vs. Hematocrit. (a) Comparison with Bugliarello In the experiment, the flow capillary number varies from 0.4 to 1.6, and the channel height is = 14.3. The plot from the coarse-grained theory is at = 1, and channel height = 14. The error bars in our theory represent different values of the hydrodynamic screening length (2.25 < < 2.75). To convert the experimental feed hematocrit to tube hematocrits, we utilize a correlation developed by Pries (b) Comparison of experimental data to the scaling relation as predicted by our coarse-grained theory.

Image of FIG. 11.
FIG. 11.

Comparison between theory and experiments: CFL vs channel height. Experimental data are from Suzuki In the experiment, the flowrate through the capillaries is 0.7 − 1.3/, corresponding to between 0.06 and 0.1 for the largest capillaries, and between 0.28 and 0.56 for the smallest capillaries. The discharge hematocrit is 16% . The plots from our theory are at = 0.5, with the hydrodynamic screening length varying in between 2.25 < < 2.75. Like before, we convert the experimental feed hematocrits to tube hematocrits via a correlation developed by Pries

Image of FIG. 12.
FIG. 12.

Cell free layer vs capillary number for red blood cells. (a) Cell-free layer vs capillary number at 10% hematocrit (ϕ = 0.1). The hydrodynamic screening length is 2.25 < < 2.75, and the channel height is = 12. (b) Collisional displacements for red blood cells. We plot the displacement per collision (δ) versus the separation of particles in the shear-gradient direction (). The two particles are in the same vorticity plane (i.e., Δ = 0). (c) Lift velocity vs. height above the wall. For all three plots, the blood cells are at matched viscosity ratio (λ = 1). The capillary numbers we examine are 0.25 < < 2, which correspond to shear rates of .

Image of FIG. 13.
FIG. 13.

Cell free layer vs. viscosity ratio for red blood cells. (a) Cell-free layer vs. viscosity ratio at 10% hematocrit (ϕ = 0.1). The hydrodynamic screening length is 2.25 < < 2.75, and the channel height is = 12. (b) Collisional displacements for red blood cells. We plot the displacement per collision (δ) vs. the separation of particles in the shear-gradient direction (). The two particles are in the same vorticity plane (i.e., Δ = 0). (c) Lift velocity vs. height above the wall. For all three plots, the blood cells are at unit capillary number ( = 1). The viscosity ratios we examine are 0.25 < λ < 3.

Image of FIG. 14.
FIG. 14.

Binary collisional displacements in the presence of a wall. (a) Simulation geometry. Two particles start at the configuration shown above, and we track their trajectories during a collision event. (b) Trajectory of two red blood cells when = 1.5, Δ = 1, and Δ = 0. The solid lines are the trajectory during the collision, while the dotted lines are the trajectories in the absence of the other particle. After a long period of time, the distance between the dotted and solid curves asymptotes to a constant value, which we denote as the displacement per collision. (c) Displacement per collisions for red blood cells as a function of the initial separation of the two particles, for the cases of no wall and wall. The collisions are in the same vorticity plane (i.e., Δ = 0), and the initial height of the lower particle is = 1.5. For all graphs, the red blood cells are at matched viscosity ratio (λ = 1) as well as unit capillary number ( = 1).

Image of FIG. 15.
FIG. 15.

Swapping trajectories in the presence of a wall. (a) Schematic of swapping motion. We denote particle 1 as the upper particle at time = 0, and particle 2 as the lower particle at time = 0. After a long period of time, particle 2 moves above particle 1. (b) Comparison between “swapping trajectories” and standard binary collisions. We plot the relative position of the particle 1 with respect to the center of mass of particle 2. Particle 2 is at a height = 1.5 above the wall, and the collisions are in the same vorticity plane (Δ = 0). When Δ = 1.0, we observe a standard binary collision, but when Δ = 0.25, we observe swapping trajectories. (c) Phase diagram of “swapping trajectories.” All collisions are in the same vorticity plane (Δ = 0). For plots (b) and (c), the red blood cells are at matched viscosity ratio (λ = 1) and unit capillary number ( = 1).

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/content/aip/journal/pof2/25/6/10.1063/1.4810808
2013-06-24
2014-04-21
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Coarse-grained theory to predict the concentration distribution of red blood cells in wall-bounded Couette flow at zero Reynolds number
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/6/10.1063/1.4810808
10.1063/1.4810808
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