Cellular flow in a small blood vessel

In the smallest capillaries, or in tubes with diameter D less than or similar to 8 mu m, flowing red blood cells are well known to organize into single-file trains, with each cell deformed into an approximately static bullet-like shape. Detailed high-fidelity simulations are used to investigate flow in a model blood vessel with diameter slightly larger than this: D = 11.3 mu m. In this case, the cells deviate from this single-file arrangement, deforming continuously and significantly. At the higher shear rates simulated ( mean velocity divided by diameter U/D greater than or similar to 50 s(-1)), the effective tube viscosity is shear-rate insensitive with mu(eff)/mu(plasma) = 1.21. This matches well with the value mu(eff)/mu(plasma) = 1.19 predicted for the same 30% cell volume fraction by an established empirical fit of high-shear-rate in vitro experimental data. At lower shear rates, the effective viscosity increases, reaching mu(eff)/mu(plasma) approximate to 1.65 at the lowest shear rate simulated ( U/D approximate to 3.7 s(-1)). Because of the continuous deformations, the cell-interior viscosity is potentially important for vessels of this size. However, most results for simulations with cell interior viscosity five times that of the plasma (lambda = 5), which is thought to be close to physiological conditions, closely match results for cases with lambda = 1. The cell-free layer that forms along the vessel walls thickens from 0.3 mu m for U/D = 3.7 s(-1) up to 1.2 mu m for U/D greater than or similar to 100 s(-1), in reasonable agreement with reported experimental results. The thickness of this cell-free layer is the key factor governing the overall flow resistance, and this in turn is shown to follow a trend expected for lubrication lift forces for shear rates between U/D approximate to 8 s(-1) and U/D approximate to 100 s(-1). Only in this same range are the cells near the vessel wall on average inclined relative to the wall, as might be expected for a lubrication mechanism. Metrics are developed to quantify the kinematics of this dense cellular flow in terms of the well-known tank-treading and tumbling behaviours often observed for isolated cells in shear flows. One notable effect of lambda = 5 versus lambda = 1 is that it suppresses treading rotation rates by a factor of about 2. The treading rate is found to scale with the velocity difference across the cell-rich core and is thus significantly slower than the overall shear rate in the flow, which is presumably why the flow is otherwise insensitive to lambda. The cells in all cases also have a similarly slow mean tumbling motion, which is insensitive to cell-interior viscosity and decreases monotonically with increasing U/D.