Quantifying the transition of blood flow to the non-continuum regime

Blood flow is usually treated as a Newtonian fluid down to diameters of about $200$ $\mu m$. We employ the dissipative particle dynamics to simulate the flow of red blood cell suspensions driven through small tubes (diameters $10-150$ $\mu m$) in the range marking the transition from venules to the large capillaries. Simulation results show that for diameters less than about $100$ $\mu m$ the suspension's stress cannot be described as a continuum, even a heterogeneous one. In tube flow the cross-stream stress gradient induces an inhomogeneous distribution of RBCs featuring a centerline cell density peak, and a cell-free layer next to the wall. The local viscosity across the section as a function of the strain rate is found to be essentially independent of tube size for the larger diameters and is determined by the local hematocrit ($H$) and shear rate. As the tube diameter decreases below about $100$ $\mu m$, the viscosity in the central region departs from the large-tube similarity function of the shear rate, since $H$ increases significantly towards the centerline. The dependence of shear stress on tube size, in addition to the expected local shear rate and local hematocrit, implies that blood flow in small tubes cannot be described as a heterogeneous continuum.