When the blood microvessel dimensions approach the dimensions of the red cell, the red cell squeezes through capillaries and the viscosity increases. To model this and predict an increase in viscosity, we will assume that the red cells pass through the capillaries as a continuous train equal to the capillary length L. The cells have a radius Re. The plasma flows in a thin gap between the cell and the vessel wall (Figure 2.43). The gap thickness is 8=R-Rc and the viscosity is u. Because the gap is so thin relative to the vessel wall thickness, assume that the velocity profile is linear and equals Vey/8 where y=R-r and 8 =R-RC. |RR V Ve FIGURE 2.43 Schematic of flow between a red blood cell and the capillary wall. a) (6%) Determine an expression for the shear stress acting on the cells if the velocity in the gap can be approximated as ve=V_y/ 8. b) (6%) Perform a force balance on the cell to relate the pressure and shear stresses. Use this result to relate the cell velocity to Ap/L. c) (6%) Relate the average velocity to the cell velocity. (Note, the gaps thickness is not negligible). d) (6%) Use the results from parts (b) and (c) to derive an expression for the effective (apparent) viscosity if the red cell suspension plepy by matching the red cell velocity to the mean velocity for Poiseuille flow in a tube. R2 AP (v) (2.10.8) 8Meff L e) (6%) Show that the ratio Mett/increases as the ratio o/R increases for S/R = 0.1, 0.2, 0.3 and 0.4