Whole blood consists of a concentrated suspension of cells in plasma. The cells are mainly red blood cells (RBCs), which are flexible, rounded disks about 2 μm thick and 8 μm in diameter. Their volume fraction ϕ, called the hematocrit, is normally about 0.45. Plasma is an aqueous solution containing electrolytes, organic solutes of low to moderate molecular weight, and proteins. It is Newtonian and has a viscosity μ_p that is approximately 1.6 times that of water. When fibrinogen (a protein involved in clotting) is present, the RBCs tend to aggregate, causing blood to have a yield stress τ₀ that is typically about 0.004 Pa. In tubes with diameters of at least a few hundred μm, blood can be modeled as a homogeneous liquid. It is represented well as a Casson fluid, a generalized Newtonian fluid for which

The parameter μ0 is the apparent viscosity at high shear rates (Merrill, 1969), given by

(a) For fully developed flow in a tube with τ > τ₀, show that the Casson equation reduces to

(b) Determine v_z(r) for blood in a tube of radius R. In the momentum equation it is convenient to use the wall shear stress τ_w instead of 

The velocity can be expressed most easily in term of a dimensionless radial position η = r/R and the parameters v_w = τ_wR/μ₀ and v₀ = τ₀R/μ₀, both of which have the dimension of velocity.

(c) Time-averaged shear rates at the walls of human blood vessels range from about 50 s^–1 in large veins to 700 s^–1 in large arteries. In which type of large vessel will the yield stress be more noticeable? Plot v_z(η)/v_z(0) for both in comparison with the result for a Newtonian fluid. You may assume that τ_w equals μ₀ times the given shear rate.

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∞, τα το και του μ 10 [1+ τ> το Στο ΜοΓ 1/2 + το 2μ.Γ (P7.17-1)