Human blood behaves as a Newtonian fluid (see Prob. 20.51) in the high shear rate region where γ > 100. In the low shear rate region where γ < 50, the red cells lend to aggregate into what are called rouleaux, which make the fluid behavior depart from Newtonian. This low shear rate region is called the Casson region, and there is a transition region between the two distinct flow regions. In the Casson region as shear rate approaches zero, the shear stress goes to a finite value, similar to a Bingham plastic, which is called the yield stress, τ_y, and this stress must be overcome in order to initiate flow in stagnate blood. Flow in the Casson region is usually plotted as the square root of shear rate verses the square root of shear stress, and follows a straight line relationship when plotted in this way. The Casson relationship is

√τ = √τ_y + K_c√y

Where K_c = consistency Index. In the table below are experimentally measured valises of γ and τ_y from a single blood sample over the Casson and Newtonian flow regions.

Find the values of K_c. and τ_y using linear regression in the Casnon region, and find μ by using linear regression in the Newtonian region. Also find the correlation coefficient for each regression analysis. Plot the two regression lines on a Casson plot (√γ versus √τ) and extend the regression lines as dashed line into adjoining regions; also include the data pointes in the plot. Limit the shear rate region to 0 < √γ < 15.

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, 1/s | 0.91 3.3 T, N/m? 0.059 0.15 0.19 0.27 0.39 0.87 1.33 1.65 2.11 3.44 4.12 7.02 10.21 13.01 Region 36 4.1 105 126 215 23 6.3 9.6 49 65 402 315 Tronsilion Newlonian Casson