Let's analyze the problem step by step to find the average velocity in the peripheral (annular) region. --- ## Given Data: - Hematocrit \( H = 40\% \) - Fibrinogen concentration \( C_f = .3\,g/100\,mL \) - Radius of artery \( R = 4\,mm = .4\,cm \) - Pressure gradient \( \Delta P / L = 6\,dyne/cm^3 \) - Total flow rate \( Q_{total} = 95.5\,cm^3/min \) - Max velocity \( V_{max} = 5.75\,cm/sec \) --- ## Step 1: Understand the flow model The **Casson model** relates shear stress and viscosity with the equation: \[ \sqrt{\tau} = \sqrt{\tau_y} + K_c \sqrt{\dot{\gamma}} \] where: - \(\tau\) is shear stress, - \(\tau_y\) is yield stress, - \(K_c\) is the Casson coefficient, - \(\dot{\gamma}\) is shear rate. The problem provides a simplified form: \[ \tau = \frac{R(\Delta P / L)}{2} \] which relates shear stress at the wall to the pressure gradient and radius. --- ## Step 2: Calculate shear stress at the wall Using the given pressure gradient: \[ \tau_{wall} = \frac{R (\Delta P / L)}{2} \] Plug in the values: \[ \tau_{wall} = \frac{.4\,cm \times 6\,dyne/cm^3}{2} = .4 \times 3 = 1.2\,dyne/cm^2 \] --- ## Step 3: Determine the flow profile and velocity The maximum velocity occurs at the center and is given as: \[ V_{max} = 5.75\,cm/sec \] In laminar flow, the velocity profile in a cylindrical vessel is parabolic, but here, with the Casson model, the flow is described differently considering yield stress and viscosity. --- ## Step 4: Find the average velocity The key is relating the total flow rate \(Q\) to the velocity profile: \[ Q = \int_{A} V(r) dA \] In the case of a flow with a known maximum velocity \(V_{max}\), the average velocity \(V_{avg}\) in laminar flow is typically: \[ V_{avg} = \frac{Q}{A} \] and for a parabolic profile: \[ V_{avg} = \frac{V_{max}}{2} \] Given: \[ V_{max} = 5.\,cm/sec \] we estimate: \[ V_{avg} \approx \frac{V_{max}}{2} = \frac{5.75}{2} = 2.875\,cm/sec \] --- ## **Final answer:** **The average velocity in the peripheral (annular) region is approximately \(\boxed{2.88\,cm/sec}\).**