3 3 35 points a stenosis is a narrowing of a blood

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3 3. (35 points) A stenosis is a narrowing of a blood vessel or valve. Stenosis of a blood vessel arises druing artherosclerosis and could occlude an artery, depriving the tissue downstream of oxygen. Further, the fluid shear stresses acting on the endothelial cells lining blood vessels mayh affect the expression of genes that regulate endothelial cell function. Such gene expression can influence whether the stenosis grows or not. Consider a symmetric stenosis as shown in the figure below. Assume that the velocity profile within the stenosis of radius Ri(z) has the same shape as the profile outside the stenosis and is represented as 𝑣𝑧𝑟 =𝑣?𝑎𝑥12?2Outside the stenosis, the radius equals Roand the maximum velocity is constant. Within the stenosis, the radius of the fluid channel R(z) equals ? 𝑧 =?010.514𝑧21/2The origin of the z axis is the midpoint of the stenosis. (a) Develop an expression for vmaxin a stenosis in terms of the volumetric flow rate Q, cylindrical tube radius Ro, and the distance along the stenosis 𝐿. Hint: You might want to begin with ?=𝑣 ∙ ? 𝑑𝐴=𝑣𝑧?02𝜋0𝑟𝑑𝑟𝑑𝜃𝑟𝐿𝑧.
4 (b) Compute the shear stress acting on the surface of the stenosis (r = Ri) at z=0 relative to the valve outside the stenosis. Hint: We can assume that the shear rate of blood is above 100 s-1in, and that in the stenosis, 21/2= 0.5?
5 4. (15 points) Flow separation and reversal are important aspects of fluid flow in small tubes like blood vessels. In the space provided below, please sketch the velocity profile for the conditions described. (a) Considering flow in a tube, please sketch the velocity profile for normal parabolic flow(you may ignore gravity). (b) Considering flow in a tube, please sketch the velocity profile for flow separation(you may ignore gravity). (c) Considering flow in a tube, please sketch the velocity profile for flow reversal(you may ignore gravity).

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