Question 10.2: We have noted various times throughout this book that the va...

We have noted various times throughout this book that the vascular endothelium responds to an altered wall shear stress \tau _{w} by increasing its production of, among other molecules, vasodilators and vasoconstrictors that induce a dilatation or constriction that restores \tau _{w} to its baseline value. Clearly, however, the endothelium cannot sustain arbitrarily large increases in wall shear stress; there must be a value at which the endothelium becomes damaged. Well before the discovery of endothelial mechanotransduction, D. L. Fry, a scientist at the National Institutes of Health at Bethesda, MD, showed in 1968 that aortic endothelial cells are damaged at values of \tau _{w} of 40 Pa and above (recall that normal values are \sim 1.5 Pa in arteries). This finding led to additional questions, such as whether a jet flow from a needle may likewise be able to damage, literally erode, the endothelium. (Note: The term used in the literature is denude, which means to lay bare.) A logical question, therefore, is: How do we design and interpret an experiment to quantify an “erosion” stress? Toward Control Volume and Semi-empirical Methods FIGURE 10.8 Experimental setup to determine the erosion stress for endothelial cells. Shown are a possible control volume for the fluid and a free-body diagram for the arterial segment. Although the fluid will exert shear stresses on the cells, we are interested primarily in the normal force R in this experiment. this end, consider the stress on the wall of the vessel created by the injection of a fluid through a needle. Find the erosion stress \sigma _{xx} on the endothelium by assuming a steady, incompressible flow with a volumetric flow rate of Q. Neglect gravity.