Consequences of Nonlmearity. The relationship between the flow F (say, in cubic

Question

Consequences of Nonlmearity. The relationship between the flow F (say, in cubic centimeters per second) of a liquid through a cylindrical tube of radius r and length L is given by the Hagen-Poiseuille equation F = DeltaP/etaL r^4, where eta is the viscocity of the fluid and DeltaP is the pressure difference in the tube. What are the consequences of this nonlinear relationship between flow and radius? Putting it Together: Consider the flow of blood with viscocity eta through an artery, which we model as a cylindrical tube of radius r, length L and a pressure difference DeltaP produced by the beating heart. Writing k = DeltaP/etaL, the Hagen-Poiseuille equation becomes F(r) = kr^4. 1. The radius of a typical open artery is 1.5 mm. In the "mild" case of CAD, plaque may line the walls so that it reduces the radius of the open lumen of this occluded artery to 0.75 mm. Assuming that the other variables affecting blood flow remain the same (that is, k in equation (2) remains the same), use equation (2) to determine the ratio of blood flow in the 50%-occluded artery vs. the open artery. Show your calculations.

Explanation / Answer

Let the flow in artery of radius 0.75 mm be F0.75

The flow in the artery of radius 1.5 mm will be F1.50

=> F0.75 = k( 0.75 )4 and

=> F1.50 = k( 1.50 )4

=> F0.75 / F1.50 = ( 0.75 )4 / ( 1.50 )4

=> F0.75 / F1.50 = 1 / 16

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