Flow in a cylinder Poiseuille’s Law is a fundamental law

QUESTION:

Flow in a cylinder Poiseuille's Law is a fundamental law of fluid dynamics that describes the flow velocity of a viscous incompressible fluid in a cylinder (it is used to model blood flow through veins and arteries). It says that in a cylinder of radius R and length L, the velocity of the fluid r \(\leq\) R units from the centerline of the cylinder is \(V=\frac{P}{4 L \nu}\left(R^{2}-r^{2}\right)\), where P is the difference in the pressure between the ends of the cylinder and vis the viscosity of the fluid (see figure). Assuming that P and v are constant, the velocity V along the centerline of the cylinder (r = 0) is V = k\(R^{2}\)/L, where k is a constant that we will take to be k = 1.

a. Estimate the change in the centerline velocity (r = 0) if the radius of the flow cylinder increases from R = 3 cm to R = 3.05 cm and the length increases from L = 50 cm to L = 50.5 cm.

b. Estimate the percent change in the centerline velocity if the radius of the flow cylinder R decreases by 1% and the length L increases by 2%.

c. Complete the following sentence: If the radius of the cylinder increases by p%, then the length of the cylinder must decrease by approximately ___% in order for the velocity to remain constant.