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Flow in a cylinder Poiseuille’s Law is a fundamental law
Chapter 10, Problem 55E(choose chapter or problem)
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.
Questions & Answers
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.
ANSWER:Solution 55E(a)Step 1 of 3:Consider the Poiseuille Law if a cylinder of length L and radius R. the velocity of the fluid r R units from the centerline of the cylinder is , where P is the difference in the pressure between the ends of the cylinder and v is the viscosity of the fluidNow P and V are constant along r = 0 , then velocity is V = So, Assume that k = 1 then V =