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 s R units from the P centerline of the cylinder is V -(R² – r²), where P is the 4 Lv difference in the pressure between the ends of the cylinder, and v is the viscosity of the fluid (see figure). Assuming P and v are constant, the velocity V along the centerline of the cylinder kR? (r = 0) is V = 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 its 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 increase by approximately_% in order for the velocity to remain constant. 7.
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 s R units from the P centerline of the cylinder is V -(R² – r²), where P is the 4 Lv difference in the pressure between the ends of the cylinder, and v is the viscosity of the fluid (see figure). Assuming P and v are constant, the velocity V along the centerline of the cylinder kR? (r = 0) is V = 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 its 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 increase by approximately_% in order for the velocity to remain constant. 7.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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 s R units from the
P
centerline of the cylinder is V
-(R² – r²), where P is the
4 Lv
difference in the pressure between the ends of the cylinder, and
v is the viscosity of the fluid (see figure). Assuming P and v
are constant, the velocity V along the centerline of the cylinder
kR?
(r = 0) is V =
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 its 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 increase
by approximately_% in order for the velocity to remain
constant.
7.
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