   Chapter 4, Problem 34P

Chapter
Section
Textbook Problem

, The volume flow rate m an artery that supplies blood to the brain is 3.5× 10-6 m3/s. If the cross-sectional area of the artery is 8.5 × 10 5 m2 ,what is the average speed of the blood through this vessel? Find the average blood speed through a narrowed section of the artery where the cross-sectional area is reduced by a factor of 3.

To determine

What is the average speed of the blood through this vessel? Find the average blood speed through a narrowed section of the artery where the cross-sectional area is reduced by a factor of 3.

Explanation

Given:

Volume flow rate in artery V=3.5×106m3/s

Corresponding cross-sectional area A=8.5 ×105m2

Concept Used:

The concept used is a consequence of Bernoulli’s Principle for steady fluid flow. Assuming an incompressible fluid, Bernoulli’s Principle says for a steady fluid flow the pressure is lower when the fluid flow is faster. This implies that the volume rate passing through each portion is the same irrespective of the area of cross-section of flow. This is applicable for fluid flow in pipes, in human blood vessels etc. So, we use the concept that the volume rate of flow of blood through the artery is the same irrespective of the fact that it flows through a widened area or a narrow area of cross-section. The relation between volume flow rate V, area of cross-section A and velocity of flow at that point v can be quantified as

V=Av.............(1)

Now when area increases or decreases, the fluid flows with a lower velocity or higher velocity respectively, since the product remains a constant i.e. the volume rate of flow is same everywhere. This can be expressed as

A1v1=A2v2.............(2)

This equation is referred as equation of continuity. This can be rearranged as

v1v2=A2A1.......................(3)

Equation (3) implies that whenever area is reduced by a factor, velocity gets multiplied by the same factor.

Calculation:

As per the given problem

Volume flow rate in artery, V=3.5×106m3/s

Also, cross-sectional area, A=8

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