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Chapter 11: Problem 56

(a) The volume flow rate in an artery supplying the brain is \(3.6 \times 10^{-6} \mathrm{m}^{3} / \mathrm{s} .\) If the radius of the artery is \(5.2\) \(\mathrm{mm},\) determine the average blood speed. (b) Find the average blood speed at a constriction in the artery if the constriction reduces the radius by a factor of \(3 .\) Assume that the volume flow rate is the same as that in part (a).

Short Answer

Expert verified
(a) Average speed is 0.0424 m/s. (b) Speed in constriction is 0.3817 m/s.

Step by step solution

01

Understand Given Variables

We are given the volume flow rate \( Q \), which is \( 3.6 \times 10^{-6} \mathrm{m}^3/\mathrm{s} \), and the radius \( r \) of the artery, which is \( 5.2 \mathrm{mm} \). First, convert the radius to meters by dividing by 1000, so \( r = 0.0052 \mathrm{m} \).
02

Calculate the Cross-sectional Area

The cross-sectional area \( A \) of the artery is given by the formula for the area of a circle: \( A = \pi r^2 \). Substituting \( r = 0.0052 \mathrm{m} \), we get \[ A = \pi \times (0.0052)^2 = \pi \times 0.00002704 \mathrm{m}^2. \]
03

Determine the Average Blood Speed in Regular Artery

The average speed \( v \) is given by the equation \( Q = A \times v \). Rearrange to solve for \( v \), \( v = \frac{Q}{A} \). Substitute \( Q = 3.6 \times 10^{-6} \mathrm{m}^3/\mathrm{s} \) and \( A = \pi \times 0.00002704 \mathrm{m}^2 \) to find \( v \).
04

Solve for Average Blood Speed in Regular Artery

Perform the calculation: \[ v = \frac{3.6 \times 10^{-6}}{\pi \times 0.00002704} = \frac{3.6 \times 10^{-6}}{8.49 \times 10^{-5}}. \] Now compute this division to get the speed, \( v \approx 0.0424 \mathrm{m/s} \).
05

Determine Radius in Constricted Artery

The constriction reduces the radius by a factor of 3. Therefore, the new radius is \( r' = \frac{0.0052}{3} = 0.001733 \mathrm{m} \).
06

Calculate New Cross-sectional Area in Constriction

Using the same cross-sectional area formula \( A' = \pi (r')^2 \), substitute \( r' = 0.001733 \mathrm{m} \) to get \[ A' = \pi \times (0.001733)^2 = \pi \times 0.000003002. \]
07

Determine the Average Blood Speed in Constricted Artery

Using the formula \( v' = \frac{Q}{A'} \), substitute \( Q = 3.6 \times 10^{-6} \mathrm{m}^3/\mathrm{s} \) and \( A' = \pi \times 0.000003002 \) to get \[ v' = \frac{3.6 \times 10^{-6}}{9.43 \times 10^{-6}}. \] Perform this division to compute \( v' \approx 0.3817 \mathrm{m/s} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume Flow Rate
In fluid dynamics, understanding the volume flow rate is crucial for analyzing how fluids such as blood move through arteries. The volume flow rate, denoted as \( Q \), is the measure of how much fluid passes through a given surface area per unit of time. In simpler terms, it tells us how fast the fluid is flowing through a pipe or vessel. The formula to calculate the volume flow rate is given by:
  • \( Q = A \cdot v \)
Here, \( A \) represents the cross-sectional area of the vessel, and \( v \) is the speed or velocity of the fluid. The unit of volume flow rate is typically cubic meters per second (\( \mathrm{m}^3/\mathrm{s} \)).In the example problem, the volume flow rate in an artery is \( 3.6 \times 10^{-6} \, \mathrm{m}^3/\mathrm{s} \). This tells us the blood continues to circulate through the artery at a constant rate. It is a key variable when assessing changes in conditions such as when the vessel is constricted.
Average Blood Speed
The average speed of blood within an artery is a significant parameter in fluid dynamics. It helps in determining how efficiently blood can be transported through the circulatory system. To find the average blood speed, we use the formula:
  • \( v = \frac{Q}{A} \)
In this equation, \( v \) is the average speed of the blood, \( Q \) is the volume flow rate, and \( A \) is the cross-sectional area of the artery.Using the provided volume flow rate and the cross-sectional area calculated from the artery's radius, the average blood speed in the example was determined to be approximately \( 0.0424 \, \mathrm{m/s} \). Understanding how to calculate this speed provides insight into potential circulatory issues or inefficiencies. The average speed helps predict how quickly oxygen and nutrients can reach various parts of the body.
Artery Constriction
Artery constriction significantly affects how blood flows through the body. It involves the narrowing of the vessel, which can change the velocity and pressure of the blood passing through it. When an artery becomes constricted, the radius decreases, affecting the cross-sectional area \( (A') \) through which the blood can flow. In our specific example, the constriction reduces the artery's radius by a factor of 3.This results in a smaller cross-sectional area. Thus, when applying the formula:
  • \( v' = \frac{Q}{A'} \)
For the constricted section, the velocity \( v' \) increases as the area \( A' \) decreases, assuming the volume flow rate \( Q \) remains constant.In this case, the calculation showed the speed increases to approximately \( 0.3817 \, \mathrm{m/s} \) due to constriction. This demonstrates how changes in vessel dimensions can significantly impact fluid dynamics, emphasizing the importance of maintaining an open and healthy circulatory path.

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Most popular questions from this chapter

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