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Chapter 11: Problem 79

In the human body, blood vessels can dilate, or increase their radii, in response to various stimuli, so that the volume flow rate of the blood increases. Assume that the pressure at either end of a blood vessel, the length of the vessel, and the viscosity of the blood remain the same, and determine the factor \(R_{\text { dilated }} / R_{\text { normal }}\) by which the radius of a vessel must change in order to double the volume flow rate of the blood through the vessel.

Short Answer

Expert verified
The radius must increase by a factor of approximately 1.189 to double the flow rate.

Step by step solution

01

Recall the Hagen-Poiseuille Law

The Hagen-Poiseuille Law explains the flow of a viscous fluid through a cylindrical pipe or vessel. According to this law, the volume flow rate \(Q\) is proportional to the fourth power of the radius \(r\) of the vessel:\[ Q = \frac{\pi r^4 (P_1 - P_2)}{8 \eta L} \]where \((P_1 - P_2)\) is the pressure difference across the vessel, \(\eta\) is the viscosity of the fluid, and \(L\) is the length of the vessel.
02

Set Up the Equation for Doubling the Flow Rate

Given that the goal is to double the volume flow rate, let the normal flow rate be \(Q_{\text{normal}}\) and the dilated flow rate be \(Q_{\text{dilated}} = 2Q_{\text{normal}}\). Substituting into the Hagen-Poiseuille equation, we have:\[ 2 \cdot \frac{\pi r_{\text{normal}}^4 (P_1 - P_2)}{8 \eta L} = \frac{\pi r_{\text{dilated}}^4 (P_1 - P_2)}{8 \eta L} \]
03

Simplify the Equation

Since the pressure difference, viscosity, and length remain constant, we can cancel these terms from both sides of the equation:\[ 2 \times r_{\text{normal}}^4 = r_{\text{dilated}}^4 \]
04

Solve for the Radius Ratio

To find the factor by which the radius must change, solve for the ratio \(R_{\text{dilated}} / R_{\text{normal}}\):\[ \left(\frac{r_{\text{dilated}}}{r_{\text{normal}}}\right)^4 = 2 \]Take the fourth root of both sides to find:\[ \frac{r_{\text{dilated}}}{r_{\text{normal}}} = 2^{1/4} \approx 1.189 \]

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Key Concepts

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

Volume Flow Rate
The volume flow rate represents the quantity of fluid flowing per unit time through a vessel or pipe. In the context of blood vessels, it describes how much blood is transported in a given period. This concept is central to understanding how the body's circulatory system adjusts blood flow according to its needs. For instance, during exercise, the body increases blood flow to muscles, requiring a change in volume flow rate.
  • The Hagen-Poiseuille Law is essential here as it connects the volume flow rate to various factors such as radius, viscosity, pressure difference, and vessel length.
  • The formula shows that the flow rate, denoted as \( Q \), is proportional to the radius raised to the fourth power \( r^4 \).
  • This dependence means even small changes in the radius can significantly affect the flow rate, allowing the body to regulate blood supply efficiently.
Blood Vessel Dilation
Blood vessel dilation refers to the process by which blood vessels increase in diameter. This physiological response occurs due to various stimuli like increased body temperature or exercise. During dilation, the vessels widen, reducing resistance and allowing more blood to flow through.
  • Dilation adjusts the radius \( r \) of the vessel, which, according to the Hagen-Poiseuille Law, has a profound effect on the flow rate due to its fourth power relation.
  • In medical conditions or during activities requiring increased blood supply, vessel dilation is a critical mechanism to enhance the flow rate efficiently.
  • By calculating the new radius needed for a specific increase in flow, doctors and researchers understand how vascular adaptations affect circulation.
Viscosity
Viscosity is a measure of a fluid's resistance to flow. In the case of blood, its viscosity affects how easily it can move through the vessels. A higher viscosity means the blood is "thicker" and flows more slowly, whereas lower viscosity means it's "thinner" and flows more freely.
  • In the Hagen-Poiseuille equation, viscosity \( \eta \) acts as a resistance factor, inversely affecting the volume flow rate.
  • The relationship highlights that for the same pressure gradient and vessel dimensions, a more viscous fluid will have a lower flow rate.
  • In practical terms, health conditions that alter blood viscosity, like dehydration, can have significant impacts on circulation efficiency.
Radius Ratio
Understanding the ratio of the dilated radius to the normal radius \( \frac{r_{\text{dilated}}}{r_{\text{normal}}} \) is crucial in predicting how changes in vessel size affect the flow rate. This ratio tells us the extent of dilation needed to achieve a desired increase in volume flow rate.
  • From the exercise, we find that to double the flow rate, the radius must increase such that \( \left(\frac{r_{\text{dilated}}}{r_{\text{normal}}}\right)^4 = 2 \).
  • Solving for the radius ratio gives us \( 2^{1/4} \approx 1.189 \), indicating about an 18.9% increase in the radius is necessary.
  • This insight is invaluable for medical fields where precise control over blood flow is required, such as in the treatment of cardiovascular diseases.

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Most popular questions from this chapter

(a) The mass and the radius of the sun are, respectively, \(1.99 \times 10^{30} \mathrm{kg}\) and \(6.96 \times 10^{8} \mathrm{m}\) . What is its density? (b) If a solid object is made from a material that has the same density as the sun, would it sink or float in water? Why? (c) Would a solid object sink or float in water if were made from a material whose density was the same as that of the planet Saturn (mass \(=5.7 \times 10^{26} \mathrm{kg},\) radius \(=6.0 \times 10^{7} \mathrm{m} ) ?\) Provide a reason for your answer.

A barber鈥檚 chair with a person in it weighs 2100 N. The output plunger of a hydraulic system begins to lift the chair when the barber鈥檚 foot applies a force of 55 N to the input piston. Neglect any height difference between the plunger and the piston. What is the ratio of the radius of the plunger to the radius of the piston?

A room has a volume of 120 \(\mathrm{m}^{3}\) . An air-conditioning system is to replace the air in this room every twenty minutes, using ducts that have a square cross section. Assuming that air can be treated as an incom- pressible fluid, find the length of a side of the square if the air speed within the ducts is \((\text { a } 3.0 \mathrm{m} / \mathrm{s} \text { and }(\mathrm{b}) 5.0 \mathrm{m} / \mathrm{s} \text { . }\)

Multiple-Concept Example 8 presents an approach to problems of this kind. The hydraulic oil in a car lift has a density of \(8.30 \times 10^{2} \mathrm{kg} / \mathrm{m}^{3}\) . The weight of the input piston is negligible. The radii of the input piston and output plunger are \(7.70 \times 10^{-3} \mathrm{m}\) and \(0.125 \mathrm{m},\) respectively. What input force \(F\) is needed to support the 24500 \(\mathrm{-N}\) combined weight of a car and the output plunger, when (a) the bottom surfaces of the piston and plunger are at the same level, and (b) the bottom surface of the output plunger is 1.30 m above that of the input piston?

A spring is attached to the bottom of an empty swimming pool, with the axis of the spring oriented vertically. An 8.00-kg block of wood \(\left(\rho=840 \mathrm{kg} / \mathrm{m}^{3}\right)\) is fixed to the top of the spring and compresses it. Then the pool is filled with water, completely covering the block. The spring is now observed to be stretched twice as much as it had been compressed. Determine the percentage of the block's total volume that is hollow. Ignore any air in the hollow space.
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