Problem 49 - University Physics with Modern Physics

Question

Viscous blood is flowing through an artery partially clogged by cholesterol. A surgeon wants to remove enough of the cholesterol to double the flow rate of blood through this artery. If the original diameter of the artery is \(D\) , what should be the new diameter (in terms of \(D\) ) to accomplish this for the same pressure gradient?

Step-by-Step Solution

Answer

The new diameter should be \( D \cdot \sqrt[4]{2} \).

Step 1: Understanding the relationship of flow rate and diameter

The flow rate of a fluid through a pipe is governed by the Hagen-Poiseuille equation for laminar flow, which states \( Q = \frac{\pi P r^4}{8 \eta L} \), where \( Q \) is the flow rate, \( P \) is the pressure difference, \( r \) is the radius of the pipe, \( \eta \) is the fluid's viscosity, and \( L \) is the length of the pipe. The flow rate is directly proportional to the fourth power of the radius.

Step 2: Establishing the relation for flow rate doubling

To double the flow rate, i.e., \( Q' = 2Q \), the radius must be adjusted since other parameters remain constant. From the equation \( Q' = \frac{\pi P (r')^4}{8 \eta L} \), where \( Q' = 2Q \), it follows that \((r')^4 = 2r^4 \).

Step 3: Solving for the new radius

To find the new radius \( r' \), solve the equation \( (r')^4 = 2r^4 \). Taking the fourth root of both sides, \( r' = r \cdot \sqrt[4]{2} \).

Step 4: Expressing diameter in terms of the original diameter

Since the diameter \( D = 2r \), the new radius \( r' = r \cdot \sqrt[4]{2} \) translates to a new diameter \( D' = 2r' = 2r \cdot \sqrt[4]{2} = D \cdot \sqrt[4]{2} \).

Step 5: Final expression for the new diameter

Substituting the expression for \( r' \) back into terms of diameter, the relationship is simplified to \( D' = D \cdot \sqrt[4]{2} \). This is the formula for the new diameter needed to double the flow rate.

Key Concepts

Laminar Flow

Laminar flow is a type of fluid movement where the fluid travels smoothly or in regular paths. In laminar flow, layers of fluid slide past one another like pages in a book. The opposite of laminar flow is turbulent flow, where the fluid moves chaotically.

This smooth flow is characterized by very low flow resistance and highly predictable behavior. When blood flows through an artery in a laminar manner, it exhibits minimal friction and turbulence, maximizing efficiency. Blood vessels are typically too narrow for turbulent flow unless there are abnormalities like blockages.

Understanding laminar flow is crucial in contexts like medical procedures or fluid dynamics, as it ensures that predictions about flow rate or pressure loss are more accurate. In the case of blood flowing through arteries, maintaining laminar flow is essential for reducing strain on the heart and ensuring efficient nutrient and oxygen delivery throughout the body.

Blood Viscosity

Viscosity is the measure of a fluid’s resistance to flow. In simple terms, it is how "thick" or "sticky" a fluid is. Think of how honey flows slowly compared to water. Blood viscosity is a vital concept when discussing blood flow, particularly in terms of how easily it moves through arteries and veins.

Several factors can influence blood viscosity, such as hematocrit levels (the proportion of red blood cells in blood), temperature, and certain diseases. High blood viscosity can increase the resistance inside blood vessels, making it tougher for the heart to pump blood, similar to the difficulty of sipping a thick milkshake through a straw.

Understanding blood viscosity helps healthcare professionals anticipate how blood may flow through arteries, especially when there are blockages or changes in blood composition. It's important for diagnosing and managing conditions like high blood pressure or clotting disorders, where viscosity plays a critical role.

Artery Diameter

The diameter of an artery is pivotal in determining how much blood can flow through it at a given time. According to the Hagen-Poiseuille equation, the flow rate is proportional to the fourth power of the radius, making even small changes in diameter significantly affect flow.

If an artery is narrow due to plaque buildup, it can severely restrict blood flow, similar to how a kink in a hose limits water output. Increasing the diameter, such as through surgical intervention, can dramatically improve flow efficiency and reduce pressure load on the heart.

In medical practice, understanding artery diameter is crucial for procedures like angioplasty, where arteries are widened to restore adequate blood flow. Keeping arteries healthy and free from excessive constriction is a key goal in preventing cardiovascular diseases.

Flow Rate

Flow rate refers to the volume of fluid that moves through a given area in a specific amount of time. It is a fundamental concept in fluid dynamics and directly affects the distribution of nutrients and oxygen in biological systems.

In cardiovascular health, the flow rate determines how well blood circulates throughout the body. The duration it takes for blood to reach various tissues affects how effectively they function. If flow rate is too low, tissues may receive insufficient oxygen, leading to fatigue or tissue damage.

Flow rate is dictated by variables like pressure difference, fluid viscosity, and vessel diameter. Doubling the diameter, for instance, can increase the flow significantly, as depicted by the Hagen-Poiseuille equation. Therefore, regulating or modifying flow rate can be crucial in treatments involving circulation issues or blood flow enhancement.