Problem 59 - College Physics

Question

Clogged artery. 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 approximately 1.189D.

Step 1: Understand the Relationship between Flow Rate and Diameter

The flow rate of a liquid through a pipe is given by Poiseuille's law, which states that the flow rate (\(Q\)) is proportional to the fourth power of the radius (\(r\)) of the pipe: \[ Q \propto r^4 \].Since the diameter \(D\) is twice the radius (\(r = \frac{D}{2}\)), the flow rate is related to the diameter as:\[ Q \propto \left(\frac{D}{2}\right)^4 = \frac{D^4}{16} \].

Step 2: Set Up the Equation for Doubling the Flow Rate

To double the flow rate, we need to find the new diameter \(D'\) such that the new flow rate \(Q'\) is twice the original flow rate \(Q\):\[ Q' = 2Q \].Using Poiseuille's law for \(Q'\):\[ Q' \propto \left(\frac{D'}{2}\right)^4 = \frac{{D'}^4}{16} \].

Step 3: Equate and Solve for the New Diameter

Since \(Q' = 2Q\), we have:\[ \frac{{D'}^4}{16} = 2 \cdot \frac{D^4}{16} \],which simplifies to:\[ {D'}^4 = 2D^4 \].Taking the fourth root of both sides:\[ D' = D \sqrt[4]{2} \].

Step 4: Simplify the Solution

We can simplify \( D' = D \sqrt[4]{2} \) further:\[ D' = D \cdot 2^{1/4} \].Using the approximation \( 2^{1/4} \approx 1.189 \), the new diameter is:\[ D' \approx 1.189D \].

Key Concepts

Flow Rate

Flow rate refers to the volume of fluid that moves through a pipe or other conduit per unit time. In medical terms, this can apply to how blood moves through an artery. Flow rate is typically measured in liters per minute or milliliters per second. It is a key factor in determining how well blood circulates through the body. Poiseuille's Law, which applies to viscous fluid flow, states that the flow rate (Q) is dependent on the fourth power of the radius (r). This can be expressed mathematically as:\[ Q \propto r^4 \]If you know the diameter, then since the diameter is twice the radius, the effect on flow rate becomes very significant as changes to radius are raised to the fourth power.

In the context of the given problem, you need a higher flow rate due to a blocked artery. To achieve this, one needs to either decrease viscosity or increase the artery's diameter, given that the pressure gradient remains constant.

Artery Diameter

The diameter of an artery is a critical factor in determining the flow rate of blood according to Poiseuille's Law. Since the flow rate is proportional to the fourth power of the radius, even small changes to the artery's size can greatly influence how blood flows through it.

• For instance, if the radius is doubled, the flow rate increases by a factor of 16.
• Conversely, if the radius is halved, the flow rate decreases significantly, making the circulation less efficient.

In this exercise, by reducing the cholesterol to increase the diameter of the artery, the flow can significantly improve. Specifically, to double the flow rate, the new diameter should be approximately 1.189 times the original diameter, calculated as \( D' \approx 1.189D \). This solution highlights the powerful effect of diameter changes due to the radius being raised to the fourth power in Poiseuille's equation.

Blood Viscosity

Blood viscosity refers to the thickness and stickiness of blood. It's an important physiological parameter that impacts how easily blood flows through the circulatory system. Higher viscosity means thicker blood, making it harder for it to move through arteries and veins, which can increase the workload on the heart.

In the context of Poiseuille’s Law, viscosity is inversely related to flow rate. As viscosity increases, the flow rate decreases for a given pressure gradient and vessel size. It's represented in the equation as a component that inversely affects the rate:\[ Q \propto \frac{1}{\text{viscosity}} \]By decreasing viscosity, you'd effectively increase flow rate, and vice-versa. While the problem scenario doesn’t change viscosity, understanding its influence helps illustrate why simply dilating the artery diameter is beneficial in achieving a higher flow rate without altering other parameters.

Pressure Gradient

The pressure gradient is the difference in pressure that drives blood flow through a vessel. It is essentially the force that pushes the blood forward. In Poiseuille's Law, the flow rate is directly proportional to this gradient when all other factors remain constant.

• If the pressure gradient increases, the flow rate also increases, assuming viscosity and vessel diameter remain unchanged.
• Conversely, a reduction in this gradient would result in decreased flow rate.

However, in this exercise, it is assumed that the pressure gradient stays constant. Thus, to achieve the desired increase in flow rate, modifications should be made to the diameter of the artery or adjustments to the viscosity, showcasing the interconnectedness and balance of these variables in maintaining a healthy circulatory system.