Problem 63 - Algebra and Trigonometry

Question

Blood Flow As blood moves through a vein or an artery, its velocity \(v\) is greatest along the central axis and decreases as the distance \(r\) from the central axis increases (see the figure). The formula that gives \(v\) as a function of \(r\) is called the law of laminar flow. For an artery with radius \(0.5 \mathrm{cm},\) we have $$ v(r)=18,500\left(0.25-r^{2}\right) \quad 0 \leq r \leq 0.5 $$ $$ \begin{array}{l}{\text { (a) Find } v(0.1) \text { and } v(0.4) .} \\ {\text { (b) What do your answers to part (a) tell you about the flow }} \\ {\text { of blood in this artery? }} \\ {\text { (c) Make a table of values of } v(r) \text { for } r=0,0.1,0.2,0.3,} \\ {0.4,0.5 .}\end{array} $$

Step-by-Step Solution

Answer

Blood flows faster near the center with decreasing velocity towards the artery walls.

Step 1: Interpret the problem

We want to calculate the velocity of blood flow, \( v(r) \), at specific distances \( r \) from the central axis of the artery. The given formula is \( v(r) = 18,500(0.25 - r^2) \). We will calculate \( v(r) \) for specific values of \( r \).

Step 2: Evaluate v(r) for r = 0.1

Using the given formula, substitute \( r = 0.1 \) into \( v(r) \): \[ v(0.1) = 18,500(0.25 - (0.1)^2) \] Calculate:\[ v(0.1) = 18,500(0.25 - 0.01) = 18,500 \times 0.24 = 4,440 \, \text{cm/s} \]

Step 3: Evaluate v(r) for r = 0.4

Substitute \( r = 0.4 \) into the formula:\[ v(0.4) = 18,500(0.25 - (0.4)^2) \]Calculate:\[ v(0.4) = 18,500(0.25 - 0.16) = 18,500 \times 0.09 = 1,665 \, \text{cm/s} \]

Step 4: Analyze Part (a) Solutions

For \( r = 0.1 \), \( v(0.1) = 4,440 \) cm/s, and for \( r = 0.4 \), \( v(0.4) = 1,665 \) cm/s. This indicates blood flows faster closer to the central axis since velocity decreases as the distance from the center increases.

Step 5: Create Value Table for v(r)

Calculate \( v(r) \) at \( r = 0, 0.1, 0.2, 0.3, 0.4, 0.5 \) using the formula: - For \( r = 0 \): \[ v(0) = 18,500(0.25 - 0) = 4,625 \] - From previous steps: \[ v(0.1) = 4,440 \] \[ v(0.4) = 1,665 \] - For \( r = 0.2 \): \[ v(0.2) = 18,500(0.25 - 0.04) = 3,885 \] - For \( r = 0.3 \): \[ v(0.3) = 18,500(0.25 - 0.09) = 2,960 \] - For \( r = 0.5 \): \[ v(0.5) = 18,500(0.25 - 0.25) = 0 \] | \( r \) | \( v(r) \) cm/s ||------|-------------|| 0 | 4,625 || 0.1 | 4,440 || 0.2 | 3,885 || 0.3 | 2,960 || 0.4 | 1,665 || 0.5 | 0 |

Key Concepts

Blood Flow Velocity

Blood flow velocity is an essential concept in understanding how blood travels through vessels such as arteries and veins. It describes how fast blood moves along the path of a vessel. The velocity of blood flow is not uniform across the cross-section of a vessel. In any given artery, the flow is fastest at the center, right along the axis, and slows as you move towards the vessel's walls.

This is because of the friction between the fluid layers, akin to how the surface of a river flows slower due to contact with the riverbed. In our artery example, the function governing flow velocity is given by
\( v(r) = 18,500(0.25 - r^2) \)
. Here, the function shows how velocity changes with the distance \( r \) from the central axis, highlighting that the farther we get from the center, the slower the flow. This model helps us understand typical blood flow dynamics and any deviations that could signal medical issues.

Radius of Artery

The radius of an artery is crucial in modeling blood flow. It refers to the distance from the center of the artery's cross-section to its wall. Changes in the radius can significantly alter blood velocity and flow rate.

In clinical and physiological contexts, the artery's radius can change due to factors like vasodilation (widening) or vasoconstriction (narrowing), both of which affect blood pressure and flow. In our problem, the artery has a fixed radius of \( 0.5 \) cm. This fixed size provides a limit on the value \( r \) can take in the velocity function,

• ensuring \( r \leq 0.5\)
• validating the conditions where laminar flow calculations are applicable

These conditions illustrate how even small changes in the radius can impact blood flow dynamics tremendously, making it a parameter of great interest in medical studies.

Laminar Flow Equation

The laminar flow equation plays a pivotal role in modeling and understanding blood flow within arteries. This type of flow is characterized by smooth, parallel layers of liquid gliding past one another without cross-currents, which is the mode of flow typically seen in small blood vessels and during minimal turbulences.

In the given exercise, the equation \( v(r) = 18,500(0.25 - r^2) \) exemplifies laminar flow in an artery. This expression helps us comprehend how blood velocity decreases with increasing \( r \), the radial distance from the center. It's a linear model that assumes ideal conditions for flow,

• ignoring complexities like vessel wall irregularities or varying blood viscosity.
• providing a clear understanding of how blood behaves under normal physiological conditions.

Such models are indispensable in predicting how diseases might affect blood flow, aiding early diagnosis and treatment strategies.

Blood Flow Modeling

Blood flow modeling is the process of using mathematical formulas to represent and predict how blood moves through the body's complex network of vessels.

This modeling is essential for various purposes:

• designing medical devices such as stents and blood pressure monitors
• understanding pathological conditions like atherosclerosis

In our context, the laminar flow equation is a fundamental tool that simplifies complex dynamics into manageable calculations.

By accurately capturing the essence of blood flow, such models assist researchers and healthcare professionals in solving intricate physiological challenges. They aid in crafting preventive measures, diagnostic tools, and personalized treatments, making it possible to predict possible disruptions in flow before they lead to significant health issues.