Problem 18 - Calculus: Early Transcendentals

Question

The velocity \( v \) of blood that flows in a blood vessel with radius \( R \) and length \( l \) at a distance \( r \) from the central axis is $$ v(r) = \frac{P}{4\eta l} (R^2 - r^2) $$ where \( P \) is the pressure difference between the ends of the vessel and \( \eta \) is the viscosity of the blood (see Example 3.7.7). Find the average velocity (with respect to \( r \)) over the interval \( 0 \le r \le R \). Compare the average velocity with the maximum velocity.

Step-by-Step Solution

Answer

The average velocity is \(\frac{PR^2}{6\eta l}\) and is two-thirds of the maximum velocity.

Step 1: Set up the Integral for Average Velocity

The average velocity, denoted as \( \bar{v} \), is calculated by integrating the velocity function \( v(r) \) over the interval \( 0 \) to \( R \), and then dividing by the length of the interval, \( R \). This is given by the formula: \[ \bar{v} = \frac{1}{R} \int_0^R v(r) \, dr. \] Substitute \( v(r) = \frac{P}{4\eta l} (R^2 - r^2) \) into the integral.

Step 2: Evaluate the Integral

Substitute \( v(r) = \frac{P}{4\eta l} (R^2 - r^2) \) into the integral and evaluate: \[ \int_0^R \frac{P}{4\eta l} (R^2 - r^2) \, dr. \] Factor out the constant \( \frac{P}{4\eta l} \) from the integral: \[ \frac{P}{4\eta l} \int_0^R (R^2 - r^2) \, dr = \frac{P}{4\eta l} \left[ R^2r - \frac{r^3}{3} \right]_0^R. \]

Step 3: Compute Definite Integral

Evaluate the integral equation from step 2: \[ \frac{P}{4\eta l} \left[ R^2R - \frac{R^3}{3} \right] = \frac{P}{4\eta l} \left( R^3 - \frac{R^3}{3} \right). \] Simplify the expression: \[ \frac{P}{4\eta l} \left( \frac{2R^3}{3} \right) = \frac{PR^3}{6\eta l}. \] This value represents the integral of the velocity function over the interval \( 0 \) to \( R \).

Step 4: Calculate Average Velocity

The average velocity is: \[ \bar{v} = \frac{1}{R} \times \frac{PR^3}{6\eta l} = \frac{PR^2}{6\eta l}. \] This result means the average velocity over the interval is \( \frac{PR^2}{6\eta l} \).

Step 5: Determine Maximum Velocity

The maximum velocity occurs at \( r = 0 \). Substitute \( r = 0 \) into the velocity formula: \[ v_{\text{max}} = \frac{P}{4\eta l} (R^2 - 0^2) = \frac{PR^2}{4\eta l}. \] This value represents the maximum velocity at the center of the blood vessel.

Step 6: Compare Average and Maximum Velocities

Compare \( \bar{v} \) and \( v_{\text{max}} \): \[ \bar{v} = \frac{PR^2}{6\eta l} \text{ and } v_{\text{max}} = \frac{PR^2}{4\eta l}. \] Find the ratio: \[ \frac{\bar{v}}{v_{\text{max}}} = \frac{\frac{PR^2}{6\eta l}}{\frac{PR^2}{4\eta l}} = \frac{2}{3}. \] Thus, the average velocity is two-thirds of the maximum velocity.

Key Concepts

average velocity calculation

To calculate the average velocity of blood flow in a vessel, you need to focus on the integral of the velocity function over its entire radius. The concept behind this is to find a single value that represents the entire range of velocities that blood particles can have as they move through the vessel. The mathematical expression for average velocity, \( \bar{v} \), is derived using:

• An integral of the velocity function, \( v(r) \), which accounts for varying velocities at different distances, \( r \), from the central axis.
• The integration is performed over the interval from the innermost point (\( r = 0 \)) to the outer edge of the vessel (\( r = R \)).
• The result is divided by the length of the interval, \( R \), to yield the average.

Once the integral of \( v(r) = \frac{P}{4\eta l} (R^2 - r^2) \) is evaluated, it gives the total extent of velocity change in terms of blood flow within the vessel. The final calculation reveals that the average velocity is \( \frac{PR^2}{6\eta l} \), which provides a single representative speed for the blood across the entire vessel's radius.

integrating velocity functions

Integrating velocity functions is crucial when determining average values, as it summarizes the fluid's behavior throughout the vessel. This function, \( v(r) \), represents each point's velocity based on its distance from the center line. By integrating this function, you are effectively pooling together all individual velocities to compute a meaningful average.To perform the integration, you'll initially substitute:

• \( v(r) = \frac{P}{4\eta l} (R^2 - r^2) \) into the integral equation.
• Extract constants such as \( \frac{P}{4\eta l} \) which simplify the integral.

Then, calculate the integral \( \int_0^R (R^2 - r^2) \, dr \). This becomes \( R^2r - \frac{r^3}{3} \), evaluated from 0 to \( R \). Substituting \( R \) simplifies it to \( \frac{2R^3}{3} \), leading to solving the integral to find the total velocity in the vessel. Consequently, the integral recognizes the full extent of variance for velocity due to pressure dynamics and vessel characteristics.

maximum velocity in cylindrical vessels

Maximum velocity in a cylindrical vessel is an essential element of understanding how blood flows efficiently through vessels. In this context, it occurs at the very center of the vessel where the resistance by the vessel walls is at its minimal. This leads to the highest velocity, aligning with principles of fluid dynamics.The formula \( v(r) = \frac{P}{4\eta l} (R^2 - r^2) \) indicates that setting \( r = 0 \), where \( r \) is the distance to the central axis, leads to maximum velocity \( v_{\text{max}} \). By evaluating at \( r = 0 \), you find:

• The term \((R^2 - r^2)\) becomes simply \( R^2 \).
• This results in \( v_{\text{max}} = \frac{PR^2}{4\eta l} \).

Understanding this concept confirms that blood velocity is affected by the centrality within the vessel, demonstrating maximum efficiency of flow under given pressure and viscosity constraints.

pressure difference and viscosity

Pressure difference and viscosity are fundamental determinants in fluid dynamics, specifically in the context of vascular blood flow. These factors dictate how blood moves through vessels, influencing both velocity and flow patterns.**Pressure Difference (\( P \))**:- It is the force driving the blood through vessels, present due to the pressure gradient between two ends of a vessel.- A higher pressure difference leads to greater potential for faster blood flow, essentially acting like an engine for the circulation.**Viscosity (\( \eta \))**:- Represents the 'thickness' or internal friction of blood. Higher viscosity corresponds to slower flow due to greater resistance within the fluid.The velocity profile, given by \( v(r) = \frac{P}{4\eta l} (R^2 - r^2) \), highlights:

• As either pressure difference increases or viscosity decreases, blood flow velocity increases.
• This dynamic relationship implies that adjustments in these two parameters can significantly affect how effectively blood can be transported throughout the circulatory system.

Understanding these primary factors clarifies how various biological and environmental conditions might alter blood flow efficiency, with pressure difference propelling movement and viscosity moderating it.