Question

A blood vessel with a circular cross section of constant radius $R$ carries blood that flows parallel to the axis of the vessel with a velocity of $v(r)=V(1-r^2/R^2),$ where $V$ is a constant and $r$ is the distance from the axis of the vessel. a. Where is the velocity a maximum? A minimum? b. Find the average velocity of the blood over a cross section of the vessel. c. Suppose the velocity in the vessel is given by $v(r)=V{(1-r^2/R^2)}^{1/p},$ where $p\ge 1.$ Graph the velocity profiles for $p=1,2,$ and 6 on the interval $0\le r\le R.$ Find the average velocity in the vessel as a function of $p.$ How does the average velocity behave as $p\to \mathrm{\infty }?$

Short answer

a. The velocity is a maximum at r = 0 and minimum at r = R. b. The average velocity of the blood over a cross-section of the vessel is V/2. c. By analyzing the integral derived from the general case in the problem, we find that as p → ∞, the blood flow becomes more streamlined, and the average velocity increases but remains less than the maximum velocity V.

Step-by-step solution

Finding the first derivative of the velocity function

Differentiate the velocity function $v(r)$ with respect to $r$: $v^{\prime }(r)=\frac{d}{dr}\left(V(1-\frac{r^2}{R^2})\right)$ Using the chain rule: $v^{\prime }(r)=V(-\frac{2r}{R^2})$

Finding the critical points of the velocity function

To find the critical points of the function, set the first derivative equal to 0 and solve for $r$: $0=V(-\frac{2r}{R^2})$ Solving for $r$, we get $r=0$.

Analyzing the critical points of the velocity function

The velocity is maximum when $r=0$, and minimum at $r=R$. b. Find the average velocity of the blood over a cross-section of the vessel.

Integrate the velocity function over the cross-sectional area

To find the average velocity over the cross-sectional area, we must first integrate the velocity function across the area given by $A=\pi R^2$. ${\int }_0^{R}v(r)\cdot 2\pi rdr$

Divide by the total area

To find the average velocity, divide the integral of the velocity by the cross-sectional area: $\frac{{\int }_0^{R}v(r)\cdot 2\pi rdr}{\pi R^2}$

Solve the integral

Now, solve the integral to find the average velocity. Substituting the given velocity function: $\frac{{\int }_0^{R}V(1-\frac{r^2}{R^2})\cdot 2\pi rdr}{\pi R^2}$ After solving the integral, we obtain: $\frac{V}{2}$ c. Graph the velocity profiles for $p=1,2,$ and 6 on the interval $0\le r\le R.$ Find the average velocity in the vessel as a function of $p.$ How does the average velocity behave as $p\to \mathrm{\infty }?$

Solve the integral for the general case

Now we have a new velocity function with the additional parameter $p$: $v(r)=V{(1-\frac{r^2}{R^2})}^{\frac{1}{p}}$ Solve the integral to find the average velocity for the general case $\frac{{\int }_0^{R}V{(1-\frac{r^2}{R^2})}^{\frac{1}{p}}\cdot 2\pi rdr}{\pi R^2}$

Analyze the behavior of the average velocity as $p\to \mathrm{\infty }$

Using the result from Step 1, investigate the behavior of the average velocity as $p$ approaches infinity. As $p\to \mathrm{\infty }$, the vessel velocity approaches $V$ along the axis and zero at the vessel wall. The blood flow becomes more streamlined, resulting in an increased average velocity. The average velocity increases but is always less than the maximum velocity $V$.

Key concepts

Velocity Function

In fluid dynamics, the velocity function describes how fast a fluid moves at any given point within a domain. Particularly in blood flow, we often want to understand how the speed of blood relates to its position within a blood vessel. For a vessel with a circular cross-sectional area, the velocity can be expressed in terms of the distance from the center, represented by the variable $r$. Here, the velocity function is given by: $$v(r)=V(1-\frac{r^2}{R^2})$$ where: - $V$ is a constant representing the maximum velocity at the center of the vessel (i.e., $r=0$) - $R$ is the radius of the vessel, indicating the maximum distance from the center to the wall - $r$ is the variable distance from the axis The velocity is highest at the center and decreases towards the wall, following a parabolic profile. At $r=0$, the velocity is maximal at $V$, and at $r=R$, it touches zero.

Blood Flow

Blood flow within vessels is crucial for transporting essential nutrients and oxygen to tissues while removing waste products. It is significantly affected by the velocity distribution across the vessel's cross-section. The distribution described by the velocity function $v(r)$, portrays a streamlined flow, with the fastest movement occurring at the center. There are several important points to note about blood flow in this context: - **Maximum Velocity**: Blood moves fastest at the center of the vessel due to lower resistance. - **Zero Velocity at Walls**: Due to friction, the velocity decreases towards the wall of the vessel, reaching zero. - **Parabolic Profile**: This type of distribution is typical of laminar flow, where the flow is smooth, consistent, and without turbulence.

Cross-sectional Area

The cross-sectional area of a vessel is an important factor in determining the overall flow characteristics. It helps in calculating the average velocity and the total volume of fluid passing through the vessel over time. For a circular vessel, the cross-sectional area is calculated as: $$A=\pi R^2$$ With this area, the average velocity of blood flow can be calculated by integrating the velocity function across it. Integration quantifies the total amount of blood flowing through each infinitesimally small section of the vessel. By dividing the integrated velocity by the total area, we achieve the average velocity, providing a useful measure for understanding overall flow dynamics.

Integral Calculus

Integral calculus is the mathematical tool used to compute the total accumulation of a quantity, such as finding the average velocity across a cross-section of a blood vessel. To find the average speed of blood flow, one needs to integrate the velocity function across the radius from 0 to $R$, as follows: First, compute the definite integral: $${\int }_0^{R}v(r)\cdot 2\pi rdr$$ This evaluates the activity or movement of blood at every infinitesimal point across the cross-section. After computing the integral, divide this result by the cross-sectional area $\pi R^2$ to get the average velocity: $$\text{Average Velocity}=\frac{{\int }_0^{R}v(r)\cdot 2\pi rdr}{\pi R^2}$$ The integral essentially quantifies the sum of velocities multiplied by their respective circular widths, helping us understand the overall flow pattern. As variations such as the introduction of the parameter $p$ affect velocity profiles, integral calculus plays an integral role in adapting these computations for complex cases, giving insights into changes in flow dynamics as parameters vary.