Chapter 5.5, Problem 14E

Reminder Round all answers to two decimal places unless otherwise indicated.

Poiseuille’s law for fluid velocities Poiseuille’s law describes the velocities of fluids flowing in a tube---for example, the flow of blood in a vein. (See Figure 5.74) This law applies when the velocities are not too large----more specifically, when the flow has no turbulence. In this case, the flow is laminar, which means that the paths of the flow are parallel to the tube walls. The law states that

$v=k\left(R^2-r^2\right)$,

where $v$ is the velocity, $k$ is a constant (which depends on the fluid, the tube, and the units used for measurement), $R$ is the radius of the tube, and $r$ is the distance from the centerline of the tube. Since $k$ and $R$ are fixed for any application, $v$ is a function at a point of distance $r$ from the centerline of the tube.

a. What is $r$ for a point along the walls of the tube? What is the velocity of the fluid along the walls of the tube?

b. Where in the tube does the fluid flow most rapidly?

c. Choose numbers for $k$ and $R$, and make a graph of $v$ as a function of $r$. Use a horizontal span of $0$ to $R$.

d. Describe your graph from part c.

e. Explain why you needed to use a horizontal span of $0$ to $R$ in order to describe the flow throughout the tube.