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In a fully developed laminar flow within a circular pipe, understanding the velocity profile is essential. The velocity profile describes how fluid velocity changes from the center of the pipe to its walls. It is generally parabolic for laminar flow, meaning the velocity is highest at the center and decreases toward the pipe wall. In this exercise, the velocity profile is mathematically represented by the equation:

• \( u(r) = 2\left(1-\frac{r^{2}}{R^{2}}\right) \)

Here, \(u(r)\) is the velocity at a distance \(r\) from the center of the pipe, while \(R\) is the pipe's inner radius. The quadratic relation indicates that as \(r\) approaches \(R\), velocity decreases to zero at the pipe wall due to the no-slip boundary condition.
Grasping the velocity profile helps predict flow behavior, which is vital in applications like fluid delivery systems.

The maximum velocity in a pipe flow scenario tells us the speed at which fluid flows at the very center of the pipe. This is where resistance due to the pipe walls is nonexistent. For the given velocity profile, maximum velocity occurs at \(r = 0\).
By substituting \(r = 0\) into the velocity equation:

• \( u_{max} = 2\left(1-\frac{0^{2}}{R^{2}}\right) = 2 \, \text{m/s} \)

Thus, the maximum velocity is straightforward to derive and equals 2 m/s. This maximum value plays a crucial role in determining the fluid's momentum and is a baseline for comparing different flow scenarios.

Getting the average velocity of fluid in a pipe gives us a sense of the overall speed at which the entire fluid is moving. Since flow velocity varies across the pipe's diameter in a parabolic manner, the average velocity is a necessary metric for understanding overall flow behavior. The formula to calculate it is:

• \( u_{avg} = \frac{2}{R^2} * \frac{2}{3} * R^2 = \frac{4}{3} \, \text{m/s} \)

In this calculation, the radius terms \(R^2\) cancel out, simplifying to \(\frac{4}{3}\) m/s.
The average velocity is less than the maximum velocity, capturing the extent of velocity variation across the pipe section. It is particularly useful in determining the volume flow rate.

The volume flow rate is a pivotal measure indicating the volume of fluid passing through a pipe per unit of time. It combines the average velocity with the cross-sectional area of the pipe.
Using the formula:

• \( Q = u_{avg} * \pi R^{2} = \frac{4}{3} * \pi * (0.02)^2 = 0.00000534 \, \text{m}^3/\text{s} \)

This expression accounts for both the area through which fluid flows and the representative velocity at which it flows. The volume flow rate is crucial for ensuring that the system meets the desired flow requirements, important in designing systems for efficient fluid distribution.