Question

In fully developed laminar flow in a circular pipe, the velocity at \(R / 2\) (midway between the wall surface and the centerline) is measured to be \(6 \mathrm{~m} / \mathrm{s}\). Determine the velocity at the center of the pipe. Answer: \(8 \mathrm{~m} / \mathrm{s}\)

Short answer

Based on the given information, the axial velocity at the center of the pipe, \(u_c\), is 8 m/s.

Step-by-step solution

Recall the velocity profile for laminar flow in a circular pipe

The axial velocity in a laminar flow inside a circular pipe varies parabolically with the radial distance r from the center of the pipe. The velocity profile can be expressed as: \(u(r) = u_c \cdot (1 - (\frac{r}{R})^2)\) where \(u(r)\) is the axial velocity at radial distance r, \(u_c\) is the axial velocity at the center of the pipe (r = 0), and R is the radius of the pipe.

Use the given information and substitute into the velocity profile equation

We are given that the velocity at the point \(R / 2\) is 6 m/s. Therefore, we can write: \(6 = u_c \cdot (1 - (\frac{R / 2}{R})^2)\) Simplify the equation: \(6 = u_c \cdot (1 - (\frac{1}{2})^2)\)

Solve the equation for \(u_c\)

Now we need to solve the equation for \(u_c\) (velocity at the center of the pipe): \(6 = u_c \cdot (1 - \frac{1}{4})\) So, \(u_c = \frac{6}{(1 - \frac{1}{4})}\) Calculate \(u_c\): \(u_c = \frac{6}{\frac{3}{4}}\) or \(u_c = 6 \cdot \frac{4}{3}\)

Calculate the velocity at the center of the pipe

Finally, we can calculate the velocity at the center of the pipe: \(u_c = 8 \mathrm{~m} / \mathrm{s}\) Answer: The velocity at the center of the pipe is \(8 \mathrm{~m} / \mathrm{s}\).

Key concepts

Velocity Profile

In fluid mechanics, understanding the velocity profile in a pipe is crucial, especially for predicting how fluids behave under different conditions. In laminar flow, the fluid moves in smooth layers or parallel sheets, without any disruptions between them. The velocity profile for laminar flow inside a circular pipe has a distinctive parabolic shape. This means that the velocity is highest at the center of the pipe and decreases as you move toward the pipe walls. This behavior can be mathematically described using the equation:\[ u(r) = u_c \cdot \left(1 - \left( \frac{r}{R} \right)^2 \right) \]where \(u(r)\) is the velocity at a distance \(r\) from the center, and \(u_c\) is the velocity at the center (when \(r = 0\)). This parabolic distribution reflects how the pipe's structure and the viscosity of the fluid influence flow behavior.

Circular Pipe

A circular pipe is one of the most common geometries used in piping systems due to its structural efficiency and ease of manufacturing. The symmetry of a circular pipe simplifies the modeling of fluid flow, especially under laminar conditions. Because of the circular shape, the velocity profile can be uniformly mapped from the center to the wall, which leads to the parabolic distribution pattern. This uniformity allows engineers to predict the flow characteristics accurately: - **Centerline:** Velocity is highest here because there is less friction from the pipe walls. - **Near the walls:** The velocity decreases, approaching zero at the wall itself because of no-slip conditions. This makes circular pipes ideal for applications where controlled and predictable flow rates are essential.

Velocity Distribution

The concept of velocity distribution explains how fluid particles in different layers of flow move with varying speeds across the cross-section of a pipe. For fully developed laminar flow, the velocity distribution is predictable and can be described by the same parabolic relationship mentioned earlier. This distribution shows that: - **Central Zone:** Fluid particles at the center travel faster because they are not slowed by friction with the pipe walls. - **Transition Zone:** As you move towards the wall, particles slow down gradually. - **Boundary Layer:** Near the wall, the velocity drops significantly due to the viscous drag. Understanding velocity distribution helps in designing efficient systems where flow rate and pressure drops are critical factors. Correctly predicting the distribution allows for better design and operation of fluid transport systems.

Fluid Mechanics

Fluid mechanics is the branch of physics that studies the behavior of fluids (liquids and gases) in motion and at rest. Key principles in fluid mechanics, such as continuity, momentum, and energy conservation, are used extensively to analyze and solve problems related to fluid flow. In the context of this exercise: - **Continuity Equation:** Helps understand how velocity and cross-section area changes maintain a constant flow rate. - **Viscosity Effects:** Determine how fluid layers resist motion against each other. - **No-Slip Condition:** Defines that fluid velocity relative to the boundary is zero, affecting velocity profile and distribution. Fluid mechanics is fundamental in predicting how changes in pipe diameter or fluid type will impact flow speed and pressure, which is essential for engineering applications ranging from water supply systems to oil pipelines.