Question

In fully developed laminar flow inside a circular pipe, the velocities at \(r=0.5 R\) (midway between the wall surface and the centerline) are measured to be 3,6 , and \(9 \mathrm{~m} / \mathrm{s}\). (a) Determine the maximum velocity for each of the measured midway velocities. (b) By varying \(r / R\) for \(-1 \leq\) \(r / R \leq 1\), plot the velocity profile for each of the measured midway velocities with \(r / R\) as the \(y\)-axis and \(V(r / R)\) as the \(x\)-axis.

Short answer

Answer: The maximum velocities at the center of the pipe for each midway velocity are: (a) 2 m/s for the 3 m/s midway velocity, (b) 4 m/s for the 6 m/s midway velocity, and (c) 6 m/s for the 9 m/s midway velocity. The velocity profiles are plotted using the Hagen-Poiseuille equation for each maximum velocity with respect to r/R from -1 to 1, with r/R on the y-axis and V(r/R) on the x-axis. The resulting graphs show the velocity profiles for each measured midway velocity.

Step-by-step solution

Understanding the Hagen-Poiseuille equation

The Hagen-Poiseuille equation is given by: \[V(r) = 2V_{max}\left(1 - \frac{r^2}{R^2}\right)\] where V(r) is the velocity at a radius r, \(V_{max}\) is the maximum velocity, and R is the radius of the circular pipe.

Determine the maximum velocity for midway velocities

We are given three midway velocities at r = 0.5R. For each midway velocity, we can use the Hagen-Poiseuille equation to find the corresponding maximum velocity V_max. The equation for \(V(0.5R)\) is: \[V(0.5R) = 2V_{max}\left(1 - \frac{(0.5R)^2}{R^2}\right)\] Let's solve for each maximum velocity: (a) Midway velocity 3 m/s: \[3 = 2V_{max}\left(1 - \frac{1}{4}\right)\] \[V_{max1} = \frac{3}{1.5} = 2 \mathrm{~m} / \mathrm{s}\] (b) Midway velocity 6 m/s: \[6 = 2V_{max}\left(1 - \frac{1}{4}\right)\] \[V_{max2} = \frac{6}{1.5} = 4 \mathrm{~m} / \mathrm{s}\] (c) Midway velocity 9 m/s: \[9 = 2V_{max}\left(1 - \frac{1}{4}\right)\] \[V_{max3} = \frac{9}{1.5} = 6 \mathrm{~m} / \mathrm{s}\]

Plot the velocity profile for each measured midway velocity

We now need to plot the velocity profile V(r/R) for each of the measured midway velocities by varying r/R from -1 to 1, with r/R on the y-axis and V(r/R) on the x-axis. We will use the Hagen-Poiseuille equation with respective maximum velocities: (a) \(V_{max1} = 2 \mathrm{~m} / \mathrm{s}\): \[V(r/R) = 4\left(1 - r^2/R^2\right)\] (b) \(V_{max2} = 4 \mathrm{~m} / \mathrm{s}\): \[V(r/R) = 8\left(1 - r^2/R^2\right)\] (c) \(V_{max3} = 6 \mathrm{~m} / \mathrm{s}\): \[V(r/R) = 12\left(1 - r^2/R^2\right)\] The next step is to plot the above equations with respect to the y-axis (r/R) and the x-axis (V(r/R)) in a graphing software or calculator. The resulting graphs will show the velocity profiles for each measured midway velocity.

Key concepts

Hagen-Poiseuille Equation

The Hagen-Poiseuille equation is a fundamental principle in fluid dynamics that describes the velocity profile of laminar flow through a circular pipe. It is crucial for understanding how fluids behave under certain conditions, particularly in systems where precision matters, like in medical devices or industrial applications.

In a laminar flow through a pipe, the fluid in the center moves faster than fluid near the edges due to frictional forces. The Hagen-Poiseuille equation, expressed as \(V(r) = 2V_{max}\big(1 - ({r^2}/{R^2})\big)\), mathematically defines this velocity distribution. Here, \(V(r)\) is the fluid velocity at a distance \(r\) from the center, \(V_{max}\) is the maximum velocity at the center of the pipe, and \(R\) is the pipe's radius.

Understanding this equation helps in predicting the flow rate and designing pipes to regulate fluid flow effectively. An insightful exercise improvement advice is to encourage students to manually calculate several velocity values at different radii to improve their hands-on understanding of the equation's implications.

Maximum Velocity Determination

Determining the maximum velocity of a fluid inside a pipe is key to many engineering applications, ranging from designing water supply systems to predicting blood flow in veins. When you have a measured velocity at any point within the pipe, this value allows you to compute the maximum velocity at the center where the fluid encounters the least resistance.

In the given exercise, velocities are known at the midpoint between the wall and the centerline, allowing for the use of the Hagen-Poiseuille equation to determine the maximum velocity \(V_{max}\). For example, when a midway velocity (\(V(0.5R)\)) of 3 m/s is given, the equation modifies to \(3 = 2V_{max}(1 - {1}/{4})\), which can be rearranged to find the \(V_{max}\). This concept not only teaches students about fluid dynamics but also reinforces algebraic manipulation skills. It's advisable to perform these calculations for different midpoint velocities to build a stronger intuition of how the maximum velocity correlates with the measured velocity at other points.

Velocity Profile Plotting

Plotting the velocity profile of a fluid flow within a circular pipe is a visually engaging way to understand laminar flow characteristics. It involves using the Hagen-Poiseuille equation to calculate the velocity at various points and then graphically representing these velocities. This graphical representation helps students better visualize how the fluid's velocity decreases from the center of the pipe to its walls.

For instance, with a maximum velocity \(V_{max}\), you can calculate the corresponding velocity \(V(r/R)\) across different radial positions by graphing the equation \(V(r/R) = 2V_{max}(1 - r^2/R^2)\) for values of \(r/R\) from -1 to 1. It's beneficial to encourage students to use different tools, like graphing calculators or software, to plot these profiles, as it enhances their digital literacy and data interpretation skills.

Visualizing Fluid Dynamics

Through velocity profile plotting, students can observe the parabolic shape of the velocity distribution—a characteristic of laminar flow in pipes. This gives a tangible form to theoretical concepts, an excellent exercise improvement advice to solidify their comprehension of fluid mechanics.