Question

Plot the velocity profile for pipeflow. A fluid that flows through a (very long) pipe has zero velocity on the pipe wall and a maximum velocity along the centerline of the pipe. The velocity \(v\) varies through the pipe cross section according to the following formula: $$ v(r)=\left(\frac{\beta}{2 \mu_{0}}\right)^{1 / n} \frac{n}{n+1}\left(R^{1+1 / n}-r^{1+1 / n}\right) $$ where \(R\) is the radius of the pipe, \(\beta\) is the pressure gradient (the force that drives the flow through the pipe), \(\mu_{0}\) is a viscosity coefficient (small for air, larger for water and even larger for toothpaste), \(n\) is a real number reflecting the viscous properties of the fluid ( \(n=1\) for water and air, \(n<1\) for many modern plastic materials), and \(r\) is a radial coordinate that measures the distance from the centerline \((r=0\) is the centerline, \(r=R\) is the pipe wall). Make a function that evaluates \(v(r) .\) Plot \(v(r)\) as a function of \(r \in[0, R]\), with \(R=1, \beta=0.02, \mu=0.02\), and \(n=0.1 .\) Thereafter, make an animation of how the \(v(r)\) curves varies as \(n\) goes from 1 and down to \(0.01\). Because the maximum value of \(v(r)\) decreases rapidly as \(n\) decreases, each curve can be normalized by its \(v(0)\) value such that the maximum value is always unity. Name of program file: plot_velocity_pipeflow.py.

Short answer

Plot velocity using a defined function and create an animation by varying \(n\). Normalize each plot by the maximum \(v(0)\) value for consistency.

Step-by-step solution

Understand the Problem

We are given a formula for the velocity profile of a fluid flowing through a pipe. The velocity, \(v(r)\), depends on various parameters such as \(R\), \(\beta\), \(\mu_0\), \(n\), and \(r\). Our task is to plot this velocity profile and create an animation for varying values of \(n\).

Define the Function for v(r)

We need to define a function in Python to calculate \(v(r)\). The formula given is:\[v(r)=\left(\frac{\beta}{2 \mu_{0}}\right)^{1 / n} \frac{n}{n+1}\left(R^{1+1 / n}-r^{1+1 / n}\right)\]We will implement this in a Python function with parameters for \(R\), \(\beta\), \(\mu_0\), \(n\), and \(r\).

Set Constant Parameters

In our problem, set the constant values \(R = 1\), \(\beta = 0.02\), and \(\mu_0 = 0.02\). These will be used as default values in our function for calculating \(v(r)\).

Plot v(r) for Fixed n

Using the function created, plot \(v(r)\) for \(n = 0.1\) over the range \(r \in [0, R]\). This can be done using Python libraries like Matplotlib.

Normalize v(r) for Animation

Since the velocity profile should be normalized, calculate \(v(0)\) and divide \(v(r)\) by \(v(0)\) to ensure the maximum value is unity before plotting. This step prepares our curves for animation.

Create Animation for Varying n

Implement a loop in Python that plots \(v(r)\) for varying values of \(n\) from 1 to 0.01. Use a library like Matplotlib to create and save the animation, normalizing each curve as done in Step 5.

Key concepts

Pipe Flow

Pipe flow is a fascinating topic in fluid dynamics. It describes how fluids move through a cylindrical conduit, typically under pressure. This flow is often driven by a pressure gradient, which pushes the fluid from regions of high pressure to low pressure. The core idea is that all the molecules in the fluid work together, comprising a continuous flow. However, they don’t all move at the same speed.
One of the critical aspects of understanding pipe flow is recognizing that the fluid adhering to the pipe walls experiences a no-slip condition. This means the fluid at the wall effectively has zero velocity. As you move toward the center of the pipe, the velocity increases, reaching its maximum at the centerline.
This velocity distribution can be affected by factors such as the pipe’s diameter, the fluid's viscosity, and the pressure gradient applied across the pipe. These aspects are crucial in many engineering fields, particularly those involving hydraulic and pneumatic systems.

Velocity Profile

The velocity profile in pipe flow depicts how the fluid's velocity changes across different points in the pipe’s cross-section. When plotting this graphically, the horizontal axis typically represents the radial distance from the centerline of the pipe, while the vertical axis indicates the velocity.
A common scenario is when the velocity profile is parabolic, often seen under laminar flow conditions when fluid viscosity and pipe dimensions are fixed. However, in non-Newtonian fluids, where viscosity can change with different shear rates, the velocity profile can take various forms.

• Maximum velocity occurs at the pipe's centerline (where the radius = 0).
• Zero velocity at the pipe wall due to the no-slip condition.
• Above conditions lead to the characteristic shape of the velocity profile.

Understanding and calculating the velocity profile is significant in optimizing flow applications and minimizing energy loss within pipes.

Python Programming

Python programming plays an essential role in modeling and visualizing pipe flow scenarios. Using Python, we can define functions to calculate the velocity profile of a fluid in a pipe based on given parameters. This is crucial for solving fluid dynamics problems efficiently.
In this context, we can leverage libraries such as Matplotlib to plot the velocity profile. By defining a function that calculates velocity based on the formula provided, we can assess changes over a specific range of parameters, such as varying viscosity or pipe radius.

• Defining functions for calculation improves reusability and clarity.
• Libraries like Matplotlib and NumPy help in data visualization and numerical operations.
• Python allows for animation creation, which can dynamically show how profiles change with varying parameters.

For students and engineers alike, mastering these aspects of Python can significantly enhance the ability to simulate and understand complex fluid flows.

Viscosity

Viscosity is a crucial property of fluids that assesses their resistance to deformation or flow. It is essentially a measure of how thick or sticky a fluid is. Water, for instance, has low viscosity, while substances like honey or toothpaste have much higher viscosity.
In the context of pipe flow, viscosity affects how quickly or slowly a fluid moves through a pipe. A highly viscous fluid will flow more slowly due to greater internal friction, whereas a less viscous fluid will flow more easily. This factor is directly incorporated into the formula for a velocity profile, influencing the variation of velocity across the pipe.

• Viscosity depends on factors like temperature and fluid composition.
• Non-Newtonian fluids have varying viscosity based on the shear rate.
• Understanding viscosity is critical in designing systems for efficient fluid transport.

Properly accounting for viscosity is essential in engineering designs to ensure systems function optimally under the desired operating conditions.