# Transient inviscid incompressible pipe flow

• bob_confused2
In summary, transient inviscid incompressible pipe flow is a type of fluid flow that occurs within a pipe with no viscosity and a constant density. It is a time-dependent phenomenon that is often studied in fluid mechanics and can be described by the Navier-Stokes equations. This type of flow is commonly used in modeling and analyzing real-world systems, such as water distribution networks and oil pipelines. The behavior of transient inviscid incompressible pipe flow is influenced by factors such as the pipe's geometry, the fluid's initial conditions, and the boundary conditions. Understanding this type of flow is crucial for engineers and scientists in various industries, as it can have significant impacts on the performance and efficiency of fluid systems.
bob_confused2
Hello
I am trying to better understand transient fluid dynamics in pipes. First, I am attempting what I believe should be relatively simple problem. I have a constant area horizontal pipe partially filled with a stationary incompressible inviscid fluid. The part of the pipe that is filled is upstream of a burst disc which separates the rest of the pipe. Upstream of the burst disc is at pressure P1 and downstream is P2. At t=0sec, the disc bursts. What is the velocity v2 of the fluid as it flows as a function of time? Here, I am assuming that I am looking only at the velocity at the location of the burst disc and that the flow is uniform. Once I understand this problem, I hope to add in friction and varying location.

I've started with F=ma=m(dv/dt)
rho(dv/dt) = -dP/dx

And I want to see how long it takes for the velocity to reach the velocity that would be calculated from Bernoulli's equation: P1-P2=rho(v2)^2/2
Any assistance/guidance would be greatly appreciated.
Thank you

This is not a simple problem, although with some care it could be simplified. The origin of the problem is the 'partially' filled part- there is a fluid-fluid interface that can *deform* as the fluid flows. This type of problem, free surface flows, is a class of problem that has resisted clean solutions since forever.

The problem can perhaps be simplied by instead of partially filling the pipe, the pipe is full and subjected, at t = 0, to a pressure spike at one end. The pressure gradient will induce fluid flow, and the pressure wave will propagate at the speed of sound. Even now this problem is very difficult to solve, but significantly easier than before. I don't know if anyone has published a solution to this simplied problem.

Thanks. I will try to take a step back even further to try the simplified problem you suggested and then try to add complexity. No wonder my head has been hurting so much as I attacked this problem.

For people checking out this post and are interested in learning more about transient 1D flow, I found this online resource:
http://www.scribd.com/doc/8717066/Fluid-Mechanics-Hydraulics-of-Pipeline-Systems

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bob_confused2 said:
constant area horizontal pipe partially filled with a stationary incompressible inviscid fluid. pressure is P2.
How does a incompressable fluid have any pressure if it partially fills a pipe? What is supplying the force? Is it gravity?

Incompressable fluids introduce all sorts of problems. For one thing the speed of information propagation (speed of sound) is infinite. If you had a partially filled container, composed of a vacuum and the incompressable fluid, then how would the fluid distribute itself in a zero g environment (assuming the fluid doesn't vaporize)? dP/dx can be infinite in an incompressable fluid such as a incompressable fluid traveling through a pipe that varies in diameter via a vertical wall (transition distance is zero).

Hmmm, I'm not sure but from your description it sounds like a 'dam break' problem. I've worked through one of these problems but that was a while ago. One difference might be that the dam break surface is completely free. Generally it's hardcore mathematicians that work on these so the maths is formidable and I seem to remember most solutions are implicit. I notice your link has characteristics and I think those are required...

Jeff, I recognise quite a lot of those assumptions, the effects you mention are generally ignored.

## 1. What is transient inviscid incompressible pipe flow?

Transient inviscid incompressible pipe flow is a type of fluid flow that occurs in a pipe where the fluid is assumed to have no viscosity (or internal friction) and is not compressed. This means that the fluid does not experience any resistance to flow and its density remains constant.

## 2. What factors affect transient inviscid incompressible pipe flow?

The key factors that affect transient inviscid incompressible pipe flow include the fluid density, the pipe diameter, the fluid velocity, and the pressure difference between the two ends of the pipe. Other factors such as the pipe roughness and fluid properties may also play a role.

## 3. How is transient inviscid incompressible pipe flow different from steady-state flow?

Transient inviscid incompressible pipe flow is a type of flow that changes with time, whereas steady-state flow remains constant over time. In transient flow, the fluid properties and flow conditions are constantly changing, while in steady-state flow, they remain constant.

## 4. What are the applications of transient inviscid incompressible pipe flow?

Transient inviscid incompressible pipe flow has many practical applications, such as in the design and analysis of pipelines for water distribution, oil and gas transportation, and irrigation systems. It is also used in the design of hydraulic systems and in the study of blood flow in the human body.

## 5. What are some common assumptions made in the analysis of transient inviscid incompressible pipe flow?

Some common assumptions made in the analysis of transient inviscid incompressible pipe flow include the neglect of pipe wall friction, the assumption that the fluid is incompressible and has a constant density, and the assumption that the flow is one-dimensional. Additionally, the flow is often assumed to be laminar and the pipe material is assumed to be rigid.

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