Lesson 6 - Principles of Fluid Mechanics 3

Welcome back. So, what were your thoughts?

Hopefully, you concluded that the pressure difference between points 1 and 2 (∆P₁₋₂) would be zero. Before we consider the important implications of this, let us work through the application of Bernoulli’s equation to make sure you agree with the conclusion.

So, what do we know about the various parameters? Well, we know the following:

• h₁ = h₂
  • We know this because the pipe is horizontal; hence both the measuring points are at the same vertical height.
• u₁ = u₂
  • We know this because the flow cross-sectional area is the same, and the density is the same at points 1 and 2, as there has been no temperature change.

Substituting these parameters into Bernoulli’s equation simply as h and u, respectively, gives:

As well as the ρgh terms cancelling out, the ρu²/2 terms will too, which leaves:

Therefore:

An important practical requirement for this situation to occur is the complete lack of friction between the fluid and the inner surface of the pipe. So, even if the pipe was constructed from ice and made as smooth as a mirror (the type of surface that we would quite easily slip on if it was an icy pavement), it would still have some friction. If a fluid flowed down it, we would expect to see a small loss of pressure between the measuring points.

This is a limitation of Bernoulli’s equation as far as we have considered it to date. But do not worry; we will consider a specific means to describe the pressure drop in a real pipe system now! We will also see that actual industrial pipes made from differing metals, plastics, and even concrete, have some degree of friction which creates a loss of pressure as fluids flow within them.