How to derive the Bernoulli's Equation - [ Fluid Mechanics]
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- Опубліковано 13 лис 2023
- What is Bernoulli's equation?
This equation will give you the powers to analyze a fluid flowing up and down through all kinds of different tubes.
What is Bernoulli's principle?
Bernoulli's principle is a seemingly counterintuitive statement about how the speed of a fluid relates to the pressure of the fluid. Many people feel like Bernoulli's principle shouldn't be correct, but this might be due to a misunderstanding about what Bernoulli's principle actually says. Bernoulli's principle states the following,
Bernoulli's principle: Within a horizontal flow of fluid, points of higher fluid speed will have less pressure than points of slower fluid speed.
Why does it have to be horizontal?
So within a horizontal water pipe that changes diameter, regions where the water is moving fast will be under less pressure than regions where the water is moving slow. This sounds counterintuitive to many people since people associate high speeds with high pressures. But, we'll show in the next section that this is really just another way of saying that water will speed up if there's more pressure behind it than in front of it. In the section below we'll derive Bernoulli's principle, show more precisely what it says, and hopefully make it seem a little less mysterious.
How can you derive Bernoulli's principle?
Incompressible fluids have to speed up when they reach a narrow constricted section in order to maintain a constant volume flow rate. This is why a narrow nozzle on a hose causes water to speed up. But something might be bothering you about this phenomenon. If the water is speeding up at a constriction, it's also gaining kinetic energy. Where is this extra kinetic energy coming from? The nozzle? The pipe?
The energy fairy?
The only way to give something kinetic energy is to do work on it. This is expressed by the work energy principle.
\[W_{external}=\Delta K=\dfrac{1}{2}mv_f^2-\dfrac{1}{2}mv_i^2\]
So if a portion of fluid is speeding up, something external to that portion of fluid must be doing work it. What force is causing work to be done on the fluid? Well, in most real world systems there are lots of dissipative forces that could be doing negative work, but we're going to assume for the sake of simplicity that these viscous forces are negligible and we have a nice continuous and perfectly laminar (streamline) flow. Laminar (streamline) flow means that the fluid flows in parallel layers without crossing paths. In laminar streamline flow there is no swirling or vortices in the fluid.
How realistic are these assumptions?
OK, so we'll assume we have no loss in energy due to dissipative forces. In that case, what non-dissipative forces could be doing work on our fluid that cause it to speed up? The pressure from the surrounding fluid will be causing a force that can do work and speed up a portion of fluid.
Enjoyed the lecture and make the equation more simpler to understand
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