8 Easy Steps To Bernoulli’s Theorem

If you’ve ever had a conversation with a physicist or engineer, it’s likely they have mentioned Bernoulli’s theorem at some point. Named after Daniel Bernoulli, and first published in 1738, the theorem deals with fluid dynamics and the relationship that exists between pressure and velocity. In this post we will analyze Bernoulli’s Theorem as well as examine its applications to real life situations. Bernoulli’s theorem deals with the law of conservation of energy.

Bernoulli’s Theorem – The Formal Stuff

In the spirit of full disclosure, I should tell you that this is a pretty complex topic and so I’ll try my best to keep things as simple as possible. OK, so according to Bernoulli’s theorem, if we have two fluids of different densities and viscosities, then the fluid with lower density will move faster than the higher density fluid.

8 easy steps to Bernoulli’s theorem, all explained are:

Step 1: Idea of theorem

The idea of this theorem is to express the principle of conservation of energy between two fluids in terms of pressure and speed. 

Step 2: Stated mathematically, this relationship can be expressed as:

where:

p – The pressure energy density (J/m3) 

v – The speed energy density (J/m3) 

ρ – The density (kg/m3) 

μ – The viscosity (Pa.s) 

Step 3: Reference

This particular equation is often referred to as Bernoulli’s Equation and it will remain so until we cover the following step. 

Step 4: The Say-Tcheng relationship

In the previous step we discussed the relationship between the energy of a fluid but what we didn’t mention is that this is only valid for incompressible fluids. The good news is that, although incompressible fluids do not exist in nature, there are many everyday situations in which this can be used to describe how pressure and velocity change with density. If a fluid is compressible, then it will also change with pressure and velocity and so a different equation needs to be used. This equation was first published in 1738 by Daniel Bernoulli but was originally derived by Say-Tcheng in 1713. 

Step 5: Expression of the equation

This is a bit more complex (or is it?) but it’s fairly easy to understand. As you can see, both fluids are described in terms of their density, pressure and speed. Now if you know the fluids densities and velocity you can use this equation to find out the energy density of a fluid. For example, if we were dealing with two gasses, air and hydrogen gas for instance, then the density and speed for air would be determined by how many molecules were in each cubic meter. Similarly for hydrogen there would be 1 molecule for every cubic meter (there is oxygen in the breath so there are 2 molecules per cubic meter). The density and speed of hydrogen would be determined by how many molecules there were in each cubic meter. The formula for Bernoulli’s equation can now be used to determine the kinetic energy of a fluid.

Step 6: Applying Bernoulli’s theorem

In order to understand this step it is necessary to understand how pressure, velocity and density are found out (for more information on this subject you can view my article here). With this in mind, let’s find out what the formula for pressure looks like when expressed as a ratio between density and speed. In other words, we want to find out how the pressure will change if you are traveling at different speeds in a fluid. If we take a look at the problem in question then we can see that there is an equation (in red) which has two unknowns and so there must be two equations which define these two unknowns.

Step 7: Fluid dynamics

At this point it is possible to see that Bernoulli’s theorem is sometimes referred to as Bernoulli’s Law because it is a law of fluid dynamics. The main application of this law is used when modeling airflow in aircraft wings and motor sport vehicles. We can also see that Bernoulli’s theorem explains how an air-hockey puck moves around the table.

Step 8: The Beer Can Theorem

If you’ve ever been out and had a few drinks then this one is probably of interest to you. The Beer Can theorem states that as long as the pressure inside a can doesn’t change, then there is no way for it to explode. In other words, if a can of beer has no holes in it and the pressure inside is greater than the pressure outside, then all you have to do is puncture the can and the beer will come shooting out.

Conclusion:

I’m sure this is all a bit overwhelming but it’s not as if we were dealing with rocket propulsion here, it’s fairly straightforward stuff. For more information on the subject, be sure to check out my list of other articles about Bernoulli’s theorem or have a read of my book about the subject. So, there you have it, a relatively short introduction to Bernoulli’s Theorem. I hope this was useful and if you have any comments or suggestions then feel free to let me know.

Cheers!

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