Buckingham Pi Dimensional Analysis - simplifying problems by eliminating units

**Alternate title: "How to make Pi"
A tutorial on the Buckingham Pi method, why dimensionless parameters are awesome (not just the Reynolds number), and how they can be used to narrow down the solution space before even a single experiment is performed.

Submitted to #SoME3
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References and images

[adissentingopinion]

Sometimes I hate videos like this because I already focused in on computer engineering and I'm super jealous I don't get to use techniques like this.

Immaculate video. You've probably made someone decide their major with this. I dream of having the kind of education ability you have.

[kylewhiteman]

I like how approachable this explanation is. It introduced me to a method that was left out in my secondary / post secondary STEM courses.

[flameofphoenix]

Oh Lord I haven't been so engaged by an explainer video in *ages*
I cannot wait for the next one and thank you for making this one!

[zatx]

i love how your videos help in clarifying the topics that I learned "raw" because I couldn't wrap my head around them

[KipIngram]

Nice. I wasn't exposed to this method in college. I came across it on my own in later years, and have the impression that it was given more emphasis in earlier decades. I think this should be in every engineer's toolbox.

[ybcnnpq]

There's a long-standing meme (at least in Russian academia) that French kids can tell that 2 + 3 = 3 + 2, but don't know how to get to the result.

[staigerh]

Outstanding video! I was sure this would have 100k+ views when I finished. The motivation was clear and the example was the perfect length.

[aaronclair]

Brilliant explanation. This is something super relevant and useful that I had basically never heard of. Awesome.

[MiroslawHorbal]

Great video. I learned something new and it was motivated and explained nicely!

[krishnapriya]

Amazing ... The easiest explanation I could ever ask for !

[francob_]

I don't know if you read the comments frequenlty, but I wanted to ask you -yeah, I know it has nothing to do with the video- which career did you study, because I'm starting my engineering bachelor and am deciding between electronics, mechanics and chemistry, and I like materials science but I don't know how much I would like the job as materials engineer. So I wanted to ask you too what do you do normally in your job - do you choose materials, design them, choose optimized shapes for the materials, ... ?
Thanks for reading me! Love your videos!

[Kram]

10:23 well, looking at it as a matrix means you can just invert it in general:
A you got a relationship like

M a + µ = 0

so a = M^-1 µ

and for that matrix you get as inverse

3 1 1
0 0 -1
1 0 0

which solves your problem for all choices of µ in one go:

a = 3 µ1 + µ2 + µ3
b = - µ3
c = µ1

You can do that exercise with a bunch of fundamental constants:

Speed of light c: Length / Time
Planck constant h: Mass Length² / Time
Gravitational Constant G: Length³ / Mass Time²
Boltzmann constant kb: Mass Length² / Temperature Time²

So you get the matrix:

| c h G kb | (Powers of)

| 1 2 3 2 | Length
| -1 -1 -2 -2 | Time
| 0 1 -1 1 | Mass
| 0 0 0 -1 | Temperature

and its inverse:

| len time mas tem (powers of)

| -3/2 -5/2 1/2 5/2 | Speed of light
| 1/2 1/2 1/2 1/2 | Planck constant
| 1/2 1/2 -1/2 -1/2 | Gravitational Constant
| 0 0 0 -1 | Boltzmann Constant

So to get any combination of powers of length, time, mass, and temperature, you can also use any combination of the speed of light, Planck constant, Gravitational Constant, and Boltzmann constant using that matrix.

A volume can be given in terms of (multiplying the vector (3, 0, 0, 0))

c^(-9/2) h^(3/2) G^(3/2)

This works out to 6.65 *10^-104 m³ or 16 Planck Volumes.
(I think it's not just 1 Planck Volume because that chose a different parametrization of the Planck Constant and/or the Gravitational Constant. (Both of those often get modified by some factor of pi and could be chosen just as well for this task)

Force works out to be 1.21*10^44 Newton (using c^4 / G)

etc.

This is what the Natural Units basically come from.

I wonder if there is something interesting to be said about those matrices in terms of their eigenvalues and eigenvectors...

They have -1 as one of the eigenvalues and the corresponding eigenvector for the inverse matrix is (6, -1, -4, 1) but the three eigenvalues/eigenvectors are more complicated.

[FranFerioli]

In grad school, I grappled with what was regarded as the most comprehensive and difficult treatment of dimensional analysis for engineers: the book "Similarity and Dimensional Methods in Mechanics" by Russian physicist L. I. Sedov. Dimensional analysis is SO RUSSIAN, I can only think the professor meant that "your" approach to scaling laws is very French.
For all Oppenheimer fans, the blast wave generated by an atomic bomb is a super-famous dimensional analysis problem. The more approachable, yet rigorous description of what happens when the bomb explodes is in the book by another distinguished Russian physicist Y. B. Zeldovich: "Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena".

[suhassheikh]

Hi, the value of the Boltzmann constant (k_B) seems to be the same as that of the vacuum permittivity (\varepsilon_0)
A minor typo, i assume?

[leocurious]

Note that in hindsight, it is easy to just assume that the drag in fluid flow scales with the Reynolds number. But you don't know that is the case until you actually do the tests, regardless of how some units might cancel out somewhere on a piece of paper.

[smergthedargon]

They always ask "Who's Albert Hung?", but never "Is Albert hung?"

[jayraldbasan]

Still, I thought it was trivial piece of information that dimensions have to cancel if the equation work BUT I GOT MY MIND BLOWN NOW THAT I KNOW THERE'S ACTUAL MATHEMATICAL RIGOR TO THE METHOD HAHA the yt algorithm has blessed me, kudos to the vid creator!

[ZaCharlemagne]

Nice video.

The Buckingham PI theorem does remind me of the Sparse Identification of Nonlinear Dynamics technique.

It's basically a technique to describe the relationship of polynomial combinations of observed variables to the state dynamics of a dynamical systems.

They're quite similar in the sense that you don't necessarily need to analyse the effect of each variable and parameter individually...

[adiaphoros]

What if the number of variables = the number of dimensions?

[gandalfthefool]

Yeah the first big thing we teach undergrads in fluid mechanics is the Buckingham pi theorem.