How Does the Buckingham Pi Theorem Apply to Homogeneous Functions in Physics?

[Cod]

This is more of a dimensional analysis or unit analysis problem than a basic analysis problem. So if I didn't post this thread in the right forum, please delete it.

Moving on, I'm having trouble grasping the concepts behind the Buckingham Pi Theorem. After reading some textbooks and doing some research on the internet, I've found out that is a theorem that correlates mathematics with physical science. Mathworld defines the Buckingham Pi Theorem like this: physical laws are independent of the form of the units; therefore, acceptable laws of physics are homogeneous in all dimensions.

Now, here's where I'm having troubles. Since the function becomes homogeneous, do you have to use Euler's Homogeneous Function Theorem? Or is there another way to do the problem once factored? When applying the Buckingham Pi Theorem, will you always end up with a homogeneous function on the way to your answer? Does the theorem ever "fail"?




Sorry if I'm missing something here, but I'm trying to learn advanced mathematics on my own so I get a better understanding of aerodynamics. So please, if I'm misinterpreting something, let me know. Any help is greatly appreciated.

 

[Wong]

Have you tried this http://www.math.ntnu.no/~hanche/notes/buckingham/buckingham-a5.pdf? It seems to give a more "friendly" introduction to the theorem.

 

[Cod]

I have not tried that link yet. I'll be sure to download the PDF when I get home from work tonight. Thanks.

 

[mathwonk]

your question does not quite make sense to me. As you stated it, the buckingham pi theorem is to me merely a restatement of the principal that any unit will suffice to state a true physical theorem, since "homogeneity" merely refers to expanding all measurements uniformly by changing the unit of length.

Moreover you did not state the problem or function that motivated your question.

Euler's theorem is a simple
fact about recovering a homogeneous function from its partial derivatives which you can easily check on any small example.

On the other hand there is some good insight in your suggestion that homogeneous statements should somehow be represented by homogeneous functions, i.e. functions in which all terms have the same degree. I never thought of that, but let's check it on some simple formulas like area and volume. Yep, pi r^2 is homogeenous in r, and so is (4/3)pi r^3, and so is V = s^3 for a cube, etc, etc... so maybe you have noticed something here!

I would say that another way to view the principle is that a change of units, by a scale factor, i.e. by a homogeneous stretching, would leave the law invariant. So true physical laws should be expressed by formulas which keep the same form when subjected to a "homogeneous change of variables", x goes to ax, y goes to by, z goes to cz, where a,b,c, are constants. etc...

 

[Cod]

Well, the reason my question doesn't make sense, is because I never should've asked it. When doing my research, I was supposed to use the theorem right below the Buckingham Pi Theorem. So everything was completely screwed up from the beginning.

Thanks for the willingness to help though :). And sorry for the mixup.

 

[abercrombiems02]

how about a simple explanation actually

basically the buckingham pi theorem states that if u have a finite number of quantites with dimensions such as a + b = c, where all these numbers have the same units, then u can divide through by the result to get an expression without any units a/c + b/c = 1. If we apply this idea to more complex functions where we have more than one kinda of unit we can simplfy the problem, as the end result maybe prove to be dependent only on certain combinations of these units. For example, when testing in a wind tunnel it would take forever to analyze what happens to an airfoil if we did the experiment over and over monitoring temperature, free stream velocity, the density of the fluid, the shape of the air foil and so on, by using the buckingham pi theorem it is possible to combine these quantities into certain forms upon which we can much easier find a resultant effect on an airfoil, when combining all these quantities specifically, the buckingham pi theorem reveals that only 2 combinations of the quantities above really effect the airfoil, Reynold's number for the airfoil, and the mach number of the fluid's freestream velocity. Imagine a combination of kg, m, and s. let's find a solution to the following quantities such that the result is dimensionless (m/s^2) * (kg/s)^a * (kg-m^2/s^2)^b * (s)^c. essentiall here we are multiply velcoity, mass flow rate, energy, and time to come up with a dimensionless quantity, so let's first solve for m. from velocity we have m and from energy we have m^2b therefore 1+2b = 0 and b = -1/2 now let's solve for kg, from mass flow rate we have kg^a and from energy we have kg^b therefore a+b = 0 if b = -1/2, then a = 1/2. finally let's solve for c, from velocity we have s^-2 from mass flow rate we have s^-a from energy we have s^-2b and from time we have s^c therefore -2 - a -2b + c = 0, solving for c yields 3/2. and our dimensionless quantity is velocity*sqrt(mass flow rate)*time^(3/2)/E. This quantity is dimensionless and most likely means nothing in the real world because i just chose a bunch of random quantities, however when a problem is analyzed and the dimensions are based on real world possibilities as opposed to arbitrary values, then the results will be a useful coefficient in an equation or a more direct relationship between two desired quantities, the application of this theorem saves millions of dollars and lots of time with experimentation as it reveals the governing dynamics of many complex systems and simplifies problems greatly.

 

[mathwonk]

abercrombie:

I like your nice application of the buck pi theorem very much, but technically it seems you are not restating the theorem, but assuming the theorem, and then applying it.

I.e. the result you state, that a homogeneous equation can always be divided through in the way you illustrate, is mathematically obvious. The point of the theorem is that the equation is homogeneous in the first place.

however i agree that you make a good case for why the theorem is interesting, if not for why it is true.