Talk:Dimensional analysis

In the most primitive form, dimensional analysis is used to check the correctness of algebraic derivations: in every physically meaningful equation, the dimensions of the two sides must be identical. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers, which is typically achieved by multiplying a certain physical quantity by a suitable constant of the inverse dimension.

I do not agree. Dimensional analysis is used to solve PDEs. The statement just describes e.g., stochiometry.

I admit that I don't know how to use dimensional analysis to solve PDE's (do you have any references?), but this paragraph was really just the beginning, showing the most primitive "dimensional analysis" as taught in college chemistry classes: make sure that the dimensions are right. I agree there's much more to Dimensional Analysis than that, and the rest of the article shows it, so I think the criticism is not justified. --AxelBoldt

What I was trying to say is the "monorail" algorithm for using units to solve stochiometry problems is not really dimensional analysis, but to be fair I will start cracking some books on this.

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The above mentioned reduction of variables uses the Buckingham Pi theorem as its central tool. This theorem describes how an equation involving several variables can be equivalently rewritten as an equation of fewer dimensionless parameters, and it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown. Two systems for which these parameters coincide are then equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one.

This is not quite correct either. The resulting dimensionless parameters generally need to be determined experimentally, or there must be some sort of experimentally verified constitutive relationship. No one as yet can predict a Froude or Mach number, we can only measure them.

That's what I was trying to say: the Pi theorem tells you how to turn the measured variables into dimensionless parameters, and then you have to empirically find the relationship between those dimensionless parameters. No one can predict a Mach number, but people can predict the proper formula for Mach numbers. How can we clarify the above paragraph? --AxelBoldt

I think you mean "proper units" for Mach numbers...

That's the 64 dollar question. People that know how to do this (e.g., Barenblatt) just smile enigmatically when asked "how you do dat?" The best that I have been able to determine is that the process is like that cartoon of the physicist at the blackboard, where in a long chain of formulas, the one in middle is labeled "magic here".

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I removed the "typed family of fields" comment, since there is no such thing in mathematics.

Such a thing can be well-defined. See G. W. Hart "Multidimensional Analysis".

Also,

Note also that the dimensionless numbers are not really dimensionless. The actual

structure of a dimensionless number is unity in the type. For example, consider the so-called dimensionless unit of strain: L/L. The L/L units are usually dropped, either implicitly or explicitly, but it is a mistake to regard strain as a physically meaningful quantity without some notion of the L in the denominator, which acts as a gauge length. For another example, consider the physical meaning (none)of adding strain (dimensionless) to Mach (dimensionless).

I don't understand this. Are you arguing that even dimensionless numbers should keep their dimensions? I can't make mathematical sense of that. Is L/L a different unit in your system than M/M? --AxelBoldt

It's not "my system", it's physics. L/L is a different unit than M/M. Yes, I am saying that dimensionless numbers should keep their dimensions. Think about it carefully. The real numbers used for computing physical quantities are meaningless without units. Velocities must be expressed in terms of L/T, whether it be meters/sec or furlongs/fortnight. The problem is that while the real numbers obey the axioms for a field, units obey group axioms. We can do math on the reals alone (analysis), the units alone (group theory) or real numbers with units attached to each quantity (typed family of fields). In the scalar world all of this is pedantic frippery. However, linear systems constructed to solved differential equations describing matter will have units attached. As it turns out, with care, units may be mixed within the system, and a solution determined using LU decomposition (say) will remain dimensionally correct. You can integrate this stuff too... (heh heh) If strain didn't keep its L/L dimensions, then strain energy per unit volume (FL/L^3) would just be F/L^2 which is units of stress.

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No further comments or analysis tolerated on dimensional analysis

Can't you see how many people wrote about it?

• Barenblatt[?], G. I., "Scaling, Self-Similarity, and Intermediate Asymptotics", Cambridge University Press, 1996
• Bridgman, P. W., "Dimensional Analysis", Yale University Press, 1937
• Langhaar, H. L., "Dimensional Analysis and Theory of Models", Wiley, 1951
• Murphy, N. F., Dimensional Analysis, Bull. V.P.I., 1949, 42(6)
• Porter, "The Method of Dimensions", Methuen, 1933
• Boucher and Alves, Dimensionless Numbers, Chem. Eng. Progress, 1960, 55, pp.55-64
• Buckingham, E., On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis, Phys. Rev, 1914, 4, p.345
• Klinkenberg A. Chem. Eng. Science, 1955, 4, pp. 130-140, 167-177
• Rayleigh, Lord, The Principle of Similitude, Nature 1915, 95, pp. 66-68
• Silberberg, I. H. and McKetta J. J., Jr., Learning How to Use Dimensional Analysis, Petrol. Refiner, 1953, 32(4), p179; (5), p.147; (6), p.101; (7), p. 129
• Van Driest, E. R., On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems, J. App. Mech, 1946, 68, A-34, March
• Perry, J. H. et al., "Standard System of Nomenclature for Chemical Engineering Unit Operations", Trans. Am. Inst. Chem. Engrs., 1944, 40, 251
• Moody, L. F., "Friction Factors for Pipe Flow", Trans. Am. Soc. Mech. Engrs., 1944, 66, 671

Who the heck do you think you are?

little guru