Concept

Buckingham's pi theorem — where it appears

The theorem that a relation among quantities can be rewritten among a smaller number of dimensionless groups, fewer by the rank of the dimension matrix. It says how many numbers a problem really has, and it never says what the relation is.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

5 quantities, 3 rows, 2 left over. The dimension matrix for the drag on a sphere: one column per quantity, one row per base dimension, and every entry an exponent. Buckingham's theorem is a statement about this matrix and nothing else — the number of independent dimensionless groups is the number of columns minus the rank, computed here by elimination. Nothing about fluids enters until somebody decides which columns to write down.

Counting what matters

Five quantities decide the drag on a sphere, and the experiment that measures it has one curve in it rather than a five-dimensional table. The reason is a rank: the matrix of dimensions has three independent rows, and what is left over is the number of dimensionless groups the answer can possibly depend on.

regimes · Dimensional

1.139 asks for a Francis. The specific-speed axis, with the four machine types on it and one duty marked: 3 m³/s at 60 m, on a shaft turning at 750 rev/min. The number is 1.1387, and the choice of runner follows from it before any blade has been drawn. What the number contains is a ratio of flow to head; what it does not contain is any size at all, which is why one axis serves a garden pump and a gigawatt turbine.

One number picks the machine

A flow rate, a head and a shaft speed contain exactly one dimensionless combination with no size in it. That combination decides whether a duty wants an impulse wheel, a Francis runner or a propeller — before anything has been drawn, sized, or costed.

applied · Specific speed

The shock radius against time, from an equation that was given no exponent. The thin-shell energy balance integrated forward from a small initial radius, on logarithmic axes. The two-fifths power is not put in: the ordinary differential equation is Ṙ = √(E/AρR³), and the straight line is what it does. The fitted slope is 0.39983 and the fitted prefactor is 0.90721 against a closed form of 0.90702 — the small residuals being the integration's memory of where it started, which the similarity solution has no equivalent of.

A radius that gives the energy away

Four quantities, three dimensions, one group. The radius of a strong blast must be a constant times (Et²/ρ)¹⁄⁵, and nothing about the device, the chemistry or the initial size can appear. The exponent is free and the constant is not.

compressible · Blast wave

The same data, in a basis the theorem allows just as much. The identical two hundred and forty points, plotted as F/(mu U d) — which is the drag coefficient times the Reynolds number, a perfectly legitimate pi group forming a complete pair with the abscissa. The result is a straight line of slope 0.9985 through five decades with an r-squared of 0.9985, and it contains no physics: the ordinate contains the abscissa.

The groups are not the only groups

Buckingham's theorem fixes how many dimensionless groups an answer can depend on and says nothing about which. Two of the infinitely many legitimate choices are used here on the same data: one manufactures a straight line through five decades out of a constant, and the other erases Stokes' law completely.

regimes · Dimensional

Named alongside it

The objects these essays reach for when they reach for this one.

Dimensional analysis Scaling Rank Correlation Dimensionless Drag coefficient Measurement Reynolds number Similarity Affinity laws Blast wave Dimensionless number

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