Buckingham π theorem

Edgar Buckingham circa 1886

In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters π₁, π₂, ..., π_p constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)

The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.

The Buckingham π theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (for example, pressure and volume are linked by Boyle's law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and the theorem would not hold.

History

Although named for Edgar Buckingham, the π theorem was first proved by French mathematician Joseph Bertrand in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem ("the method of dimensions") became widely known due to the works of Rayleigh. The first application of the π theorem in the general case to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892, a heuristic proof with the use of series expansions, to 1894.

Formal generalization of the π theorem for the case of arbitrarily many quantities was given first by A. Vaschy in 1892, then in 1911—apparently independently—by both A. Federman and D. Riabouchinsky, and again in 1914 by Buckingham. It was Buckingham's article that introduced the use of the symbol "$\pi _{i}$" for the dimensionless variables (or parameters), and this is the source of the theorem's name.

Statement

More formally, the number $p$ of dimensionless terms that can be formed is equal to the nullity of the dimensional matrix, and $k$ is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.

In mathematical terms, if we have a physically meaningful equation such as $$f(q_{1},q_{2},\ldots ,q_{n})=0,$$ where $q_{1},\ldots ,q_{n}$ are the $n$ independent physical variables, and they are expressed in terms of $k$ independent physical units, then the above equation can be restated as $$F(\pi _{1},\pi _{2},\ldots ,\pi _{p})=0,$$ where $\pi _{1},\ldots ,\pi _{p}$ are dimensionless parameters constructed from the $q_{i}$ by $p=n-k$ dimensionless equations — the so-called Pi groups — of the form $$\pi _{i}=q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}},$$ where the exponents $a_{i}$ are rational numbers. (They can always be taken to be integers by redefining $\pi _{i}$ as being raised to a power that clears all denominators.)

Significance

The Buckingham π theorem provides a method for computing sets of dimensionless parameters from given variables, even if the form of the equation remains unknown. However, the choice of dimensionless parameters is not unique; Buckingham's theorem only provides a way of generating sets of dimensionless parameters and does not indicate the most "physically meaningful".

Two systems for which these parameters coincide are called similar (as with similar triangles, they differ only in scale); they are 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. Most importantly, Buckingham's theorem describes the relation between the number of variables and fundamental dimensions.

Proof

Outline

It will be assumed that the space of fundamental and derived physical units forms a vector space over the rational numbers, with the fundamental units as basis vectors, and with multiplication of physical units as the "vector addition" operation, and raising to powers as the "scalar multiplication" operation: represent a dimensional variable as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). For instance, the standard gravity $g$ has units of $D/T^{2}=D^{1}T^{-2}$ (distance over time squared), so it is represented as the vector $(1,-2)$ with respect to the basis of fundamental units (distance, time).

Making the physical units match across sets of physical equations can then be regarded as imposing linear constraints in the physical-units vector space.

Formal proof

Given a system of $n$ dimensional variables $q_{1},\ldots ,q_{n}$ (with physical dimensions) in $k$ fundamental (basis) dimensions, the dimensional matrix is the $k\times n$ matrix $M$ whose $k$ rows are the fundamental dimensions and whose $n$ columns are the dimensions of the variables: the $(i,j)$th entry (where $1\leq i\leq k$ and $1\leq j\leq n$) is the power of the $i$th fundamental dimension in the $j$th variable. The matrix can be interpreted as taking in a combination of the dimensions of the variable quantities and giving out the dimensions of this product in fundamental dimensions. So the $k\times 1$ (column) vector that results from the multiplication $$M{\begin{bmatrix}a_{1}\\\vdots \\a_{n}\end{bmatrix}}$$ consists of the units of $$q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}}$$ in terms of the $k$ fundamental independent (basis) units.

A dimensionless variable is a quantity that has all of its fundamental dimensions raised to the zeroth power (the zero vector of the vector space over the fundamental dimensions). The dimensionless variables are exactly the vectors that belong to the kernel $\ker M$ of this matrix.

By the rank–nullity theorem, a system of $n$ vectors (matrix columns) in $k$ linearly independent dimensions (the rank of the matrix is the number of fundamental dimensions) leaves a nullity $p=\dim(\ker M)$ satisfying $p=n-k,$ where the nullity is the number of extraneous dimensions which may be chosen to be dimensionless.

The dimensionless variables can always be taken to be integer combinations of the dimensional variables (by clearing denominators). There is mathematically no natural choice of dimensionless variables; some choices of dimensionless variables are more physically meaningful, and these are what are ideally used.

The International System of Units defines $k=7$ base units, which are the ampere, kelvin, second, metre, kilogram, candela and mole. It is sometimes advantageous to introduce additional base units and techniques to refine the technique of dimensional analysis. (See orientational analysis and reference.)

Examples

Speed

This example is elementary but serves to demonstrate the procedure.

Suppose a car is driving at 100 km/h; how long does it take to go 200 km?

This question considers $n=3$ dimensioned variables: distance $d,$ time $t,$ and speed $v,$ and we are seeking some law of the form $t=\operatorname {Duration} (v,d).$ These variables admit a basis of $k=2$ dimensions: time dimension $T$ and distance dimension $D.$ Thus there is $p=n-k=3-2=1$ dimensionless quantity.

The dimensional matrix is $$M={\begin{bmatrix}1&0&\;\;\;1\\0&1&-1\end{bmatrix}}$$ in which the rows correspond to the basis dimensions $D$ and $T,$ and the columns to the considered dimensions $D,T,{\text{ and }}V,$ where the latter stands for the speed dimension. The elements of the matrix correspond to the powers to which the respective dimensions are to be raised. For instance, the third column $(1,-1),$ states that $V=D^{0}T^{0}V^{1},$ represented by the column vector $\mathbf {v} =[0,0,1],$ is expressible in terms of the basis dimensions as $V=D^{1}T^{-1}=D/T,$ since $M\mathbf {v} =[1,-1].$

For a dimensionless constant $\pi =D^{a_{1}}T^{a_{2}}V^{a_{3}},$ we are looking for vectors $\mathbf {a} =[a_{1},a_{2},a_{3}]$ such that the matrix-vector product $M\mathbf {a}$ equals the zero vector $[0,0].$ In linear algebra, the set of vectors with this property is known as the kernel (or nullspace) of (the linear map represented by) the dimensional matrix. In this particular case its kernel is one-dimensional. The dimensional matrix as written above is in reduced row echelon form, so one can read off a non-zero kernel vector to within a multiplicative constant: $$\mathbf {a} ={\begin{bmatrix}-1\\\;\;\;1\\\;\;\;1\\\end{bmatrix}}.$$

If the dimensional matrix were not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant, replacing the dimensions by the corresponding dimensioned variables, may be written: $$\pi =d^{-1}t^{1}v^{1}=tv/d.$$

Since the kernel is only defined to within a multiplicative constant, the above dimensionless constant raised to any arbitrary power yields another (equivalent) dimensionless constant.

Dimensional analysis has thus provided a general equation relating the three physical variables: $$F(\pi )=0,$$ or, letting $C$ denote a zero of function $F,$ $$\pi =C,$$ which can be written in the desired form (which recall was $t=\operatorname {Duration} (v,d)$) as $$t=C{\frac {d}{v}}.$$

The actual relationship between the three variables is simply $d=vt.$ In other words, in this case $F$ has one physically relevant root, and it is unity. The fact that only a single value of $C$ will do and that it is equal to 1 is not revealed by the technique of dimensional analysis.

The simple pendulum

We wish to determine the period $T$ of small oscillations in a simple pendulum. It will be assumed that it is a function of the length $L,$ the mass $M,$ and the acceleration due to gravity on the surface of the Earth $g,$ which has dimensions of length divided by time squared. The model is of the form $$f(T,M,L,g)=0.$$

(Note that it is written as a relation, not as a function: $T$ is not written here as a function of $M,L,{\text{ and }}g.$)

There are $k=3$ fundamental physical dimensions in this equation: time $t,$ mass $m,$ and length $\ell ,$ and $n=4$ dimensional variables, $T,M,L,{\text{ and }}g.$ Thus we need only $p=n-k=4-3=1$ dimensionless parameter, denoted by $\pi ,$ and the model can be re-expressed as $$F(\pi )=0,$$ where $\pi$ is given by $$\pi =T^{a_{1}}M^{a_{2}}L^{a_{3}}g^{a_{4}}$$ for some values of $a_{1},a_{2},a_{3},a_{4}.$

The dimensions of the dimensional quantities are: $$T=t,M=m,L=\ell ,g=\ell /t^{2}.$$

The dimensional matrix is: $$\mathbf {M} ={\begin{bmatrix}1&0&0&-2\\0&1&0&0\\0&0&1&1\end{bmatrix}}.$$

(The rows correspond to the dimensions $t,m,$ and $\ell ,$ and the columns to the dimensional variables $T,M,L,{\text{ and }}g.$ For instance, the 4th column, $(-2,0,1),$ states that the $g$ variable has dimensions of $t^{-2}m^{0}\ell ^{1}.$)

We are looking for a kernel vector $a=\left[a_{1},a_{2},a_{3},a_{4}\right]$ such that the matrix product of $\mathbf {M}$ on $a$ yields the zero vector $[0,0,0].$ The dimensional matrix as written above is in reduced row echelon form, so one can read off a kernel vector within a multiplicative constant: $$a={\begin{bmatrix}2\\0\\-1\\1\end{bmatrix}}.$$

Were it not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant may be written: $${\begin{aligned}\pi &=T^{2}M^{0}L^{-1}g^{1}\\&=gT^{2}/L\end{aligned}}.$$ In fundamental terms: $$\pi =(t)^{2}(m)^{0}(\ell )^{-1}\left(\ell /t^{2}\right)^{1}=1,$$ which is dimensionless. Since the kernel is only defined to within a multiplicative constant, if the above dimensionless constant is raised to any arbitrary power, it will yield another equivalent dimensionless constant.

In this example, three of the four dimensional quantities are fundamental units, so the last (which is $g$) must be a combination of the previous. Note that if $a_{2}$ (the coefficient of $M$) had been non-zero then there would be no way to cancel the $M$ value; therefore $a_{2}$ must be zero. Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass $M.$ (In the 3D space of powers of mass, time, and distance, we can say that the vector for mass is linearly independent from the vectors for the three other variables. Up to a scaling factor, ${\vec {g}}+2{\vec {T}}-{\vec {L}}$ is the only nontrivial way to construct a vector of a dimensionless parameter.)

The model can now be expressed as: $$F\left(gT^{2}/L\right)=0.$$

Assuming the zeroes of $F$ are discrete and that they are labelled $C_{1},C_{2},\ldots ,$ then this implies that $gT^{2}/L=C_{i}$ for some zero $C_{i}$ of the function $F.$ If there is only one zero, call it $C,$ then $gT^{2}/L=C.$ It requires more physical insight or an experiment to show that there is indeed only one zero and that the constant is in fact given by $C=4\pi ^{2}.$

For large oscillations of a pendulum, the analysis is complicated by an additional dimensionless parameter, the maximum swing angle. The above analysis is a good approximation as the angle approaches zero.

Other examples

An example of dimensional analysis can be found for the case of the mechanics of a thin, solid and parallel-sided rotating disc. There are five variables involved which reduce to two non-dimensional groups. The relationship between these can be determined by numerical experiment using, for example, the finite element method.

The theorem has also been used in fields other than physics, for instance in sport sciences.

See also

• Blast wave
• Dimensionless quantity
• Natural units
• Similitude (model)
• Reynolds number

References

Bibliography

• Hanche-Olsen, Harald (2004). "Buckingham's pi-theorem" (PDF). NTNU. Retrieved April 9, 2007.
• Hart, George W. (March 1, 1995). Multidimensional Analysis: Algebras and Systems for Science and Engineering. Springer-Verlag. ISBN 978-0-387-94417-3.
• Kline, Stephen J. (1986). Similitude and Approximation Theory. Springer-Verlag, New York. ISBN 978-0-387-16518-9.
• Hartke, Jan-David (2019). "On Buckingham's Π-theorem". arXiv:1912.08744.
• Wan, Frederic Y.M. (1989). Mathematical Models and their Analysis. Harper & Row Publishers, New York. ISBN 978-0-06-046902-3.
• Vignaux, G.A. (1991). "Dimensional analysis in data modelling" (PDF). Victoria University of Wellington. Retrieved December 15, 2005.
• Mike Sheppard, 2007 Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants
• Gibbings, J.C. (2011). Dimensional Analysis. Springer. ISBN 978-1-84996-316-9.

Original sources

• Vaschy, A. (1892). "Sur les lois de similitude en physique". Annales Télégraphiques. 19: 25–28.
• Buckingham, E. (1914). "On physically similar systems; illustrations of the use of dimensional equations". Physical Review. 4 (4): 345–376. Bibcode:1914PhRv....4..345B. doi:10.1103/PhysRev.4.345. hdl:10338.dmlcz/101743.
• Buckingham, E. (1915). "The principle of similitude". Nature. 96 (2406): 396–397. Bibcode:1915Natur..96..396B. doi:10.1038/096396d0. S2CID 3956628.
• Buckingham, E. (1915). "Model experiments and the forms of empirical equations". Transactions of the American Society of Mechanical Engineers. 37: 263–296.
• Taylor, Sir G. (1950). "The Formation of a Blast Wave by a Very Intense Explosion. I. Theoretical Discussion". Proceedings of the Royal Society A. 201 (1065): 159–174. Bibcode:1950RSPSA.201..159T. doi:10.1098/rspa.1950.0049. S2CID 54070514.
• Taylor, Sir G. (1950). "The Formation of a Blast Wave by a Very Intense Explosion. II. The Atomic Explosion of 1945". Proceedings of the Royal Society A. 201 (1065): 175–186. Bibcode:1950RSPSA.201..175T. doi:10.1098/rspa.1950.0050.

External links

• Some reviews and original sources on the history of pi theorem and the theory of similarity (in Russian)