Buckingham π theorem

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The Buckingham π theorem is a key theorem in dimensional analysis. The theorem loosely states that if we have a physically meaningful equation involving a certain number, n, of physical variables, and these variables are expressible in terms of k  independent fundamental physical quantities, then the original expression is equivalent to an equation involving a set of p = n − k  dimensionless variables constructed from the original variables. More accurately, the number of dimensionless terms that can be formed, p, is equal to the nullity of the dimensional matrix. For the purposes of the experimenter, different systems which 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 the qi  are the n  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 the πi are dimensionless parameters constructed from the qi  by p = n − k  equations of the form

\pi_i=q_1^{m_1}\,q_2^{m_2}\ldots q_n^{m_n}

where the exponents mi  are rational numbers. The use of the πi as the dimensionless parameters was introduced by Edgar Buckingham in his original 1914 paper on the subject from which the theorem draws its name.

Most importantly, the Buckingham π theorem provides a method for computing sets of dimensionless parameters from the given variables, even if the form of the equation is still unknown. However, the choice of dimensionless parameters is not unique: Buckingham's theorem only provides a way of generating sets of dimensionless parameters, and will not choose the most 'physically meaningful'.

Contents

[edit] Proving the π theorem

Proofs of the π theorem often begin by considering the space of fundamental and derived physical units as a vector space, 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.

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

The π-theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − k dimensionless parameters, where k is the number of fundamental units used. Furthermore, and most importantly, 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 called similar; 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.

The π-theorem uses linear algebra: the space of all possible physical units can be seen as a vector space over the rational numbers if we represent a unit as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). Multiplication of physical units is then represented by vector addition within this vector space. The algorithm of the π-theorem is essentially a Gauss-Jordan elimination carried out in this vector space.

[edit] Examples

[edit] 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 units of length divided by time squared. The model is of the form

f(T,M,L,g) = 0.\,

There are only three fundamental physical units in this equation: mass, time, and length. Thus we need only 4−3=1 dimensionless parameter, denoted π, and the model can be re-expressed as

f(\pi) = 0\,

where π is given by

\pi\,\! =(T)^{m_1}(M)^{m_2}(L)^{m_3}(g)^{m_4}\,
=(T)^{m_1}(M)^{m_2}(L)^{m_3}(L/T^2)^{m_4}\,

for some values of m1…m4. With a little thought or experimentation it can be found that only

\pi\,\! =(T)^2(M)^0(L)^{-1}(L/T^2)^1\,
=gT^2/L\,

(or some power thereof) satisfies this requirement. Note that if m2 were non-zero there would be no way to cancel the M value—therefore m2 must be zero. Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass.

The model can now be expressed as

f(gT^2/L) = 0.\,

Assuming the zeroes of f  are discrete, we can say gT2/L = Kn  where Kn  is the nth zero. If there is only one zero, then gT2/L=K . 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 K=4π2 .

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 in the limit that this angle is zero.

[edit] The Atomic bomb

In 1941, Sir Geoffrey I. Taylor used dimensional analysis to estimate the energy released in an atomic bomb explosion (Taylor, 1950a,b). The first atomic bomb was detonated near Alamogordo, New Mexico on July 16, 1945. In 1947, movies of the explosion were declassified, allowing Sir Geoffrey to complete the analysis and estimate the energy released in the explosion, even though the energy release was still classified. The actual energy released was later declassified and its value was remarkably close to Taylor's estimate.

Taylor supposed that the description of the process was adequately described by five physical quantities, the time t  since the detonation, the energy E  which is released at a single point in space at detonation, the radius R  of the shock wave at time t , the atmospheric pressure p  and the ambient density ρ. There are only three fundamental physical units in this equation: mass, time, and length. Thus we need only 5−3=2 dimensionless parameters, which can be found to be

\pi_0=R\,\left(\frac{\rho}{Et^2}\right)^{1/5}

and

\pi_1=p\,\left(\frac{t^6}{E^2\rho^3}\right)^{1/5}.

The process can now be described by an equation of the form

f(\pi_0,\pi_1)=0,\,

or, equivalently

R=\left(\frac{Et^2}{\rho}\right)^{1/5}g(\pi_1),

where g(π1) is some function of π1. The energy in the explosion is expected to be huge, so that for times of the order of a second after the explosion, we can estimate π1 to be approximately zero, and experiments using light explosives can be conducted to determine that g(0) is on the order of unity so that

R\approx\left(\frac{Et^2}{\rho}\right)^{1/5}.

This is Taylor's equation which, once he knew the radius of the explosion as a function of the time, allowed him to calculate the energy of the explosion. (Wan, 1989)

[edit] See also

[edit] References

  • Buckingham, E. (1915). "The principle of similitude". Nature 96: 396-397. 
  • Buckingham, E. (1915). "Model experiments and the forms of empirical equations". Trans. A.S.M.E 37: 263-296. 
  • Kline, Stephen J. (1986). Similitude and Approximation Theory. Springer-Verlag, New York. ISBN 0387165185. 
  • Taylor, Sir G. (1950). "The Formation of a Blast Wave by a Very Intense Explosion. I. Theoretical Discussion". Proc. Roy. Soc. A 201: 159-174. 
  • Taylor, Sir G. (1950). "The Formation of a Blast Wave by a Very Intense Explosion. II. The Atomic Explosion of 1945". Proc. Roy. Soc. A 201: 175-186. 
  • Wan, Frederic Y.M. (1989). Mathematical Models and their Analysis. Harper & Row Publishers, New York. ISBN 0060469021. 


To do: continue proof sketch...

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