Buckingham π theorem

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The Vaschy-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 = nk  dimensionless parameters constructed from the original variables: it is a scheme for nondimensionalization. This 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: Vaschy-Buckingham's theorem only provides a way of generating sets of dimensionless parameters, and will not choose the most 'physically meaningful'.

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Statement

More formally, the number of dimensionless terms that can be formed, p, is equal to the nullity of the dimensional matrix, and k is the rank. 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

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

where the πi are dimensionless parameters constructed from the qi  by p = nk  equations of the form

where the exponents mi  are rational numbers (they can always be taken to be integers: just raise it to a power to clear denominators).

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.

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