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The Buckingham’s ‘ Pi ’ theorem is a key theorem in dimensional analysis. The theorem 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 qualities, then the original expression is equivalent to an equation involving a set of p = n – k dimensionless variables constructed from the original variables. For the purpose of the experimenter, different systems which share the same description in terms of these dimensionless numbers are equivalent.

**Buckingham’s ‘ Pi ’ theorem.**

The **Buckingham’s ‘ Pi ’ theorem** is a key theorem in
dimensional analysis. The theorem 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 qualities, then the original expression is
equivalent to an equation involving a set of p = n – k dimensionless variables
constructed from the original variables. For the purpose 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 ( q1, q2,……..qn ) = 0

where the qi are the n physical variables and
they are expressed in terms of k independent physical units, the above equation
can be restated as

F (?1, ?2………. ?p) = 0

Where, the
?i are the dimensionless parameters constructed from the qi by p = n–k equations of the form

?i = q1^{m}, q2,^{m}……..qn^{m}

Where the exponents mi are the
rational numbers. The use of the ?i the dimensionless parameters was introduced by Edge Buckingham in
his original 1914 paper on the subject from which the theorem draws its name.

Most importantly, the Buckingham ? theorem provides a method for
computing sets 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’.

Proofs of the **?** theorem often begin by considering the space
of fundamental and derived physical unit’s 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 the regarded as imposing linear constraints in the physical units vector
space.

The theorem describes how every physical meaningful equation
involving **n **variables can be equivalently rewritten as an equation of n
– m dimensionless** **parameters, where **m** 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 experimental list who wants to
determine the form of the equation can choose the most convenient one.

The **?** 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 ?

theorem is essentially a Gauss – Jordan
elimination carried out in this vector space.

Tags : Civil - Mechanics Of Fluids - Dimensional Analysis And Model Studies

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