Editing Buckingham π theorem (section)

=== Electric power ===
To demonstrate the application of the {{pi}} theorem, consider the [[electric power|power]] consumption of a [[Electric generator|stirrer]] with a given shape.
The power, ''P'', in dimensions [M · L<sup>2</sup>/T<sup>3</sup>], is a function of the [[density]], ''ρ'' [M/L<sup>3</sup>], and the [[viscosity]] of the fluid to be stirred, ''μ'' [M/(L · T)], as well as the size of the stirrer given by its [[diameter]], ''D'' [L], and the [[Angular velocity|angular speed]] of the stirrer, ''n'' [1/T]. Therefore, we have a total of ''n'' = 5 variables representing our example.  Those ''n'' = 5 variables are built up from ''k'' = 3 independent dimensions, e.g., length: L ([[SI]] units: [[meters|m]]), time: T ([[second|s]]), and mass: M ([[kilograms|kg]]).

According to the {{pi}}-theorem, the ''n'' = 5 variables can be reduced by the ''k'' = 3 dimensions to form ''p'' = ''n'' − ''k'' = 5 − 3 = 2 independent dimensionless numbers. Usually, these quantities are chosen as <math display=inline>\mathrm{Re} = {\frac{\rho n D^2}{\mu}}</math>, commonly named the [[Reynolds number]] which describes the fluid flow regime, and <math display=inline>N_\mathrm{p} = \frac{P}{\rho n^3 D^5}</math>, the [[power number]], which is the dimensionless description of the stirrer.

Note that the two dimensionless quantities are not unique and depend on which of the ''n'' = 5 variables are chosen as the ''k'' = 3 dimensionally independent basis variables, which, in this example, appear in both dimensionless quantities. The Reynolds number and power number fall from the above analysis if <math display=inline>\rho</math>, ''n'', and ''D'' are chosen to be the basis variables. If, instead, <math display=inline>\mu</math>, ''n'', and ''D'' are selected, the Reynolds number is recovered while the second dimensionless quantity becomes <math display=inline>N_\mathrm{Rep} = \frac{P}{\mu D^3 n^2}</math>. We note that <math display=inline>N_\mathrm{Rep}</math> is the product of the Reynolds number and the power number.