Buckingham-Pi TheoremEdit


The Buckingham theorem, or also called the Pi theorem, is a fundamental theorem regarding dimensional analysis of a physical problem. Its formulation stems from the principle of dimensional invariance. Its applications are various and numerous. Particularly, it is commonly used in thermodynamics and fluid mecanics.

Mathematical formulationEdit

If we consider the physical problem described by the following equation:

Formule 1

Then every xi represents n physics variables involving r independent fundamental units. Then we can rewrite the equation with p=n-r dimensionless parameters :

Formule 2

Where every parameter is built with the original xi variables and each mi is double.

Example of the Buckingham theorem applicationEdit

Let's consider a simple physics problem. We want to establish the drag coefficient of this 2D cylinder in a permanent fluid flow.


According to this configuration, an intuitive relaltionship can be assumed under the following form :

Formule 3

Now we have to decompose the physics variables into independent fundamental parameters. For example, with a dimensional analysis, we can say : [Fx]=[L]1[M]1[T]-2 The dimensional matrix is a summary of the problem :


We can deduce the rank of this matrix r=3. According to the theorem, 3 dimensionless parameters enable us to transform this equation.

Formule 4

In this example, p=n-r=6-3=3

As a consequence we can rewrite the equation under its new form :

Formule 5

Consequences of the theoremEdit

In the litterature every experiment carried is usually related to dimensionless numbers as inputs. Among this dimensionless numbers, the most spread are the Reynolds Number, the Mach Number, the Prandtl number and the Strouhal number, among others.

It is absolutly not recommended to individually vary the parameters ρ, V, L or μ since they all appear in the Reynolds Number:

In the same manner, forces are reported to undimensional coefficients like the lift coefficient or the drag coefficient.

For our example regarding the cylinder problem, with a low Mach number to avoid any compressible effects, the result of the drag coefficient can be illustrated as follows:

Schéma 2


[1] Characteristics Quantities and Dimensional Analysis, by G. Grimvall, April 2008, in Scientific Modeling and simulations. Page 17-47

[2] Dimensional Analysis, by W.D. Curtis, Linear Algebra Application

[3] A generalization of the Pi-theorem and dimensional analysis, by Ain A. Sonin