Illustration of Pi Theorem

Let us consider the following example to illustrate the procedure of determining the various steps in the pi theorem.

Example (Pressure drop in a pipe flow)

Consider a steady flow of an incompressible Newtonian fluid through a long, smooth walled, horizontal circular pipe. It is required to measure the pressure drop per unit length of the pipe and find the number of non-dimensional parameters involved in the problem. Also, it is desired to know the functional relation among these dimensionless parameters.

Step I: Let us express all the pertinent variables involved in the experimentation of pressure drop per unit length of the pipe, in the following form;

(6.1.3)

where, D is the pipe diameter, ρ is the fluid density, μ is the viscosity of the fluid and V is the mean velocity at which the fluid is flowing through the pipe.

Step II: Next step is to express all the variables in terms of basic dimensions i.e. . It then follows that

(6.1.4)

Step III: Apply Buckingham theorem to decide the number of pi terms required. There are five variables (including the dependent variable ) and three reference dimensions. Since, , only two pi terms are required for this problem.

Step IV: The repeating variables to form pi terms, need to be selected from the list . It is to be noted that the dependent variable should not be used as one of the repeating variable. Since, there are three reference dimensions involved, so we need to select three repeating variable. These repeating variables should be dimensionally independent, i.e. dimensionless product cannot be formed from this set. In this case, may be chosen as the repeating variables.