Nondimensionalization - Nondimensionalization Steps

Nondimensionalization Steps

To nondimensionalize a system of equations, one must do the following:

1. Identify all the independent and dependent variables;
2. Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined;
3. Divide through by the coefficient of the highest order polynomial or derivative term;
4. Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1;
5. Rewrite the system of equations in terms of their new dimensionless quantities.

The last three steps are usually specific to the problem where nondimensionalization is applied. However, almost all systems require the first two steps to be performed.

As an illustrative example, consider a first order differential equation with constant coefficients:

1. In this equation the independent variable here is t, and the dependent variable is x.
2. Set . This results in the equation

3. The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives

4. The coefficient in front of χ only contains one characteristic variable t_c, hence it is easiest to choose to set this to unity first:

   Subsequently,

5. The final dimensionless equation in this case becomes completely independent of any parameters with units: