On the nondimensionalization of coupled, nonlinear ordinary differential equations

Abstract

The first step to simplify the analysis of a mathematical model is to search the dimensionless groups that control its solution patterns since the solution of any equation or system of equations that define the laws that rule a physical or engineering problem can be represented as a relation between these groups. One of the current techniques used to attain this aim is the nondimensionalization of the governing equations and boundary conditions of the problem. However, while its application to problems whose models are set by coupled, partial differential equations is extensively used in the scientific literature, it is very difficult to find works in which the nondimensionalization procedure is applied to coupled, ordinary differential equations. The main reason of this no application probably rests in the difficulty of a good definition of the reference quantities that need to be chosen to make dimensionless the dependent and independent variables of the problem. Providing a physical justification to the choice of the reference quantities, two illustrative applications are studied in this work. The dimensionless groups derived from nondimensionalization are checked by solving numerically these problems.