INTRODUCTION

The balance equations derived in the preceding chapter are the basis for all of the modeling work to come. When combined with constitutive equations and initial or boundary conditions, they provide the governing equations for most models. However, these governing equations are often difficult to solve in their general form, because they are multidimensional and nonlinear.

In this chapter we discuss a method for systematically simplifying the governing equations by determining which terms can be safely neglected in a given problem. We use the values of the physical parameters in the problem to estimate the relative importance of each term in the governing equations, using a particular procedure we call scaling. We then simplify the equations by neglecting terms that are “small” in comparison to terms that are “large.” This produces governing equations that are often simpler than the original forms, but they still reflect all the important phenomena of the problem.

Scaling analysis produces other important results as well. Through scaling we learn the characteristic values of all of the problem variables. We also derive dimensionless parameters that have physical meaning for the particular problem. We can even determine whether the solution is likely to contain a boundary layer, and how large that layer will be. This makes scaling analysis a richer and more useful tool than traditional dimensional analysis. That is why we regard scaling as an essential step in model development.