Characteristic Values in the Scaling of Differential Equations in Engineering

Abstract

This work introduces, for the first time, a formal approach to the estimation of characteristic values of differential and other related expressions in the scaling of engineering problems. The methodology introduced aims at overcoming the inability of the traditional approach to match the exact solution of asymptotic cases. This limitation of the traditional approach often leaves in doubt whether the scaling laws obtained actually represent the desired phenomena. The formal approach presented yields estimates with smaller error than traditional approaches; these improved estimates converge to the exact solution in simple asymptotic cases and do not diverge from the exact solution in cases in which the error of traditional approaches is unbounded. The significance of this contribution is that it extends the range of applicability of scaling estimates to problems for which traditional approaches were deemed unreliable, for example, cases in which the curvature of functions is large, or complex cases in which the accumulation of estimation errors exceeds reasonable limits. This research is part of a larger effort towards a computational implementation of scaling, and it is especially valuable for approximating multicoupled, multiphysics problems in continuum mechanics (e.g., coupled heat transfer, fluid flow, and electromagnetics) that are often difficult to analyze numerically or empirically.