Asymptotic Analysis — An Introduction

Abstract

The extreme generality of the full Navier-Stokes-Fourier system whereby the equations describe the entire spectrum of possible motions — ranging from sound waves, cyclone waves in the atmosphere, to models of gaseous stars in astrophysics — constitutes a serious defect of the equations from the point of view of applications. Eliminating unwanted or unimportant modes of motion, and building in the essential balances between flow fields, allow the investigator to better focus on a particular class of phenomena and to potentially achieve a deeper understanding of the problem. Scaling and asymptotic analysis play an important role in this approach. By scaling the equations, meaning by choosing appropriately the system of the reference units, the parameters determining the behavior of the system become explicit. Asymptotic analysis provides a useful tool in the situations when certain of these parameters called characteristic numbers vanish or become infinite.