Mathematical properties of the fluid dynamic equations

Abstract

The governing equations of fluid dynamics derived in Chap. 2 are either integralforms (such as Eq. (2.23) obtained directly from a finite control volume) or partialdifferential equations (such as Eqs (2.36a-c) obtained directly from an infinitesimalfluid element). The governing equations in the form of partial differential equationsare by far the most prevalent form used in computational fluid dynamics. Therefore,before taking up a study of numerical methods for the solution of these equations,it is useful to examine some mathematical properties of partial differential equationsthemselves. Any valid numerical solution of the equations should exhibit theproperty of obeying the general mathematical properties of the governing equations.Examine the governing equations of fluid dynamics as derived in Chap. 2. Notethat in all cases the highest order derivatives occur linearly, i.e. there are no productsor exponentials of the highest order derivatives-they appear by themselves, multipliedby coefficients which are functions of the dependent variables themselves.Such a system of equations is called a quasilinear system. For example, for inviscidflows, examining the equations in Sect. 2.8.2 we find that the highest order derivativesare first order, and all of them appear linearly. For viscous flows, examining theequations in Sect. 2.8.1 we find the highest order derivatives are second order, andthey always occur linearly. For this reason, in the next section, let us examine someproperties of a system of quasilinear partial differential equations. In the process,we will establish a classification of three types of partial differential equations-allthree of which are encountered in fluid dynamics. © Springer-Verlag Berlin Heidelberg 2009.