1.1 Introduction

A few basic laws are fundamental to the subject of fluid mechanics: the law of conservation of mass, Newton s laws, and the laws of thermodynamics. These laws bear a remarkable similarity to one another in their structure. They all state that if a given volume of the fluid is investigated, quantities such as mass, momentum, and energy will change due to internal causes, net change in that quantity entering and leaving the volume, and action on the surface of the volume due to external agents. In fluid mechanics, these laws are best expressed in rate form.

Since these laws have to do with some quantities entering and leaving the volume and other quantities changing inside the volume, in applying these fundamental laws to a finite-size volume it can be expected that both terms involving surface and volume integrals would result. In some cases a global description is satisfactory for carrying out further analysis, but often a local statement of the laws in the form of differential equations is preferred to obtain more detailed information on the behavior of the quantity under investigation.

There is a way to convert certain types of surface integrals to volume integrals that is extremely useful for developing the derivations. This is the divergence theorem, expressed in the form

In this expression U ^[1] is an arbitrary vector and n is a unit normal to the surface S. The closed surface completely surrounding the volume V is S.