Conservation Equations

Conservation laws can be seen as statements about the fact that some measurable quantities do not change with time when the system under investigation is isolated. In fluid mechanics there are at least two ways to mathematically state the conservation of quantities like mass, chemical species, momentum and energy. The first one states the conservation law over any finite portion of fluid in a so-called integral form; the second one states the law for any point inside the fluid volume and yields the differential form of the conservation law. The two forms are just two different ways to state the same law and they can be alternatively used, although for analytical approaches to the solution of thermo-fluid problems the second one is certainly the most convenient. The tools for switching from one form to the other rely on two fundamental theorems: the divergence theorem and the transport theorem, and both will be revisited in the first sections of this chapter. Conservation equations can be derived from first principles, and in this respect the integral forms are the easiest way to do it; in this chapter the conservation of mass, chemical species, momentum and energy will be formulated in both forms with one important simplification: since chemical reactions will not be considered in this book, the source terms in the species and energy conservation equations will be dropped.