Integral Forms of the Basic Equations

In Chap. 5, the basic equations of fluid mechanics were derived in a form valid for all flow problems of Newtonian fluids. In order to obtain this generally valid form of the equations, they were formulated as differential equations for field quantities. They represent local formulations of mass, momentum and energy conservations. Applying these equations to special flow problems, it is advantageous and often essential to derive and employ the integral forms of these equations. These are derived in this chapter from the basic differential equations stated in Chap. 5 for a point in space, i.e. they are locally formulated and are valid per unit volume. The integral form of these equations is derived by integration over a pre-defined control volume. In the preceding chapters these derivations take place separately for the continuity equation, the momentum equation in direction j and the mechanical and caloric forms of the energy equation. In the following sections exemplary applications are described to make it clear how the applications of the derived integral forms of the fluid-mechanical basic equations take place. It will thus be shown how it is possible to solve fluid-mechanical problems in a somewhat engineering manner. As in this book only an introduction to the solution of problems is given, simplified assumptions are made in the course of the solutions. Attention is drawn to these simplifications in order to ensure that the reader is aware of the limits of the validity of the derived results. On the basis of the exemplary applications, independent solutions of more extensive and complicated problems should be possible by readers of this book. Depending on the problem, the integral form of the momentum equation or the mechanical energy equation can be employed, since the latter results from the former, as shown in Chap. 5.