Properties of the Defining Equations

The following observations are made concerning the equations that are derived in this chapter. We assume for now that only reversible thermodynamic processes are involved in the main part of the flow field of interest. If this is not so, additional equations may be required. For example, if it is required to study flow regions in which there are rapid changes in temperature, density, and pressure (e. g., within shock waves in compressible flow), then it may be necessary to supplement the equations already derived with an additional one containing consequences of the Second Law of Thermodynamics (Eq. 3.13). It may be necessary to include an additional variable like the entropy, s, to properly represent such situations. Simi­larly, there are cases in which noninertial control volumes and angular-momentum effects may be important. Because this does not happen often in this textbook, we propose to introduce such complexities only where they are necessary. If students have studied the material carefully, they will see that all of the mathematical tech­niques were presented to enable them to easily write any additional equations that may be necessary.

With this in mind, the problems we treat have the following characteristics from a mathematical point of view:

1. Six conservation equations—continuity (one), momentum (three), energy (one), and the equation of state (one)—provide the necessary set of equations to solve for the six unknowns of a general flow field—namely, u, v, w, p, p, and T, where u, v, and w can be replaced by any three orthogonal velocity components appro­priate to the coordinate system most convenient to the geometry of the problem.

2. The integral conservation equations are useful mainly when the flow is steady. For unsteady flow, evaluation of the volume integral in the equations requires detailed knowledge of the physical properties of the fluid inside the control volume.

3. Even without the considerable complication of including viscous forces and thermal-conductivity effects, the general solution of the complete differential – conservation equations is a formidable task. In fact, it is usually pointed out that general solutions do not exist because solution of a set of nonlinear par­tial-differential equations is required. Notice that several nonlinear terms are present; in particular, the convective acceleration terms are nonlinear. For

example, quantities such as udu appear that involve products of the variables


and their derivatives. The mathematical consequence is that there exist no general solutions of the complete set of equations. Analytical solutions require sets of assumptions that usually lead to some form of linearization. Much of this book is devoted to devising useful solutions of this type.