Overview

Thermodynamics is largely a study of the equilibrium states of fluids having spatially uniform properties such as temperature, pressure, and density. The notion of equilibrium refers ultimately to the idea that the molecules composing the fluid are in a statistical equilibrium insofar as the distribution of molecular velocities and other characteristics is concerned. At first sight, it may seem that the laws of thermodynamics should be incompatible with the analysis of flowing fluids where the pressure varies everywhere and there very well might be spatial gradients of temperature and density. Nonetheless, by careful consideration of how the laws of thermodynamics pertain to material fluid elements, the relevancy and application of thermodynamics to flowing fluids can be well justified.

Changes to a thermodynamic system over time that maintain the equilibrium are reversible processes, whereas those that disrupt the equilibrium, say, by the addition of currents evident in a moving fluid, are irreversible. For a thermodynamic system, such as a fluid in a closed container, to remain in equilibrium despite outside influences, such as compression caused by the movement of a piston or the purposeful addition of heat, the mechanism by which the molecules adjust to the imposed changes must act faster than the changes themselves. In gases, with each molecule experiencing approximately 1010 molecular collisions per second, information about changing circumstances can spread extremely quickly throughout the gas. Apart from extreme conditions, as in a shock wave, the gas has the capacity to maintain equilibrium during the change of its gross properties such as pressure. Similarly, in a liquid, the strong intermolecular forces bring about rapid passage of information throughout the fluid that maintain equilibrium.

Although it is not expected that thermodynamic equilibrium is maintained within a moving fluid if it is taken as a whole, it is not unreasonable to expect that equilibrium can be maintained for individual material fluid elements if they are small enough.

Although the differential equations of fluid flow summarized in Section 16.1 contain all the physics that is needed to uniquely determine fluid motion, these equations do not as a general rule lend themselves to solution by any means other than numerical approximation. The problem mostly stems from the nonlinear convective term wherein the rate of change of the momentum of fluid particles depends on the momentum of the fluid particles themselves. As momentum changes, so too does the rate of change of momentum. Analytical solutions containing this physics are a rarity, so most examples of exact closed form solutions to fluid flow are in situations where the nonlinear term happens to be identically zero or near zero or can be simplified in some way.

A large part of the physics of the momentum equation is tied up in the relative importance of the viscous forces versus the inertia of the fluid particles. In fact, the pressure force can be generally expected to align with whichever of the remaining forces are dominant. Consequently, the question of whether the nonlinear terms in the Navier-Stokes equation can be ignored boils down to a discussion of whether these terms are dominant or subordinate to the viscous terms. A dimensionless parameter known as the Reynolds number expresses this ratio of forces and is thus an arbiter in making decisions as to whether the viscous and convective terms need to be kept or omitted in different situations.

The next section provides a discussion of the Reynolds number that will be useful in setting up a context for making approximations that lead to analytical solutions of viscous flow problems. Following this, several examples of flows are considered wherein the convective terms are identically zero or else close to zero because of the geometry of the flow or because the Reynolds number permits this approximation. The last section will briefly consider some of the issues encountered in solving for the motion of bodies when analytical solutions are not available.

While fluids are most often encountered in a state of motion, there are times and circumstances when from the perspective of a particular observer the fluid is at rest. In this case it can often be relatively easy to determine attributes of the fluid, such as the pressure distribution, that are needed in such practical applications as determining the force of water on a dam or submerged body. Several standard problems where fluids are at rest are illustrated in this chapter, including, in the last two sections, problems where the fluid is stationary only from the perspective of an accelerating observer. In these cases, the results of the previous chapter play an important role in the analysis.

Forces in a Fluid at Rest

The essential importance of Newton's laws of motion to computing fluid flow was noted in Eq. (1.10).When the fluid is not moving, there is no change to the momentum of fluid elements, so the left-hand side of Newton's law is zero and the relevant physics consists of a zero sum of the forces acting on fluid elements.

Consider a material fluid element such as was discussed in Section 1.4 and is now illustrated in Fig. 9.1. The forces acting on this fluid element consist of body forces, such as gravity, that act everywhere over the volume occupied by the element as well as surface forces that act at the bounding area of the volume. For a stationary fluid, the only surface force acting on material elements is the pressure, and this must act normally, because if it did not, then there would be an implied shearing force that would lead to fluid motion as discussed in Section 1.1. How fluid motion gives rise to contributions to the surface forces other than pressure will be considered later in Chapters 11 and 12. In the present case, it is only the pressure force that balances with the body force so as to result in a nonmoving fluid in equilibrium.

Three classical problems that are amenable to solution without solving the full equations of motion are considered here. These help to illustrate situations wherein the approximate techniques based on Bernoulli's equation and the use of control volumes can be used to great advantage. Many other interesting cases can be found in the literature. A number of other applications may be found in the problems at the end of the chapter.

Fluid Impinging on a Plate

A careful blend of the control volume approach with the Bernoulli equation can be used effectively to explain a considerable amount of the physics entailed in the impingement of a uniform, planar, liquid jet striking a tilted surface, as shown in Fig. 17.1. The surface is held on a hinge located opposite the midpoint of the incoming jet and is free to rotate. Which way will the plate move, and what will be its final equilibrium position? Such questions can be answered by obtaining information about the total forces and moments acting on the plate, an objective that can be met using the control volume approach without solving for the detailed flow field.

At a sufficient distance upstream of the plate, the impinging jet is assumed to have speed U uniformly across its width. On encountering the plate, the jet is expected to divide with part going upward and part downward. In what proportion it divides is one of the results that is to be found. Each portion of the split jet at some distance downstream from the pivot point may be assumed to reach a state where the motion is parallel to the plate. To enable the use of the Bernoulli equation, it is further assumed that the flow in the jet is frictionless, an assumption that can be expected to be reasonably well satisfied away from the immediate surface of the plate where viscous action is inevitable.

The pressure outside the liquid jet can be assumed to be atmospheric, say, p∞, everywhere. In particular, the two outer streamlines of the jet that define its shape must have pressure p∞.