The most teachable book on incompressible flow-- now fully revised, updated, and expanded
Incompressible Flow, Fourth Edition is the updated and revised edition of Ronald Panton's classic text. It continues a respected tradition of providing the most comprehensive coverage of the subject in an exceptionally clear, unified, and carefully paced introduction to advanced concepts in fluid mechanics. Beginning with basic principles, this Fourth Edition patiently develops the math and physics leading to major theories. Throughout, the book provides a unified presentation of physics, mathematics, and engineering applications, liberally supplemented with helpful exercises and example problems.
Revised to reflect students' ready access to mathematical computer programs that have advanced features and are easy to use, Incompressible Flow, Fourth Edition includes:
* Several more exact solutions of the Navier-Stokes equations
* Classic-style Fortran programs for the Hiemenz flow, the Psi-Omega method for entrance flow, and the laminar boundary layer program, all revised into MATLAB
* A new discussion of the global vorticity boundary restriction
* A revised vorticity dynamics chapter with new examples, including the ring line vortex and the Fraenkel-Norbury vortex solutions
* A discussion of the different behaviors that occur in subsonic and supersonic steady flows
* Additional emphasis on composite asymptotic expansions
Incompressible Flow, Fourth Edition is the ideal coursebook for classes in fluid dynamics offered in mechanical, aerospace, and chemical engineering programs.
RONALD L. PANTON is the J. H. Herring Centennial Professor Emeritus in the Department of Mechanical Engineering at The University of Texas at Austin.
The science of fluid dynamics describes the motions of liquids and gases and their interaction with solid bodies. There are many ways to further subdivide fluid dynamics into special subjects. The plan of this book is to make the division into compressible and incompressible flows. Compressible flows are those where changes in the fluid density are important. A major specialty concerned with compressible flows, gas dynamics, deals with high-speed flows where density changes are large and wave phenomena occur frequently. Incompressible flows, of either gases or liquids, are flows where density changes in the fluid are not an important part of the physics. The study of incompressible flow includes such subjects as hydraulics, hydrodynamics, lubrication theory, aerodynamics, and boundary layer theory. It also contains background information for such special subjects as hydrology, stratified flows, turbulence, rotating flows, and biological fluid mechanics. Incompressible flow not only occupies the central position in fluid dynamics but is also fundamental to the practical subjects of heat and mass transfer.
Figure 1.1 shows a ship's propeller being tested in a water tunnel. The propeller is rotating, and the water flow is from left to right. A prominent feature of this photograph is the line of vapor that leaves the tip of each blade and spirals downstream. The vapor marks a region of very low pressure in the core of a vortex that leaves the tip of each blade. This vortex would exist even if the pressure were not low enough to form water vapor. Behind the propeller one can note a convergence of the vapor lines into a smaller spiral, indicating that the flow behind the propeller is occupying a smaller area and thus must have increased velocity.
Figure 1.1 Water tunnel test of a ship's propeller. Cavitation vapor marks the tip vortex. Photograph taken at the Garfield Thomas Water Tunnel, Applied Research Laboratory, Pennsylvania State University; supplied with permission by B. R. Parkin.
An airplane in level flight is shown in Fig. 1.2. A smoke device has been attached to the wingtip so that the core of the vortex formed there is made visible. The vortex trails nearly straight back behind the aircraft. From the sense of the vortex we may surmise that the wing is pushing air down on the inside while air rises outside the tip.
Figure 1.2 Aircraft wingtip vortices. Smoke is introduced at the wingtip to mark the vortex cores. Photograph by W. L. Oberkampf.
There are obviously some differences in these two situations. The wing moves in a straight path, whereas the ship's propeller blades are rotating. The propeller operates in water, a nearly incompressible liquid, whereas the wing operates in air, a very compressible gas. The densities of these two fluids differ by a factor of 800 : 1. Despite these obvious differences, these two flows are governed by the same laws, and their fluid dynamics are very similar. The purpose of the wing is to lift the airplane; the purpose of the propeller is to produce thrust on the boat. The density of the air as well as that of the water is nearly constant throughout the flow. Both flows have a vortex trailing away from the tip of the surface. This and many other qualitative aspects of these flows are the same. Both are incompressible flows.
In this book we shall learn many characteristics and details of incompressible flows. Equally important, we shall learn when a flow may be considered as incompressible and in exactly what ways the physics of a general flow simplifies for the incompressible case. This chapter is the first step in that direction.
1.1 Continuum Assumption
Fluid mechanics, solid mechanics, electrodynamics, and thermodynamics are all examples of physical sciences in which the world is viewed as a continuum. The continuum assumption simply means that physical properties are imagined to be distributed throughout space. Every point in space has finite values for such properties as velocity, temperature, stress, and electric field strength. From one point to the next, the properties may change value, and there may even be surfaces where some properties jump discontinuously. For example, the interface between a solid and a fluid is imagined to be a surface where the density jumps from one value to another. On the other hand, the continuum assumption does not allow properties to become infinite or to be undefined at a single isolated point.
Sciences that postulate the existence of a continuum are essentially macroscopic sciences and deal, roughly speaking, with events that may be observed with the unaided eye. Events in the microscopic world of molecules, nuclei, and elementary particles are not governed by continuum laws, nor are they described in terms of continuum ideas. However, there is a connection between the two points of view. Continuum properties may be interpreted as averages of events involving a great number of microscopic particles. The construction of such an interpretation falls into the disciplines of statistical thermodynamics (statistical mechanics) and kinetic theory. From time to time we shall discuss some of the simpler microscopic models that are used for continuum events. This aids in a deeper understanding of continuum properties, but in no way does it make the ideas "truer." The fundamental assumptions of continuum mechanics stand by themselves without reference to the microscopic world.
The continuum concept developed slowly over the course of many years. Leonhard Euler (Swiss mathematician, 1707-1783) is generally credited with giving a firm foundation to the ideas. Previously, scientists had not distinguished clearly between the idea of a point mass and that of a continuum. In his major contributions, Sir Isaac Newton (1642-1727) actually used a primitive form of the point mass as an underlying assumption (he did at times, however, also employ a continuum approach). What we now call Newton's mechanics or classical mechanics refers to the motion of point masses. In the several centuries following Newton, problems concerning the vibration of strings, the stresses in beams, and the flow of fluids were attacked. In these problems it was necessary to generalize and distinguish point mass properties from continuum properties. The continuum assumption is on a higher level of abstraction and cannot be derived mathematically from the point mass concept. On the other hand, by integration and by introducing notions such as the center of mass and moments of inertia, we can derive laws governing a macroscopic point mass from the continuum laws. Hence, the continuum laws include, as a special case, the laws for a point mass.
1.2 Fundamental Concepts, Definitions, and Laws
It is hard to give a precise description of a fundamental concept such as mass, energy, or force. They are hazy ideas. We can describe their characteristics, state how they act, and express their relation to other ideas, but when it comes to saying what they are, we must resort to vague generalities. This is not really a disadvantage, because once we work with a fundamental concept for a while and become familiar with its role in physical processes, we have learned the essence of the idea. This is actually all that is required.
Definitions, on the contrary, are very precise. For example, pressure may be defined precisely after we have the ideas of force and area at hand. Once we have made a definition of a certain physical quantity, we may explore its characteristics and deduce its exact relation to other physical quantities. There is no question how pressure is related to force, but there is a certain haziness about what a force is.
The situation is analogous to the task of writing a dictionary. How can we write out the meaning of the first word? By the very nature of a dictionary we must use other words in defining the first word. The dilemma is that those words have not yet been defined. The second word is not much easier than the first. However, after the meanings of a few key words are established, the task becomes much simpler. Word definitions can then be formulated exactly, and subtle distinctions between ideas may be made. As we use the language and see a word in different contexts, we gain a greater appreciation of its essence. At this stage, the problem of which words were the very first to be defined is no longer important. The important thing is the role the word plays in our language and the subtle differences between it and similar words.
Stretching the analogy between a continuum and a dictionary a little bit further, we can draw a correspondence between the molecules of a continuum and the letters of a word. The idea conveyed by the word is essentially independent of our choice of the language and letters to form the word. In the same way, the continuum concepts are essentially independent of the microscopic particles. The microscopic particles are necessary but unimportant.
The mathematical rules by which we predict and explain phenomena in continuum mechanics are called laws. Some restricted laws apply only to special situations. The equation of state for a perfect gas and Hooke's law of elasticity are examples of this type of law. We shall distinguish laws that apply to all substances by calling them basic laws. There are many forms for the basic laws of continuum mechanics, but in the last analysis they may all be related to four laws: the three independent conservation principles for mass, momentum, and energy plus a fundamental equation of thermodynamics. These suffice when the continuum...