Fluid Mechanics with Multimedia DVD, Fourth Edition

Fluid mechanics, the study of how fluids behave and interact under various forces and in various applied situations—whether in the liquid or gaseous state or both—is introduced and comprehensively covered in this widely adopted text. Fully revised and updated with the addition of a new chapter on biofluid mechanics, Fluid Mechanics, Fourth Edition is suitable for both a first or second course in fluid mechanics at the graduate or advanced undergraduate level. The leading advanced general text on fluid mechanics, Fluid Mechanics, 4e guides students from the fundamentals to the analysis and application of fluid mechanics, including compressible flow and such diverse applications as hydraulics and aerodynamics. Updates to several chapters and sections, including Boundary Layers, Turbulence, Geophysical Fluid Dynamics, Thermodynamics and Compressibility. Fully revised and updated chapter on Computational Fluid Dynamics. New chapter on Biofluid Mechanics by Professor Portonovo Ayyaswamy, the Asa Whitney Professor of Dynamical Engineering at the University of Pennsylvania. New Visual Resources appendix provides a list of fluid mechanics films available for viewing online. Additional worked-out examples and end-of-chapter problems. Updated online Solutions Manual for adopting instructors.
Chapter One Introduction

1. Fluid Mechanics 1 2. Units of Measurement 2 3. Solids, Liquids, and Gases 3 4. Continuum Hypothesis 4 5. Transport Phenomena 5 6. Surface Tension 8 7. Fluid Statics 9 Example 1.1 12 8. Classical Thermodynamics 12 First Law of Thermodynamics 13 Equations of State 13 Specific Heats 14 Second Law of Thermodynamics 15 TdS Relations 15 Speed of Sound 16 Thermal Expansion Coefficient 16 9. Perfect Gas 16 10. Static Equilibrium of a Compressible Medium 18 Potential Temperature and Density 20 Scale Height of the Atmosphere 22 Exercises 22 Literature Cited 24 Supplemental Reading 24

1. Fluid Mechanics

Fluid mechanics deals with the flow of fluids. Its study is important to physicists, whose main interest is in understanding phenomena. They may, for example, be interested in learning what causes the various types of wave phenomena in the atmosphere and in the ocean, why a layer of fluid heated from below breaks up into cellular patterns, why a tennis ball hit with "top spin" dips rather sharply, how fish swim, and how birds fly. The study of fluid mechanics is just as important to engineers, whose main interest is in the applications of fluid mechanics to solve industrial problems. Aerospace engineers may be interested in designing airplanes that have low resistance and, at the same time, high "lift" force to support the weight of the plane. Civil engineers may be interested in designing irrigation canals, dams, and water supply systems. Pollution control engineers may be interested in saving our planet from the constant dumping of industrial sewage into the atmosphere and the ocean. Mechanical engineers may be interested in designing turbines, heat exchangers, and fluid couplings. Chemical engineers may be interested in designing efficient devices to mix industrial chemicals. The objectives of physicists and engineers, however, are not quite separable because the engineers need to understand and the physicists need to be motivated through applications.

Fluid mechanics, like the study of any other branch of science, needs mathematical analyses as well as experimentation. The analytical approaches help in finding the solutions to certain idealized and simplified problems, and in understanding the unity behind apparently dissimilar phenomena. Needless to say, drastic simplifications are frequently necessary because of the complexity of real phenomena. A good understanding of mathematical techniques is definitely helpful here, although it is probably fair to say that some of the greatest theoretical contributions have come from the people who depended rather strongly on their unusual physical intuition, some sort of a "vision" by which they were able to distinguish between what is relevant and what is not. Chess player, Bobby Fischer (appearing on the television program "The Johnny Carson Show," about 1979), once compared a good chess player and a great one in the following manner: When a good chess player looks at a chess board, he thinks of 20 possible moves; he analyzes all of them and picks the one that he likes. A great chess player, on the other hand, analyzes only two or three possible moves; his unusual intuition (part of which must have grown from experience) allows him immediately to rule out a large number of moves without going through an apparent logical analysis. Ludwig Prandtl, one of the founders of modern fluid mechanics, first conceived the idea of a boundary layer based solely on physical intuition. His knowledge of mathematics was rather limited, as his famous student von Karman (1954, page 50) testifies. Interestingly, the boundary layer technique has now become one of the most powerful methods in applied mathematics!

As in other fields, our mathematical ability is too limited to tackle the complex problems of real fluid flows. Whether we are primarily interested either in understanding the physics or in the applications, we must depend heavily on experimental observations to test our analyses and develop insights into the nature of the phenomenon. Fluid dynamicists cannot afford to think like pure mathematicians. The well-known English pure mathematician G. H. Hardy once described applied mathematics as a form of "glorified plumbing" (G. I. Taylor, 1974). It is frightening to imagine what Hardy would have said of experimental sciences!

This book is an introduction to fluid mechanics, and is aimed at both physicists and engineers. While the emphasis is on understanding the elementary concepts involved, applications to the various engineering fields have been discussed so as to motivate the reader whose main interest is to solve industrial problems. Needless to say, the reader will not get complete satisfaction even after reading the entire book. It is more likely that he or she will have more questions about the nature of fluid flows than before studying this book. The purpose of the book, however, will be well served if the reader is more curious and interested in fluid flows.

2. Units of Measurement

For mechanical systems, the units of all physical variables can be expressed in terms of the units of four basic variables, namely, length, mass, time, and temperature. In this book the international system of units (Système international d' unités) and commonly referred to as SI units, will be used most of the time. The basic units of this system are meter for length, kilogram for mass, second for time, and kelvin for temperature. The units for other variables can be derived from these basic units. Some of the common variables used in fluid mechanics, and their SI units, are listed in Table 1.1. Some useful conversion factors between different systems of units are listed in Section A1 in Appendix A.

To avoid very large or very small numerical values, prefixes are used to indicate multiples of the units given in Table 1.1. Some of the common prefixes are listed in Table 1.2.

Strict adherence to the SI system is sometimes cumbersome and will be abandoned in favor of common usage where it best serves the purpose of simplifying things. For example, temperatures will be frequently quoted in degrees Celsius (°C), which is related to kelvin (K) by the relation °C = K - 273.15. However, the old English system of units (foot, pound, °F) will not be used, although engineers in the United States are still using it.

3. Solids, Liquids, and Gases

Most substances can be described as existing in two states—solid and fluid. An element of solid has a preferred shape, to which it relaxes when the external forces on it are withdrawn. In contrast, a fluid does not have any preferred shape. Consider a rectangular element of solid ABCD (Figure 1.1a). Under the action of a shear force F the element assumes the shape ABC'D'. If the solid is perfectly elastic, it goes back to its preferred shape ABCD when F is withdrawn. In contrast, a fluid deforms continuously under the action of a shear force, however small. Thus, the element of the fluid ABCD confined between parallel plates (Figure 1.1b) deforms to shapes such as ABC'D' and ABC"D" as long as the force F is maintained on the upper plate. Therefore, we say that a fluid flows.

The qualification "however small" in the forementioned description of a fluid is significant. This is because most solids also deform continuously if the shear stress exceeds a certain limiting value, corresponding to the "yield point" of the solid. A solid in such a state is known as "plastic." In fact, the distinction between solids and fluids can be hazy at times. Substances like paints, jelly, pitch, polymer solutions, and biological substances (for example, egg white) simultaneously display the characteristics of both solids and fluids. If we say that an elastic solid has "perfect memory" (because it always relaxes back to its preferred shape) and that an ordinary viscous fluid has zero memory, then substances like egg white can be called viscoelastic because they have "partial memory."

Although solids and fluids behave very differently when subjected to shear stresses, they behave similarly under the action of compressive normal stresses. However, whereas a solid can support both tensile and compressive normal stresses, a fluid usually supports only compression (pressure) stresses. (Some liquids can support a small amount of tensile stress, the amount depending on the degree of molecular cohesion.)

Fluids again may be divided into two classes, liquids and gases. A gas always expands and occupies the entire volume of any container. In contrast, the volume of a liquid does not change very much, so that it cannot completely fill a large container; in a gravitational field a free surface forms that separates the liquid from its vapor.

4. Continuum Hypothesis

A fluid, or any other substance for that matter, is composed of a large number of molecules in constant motion and undergoing collisions with each other. Matter is therefore discontinuous or discrete at microscopic scales. In principle, it is possible to study the mechanics of a fluid by studying the motion of the molecules themselves, as is done in kinetic theory or statistical mechanics. However, we are generally interested in the gross behavior of the fluid, that is, in the average manifestation of the molecular motion. For example, forces are exerted on the boundaries of a container due to the constant bombardment of the molecules; the statistical average of this force per unit area is called pressure, a macroscopic property. So long as we are not interested in the mechanism of the origin of pressure, we can ignore the molecular motion and think of pressure as simply "force per unit area."

It is thus possible to ignore the discrete molecular structure of matter and replace it by a continuous distribution, called a continuum. For the continuum or macroscopic approach to be valid, the size of the flow system (characterized, for example, by the size of the body around which flow is taking place) must be much larger than the mean free path of the molecules. For ordinary cases, however, this is not a great restriction, since the mean free path is usually very small. For example, the mean free path for standard atmospheric air is ≈5 × 10-8 m. In special situations, however, the mean free path of the molecules can be quite large and the continuum approach breaks down. In the upper altitudes of the atmosphere, for example, the mean free path of the molecules may be of the order of a meter, a kinetic theory approach is necessary for studying the dynamics of these rarefied gases.

5. Transport Phenomena

Consider a surface area AB within a mixture of two gases, say nitrogen and oxygen (Figure 1.2), and assume that the concentration ITLITL of nitrogen (kilograms of nitrogen per cubic meter of mixture) varies across AB. Random migration of molecules across AB in both directions will result in a net flux of nitrogen across AB, from the region of higher ITLITL toward the region of lower ITLITL. Experiments show that, to a good approximation, the flux of one constituent in a mixture is proportional to its concentration gradient and it is given by

qm = -km ∇ITLITL. (1.1)

Here the vector qm is the mass flux (kg m-2 s-1) of the constituent, ∇ITLITL is the concentration gradient of that constituent, and km is a constant of proportionality that depends on the particular pair of constituents in the mixture and the thermodynamic state. For example, km for diffusion of nitrogen in a mixture with oxygen is different than km for diffusion of nitrogen in a mixture with carbon dioxide. The linear relation (1.1) for mass diffusion is generally known as Fick's law. Relations like these are based on empirical evidence, and are called phenomenological laws. Statistical mechanics can sometimes be used to derive such laws, but only for simple situations.

The analogous relation for heat transport due to temperature gradient is Fourier's law and it is given by

q = -k∇T, (1.2)

where q is the heat flux (J m-2 s-1), ∇T is the temperature gradient, and k is the thermal conductivity of the material.

Next, consider the effect of velocity gradient du/dy (Figure 1.3). It is clear that the macroscopic fluid velocity u will tend to become uniform due to the random motion of the molecules, because of intermolecular collisions and the consequent exchange of molecular momentum. Imagine two railroad trains traveling on parallel tracks at different speeds, and workers shoveling coal from one train to the other. On the average, the impact of particles of coal going from the slower to the faster train will tend to slow down the faster train, and similarly the coal going from the faster to the slower train will tend to speed up the latter. The net effect is a tendency to equalize the speeds of the two trains. An analogous process takes place in the fluid flow problem of Figure 1.3. The velocity distribution here tends toward the dashed line, which can be described by saying that the x-momentum (determined by its "concentration" u) is being transferred downward. Such a momentum flux is equivalent to the existence of a shear stress in the fluid, just as the drag experienced by the two trains results from the momentum exchange through the transfer of coal particles. The fluid above AB tends to push the fluid underneath forward, whereas the fluid below AB tends to drag the upper fluid backward. Experiments show that the magnitude of the shear stress τ along a surface such as AB is, to a good approximation, related to the velocity gradient by the linear relation

τ = μ du/dy, (1.3)

which is called Newton's law of friction. Here the constant of proportionality μ (whose unit is kg m-1 s-1) is known as the dynamic viscosity, which is a strong function of temperature T. For ideal gases the random thermal speed is roughly proportional to [square root of T]; the momentum transport, and consequently μ, also vary approximately as [square root of T]. For liquids, on the other hand, the shear stress is caused more by the intermolecular cohesive forces than by the thermal motion of the molecules. These cohesive forces, and consequently μ for a liquid, decrease with temperature.

Although the shear stress is proportional to μ, we will see in Chapter 4 that the tendency of a fluid to diffuse velocity gradients is determined by the quantity ν ≡ μ/ρ, (1.4)

where ρ is the density (kg/m3) of the fluid. The unit of ν is m2/s, which does not involve the unit of mass. Consequently, is frequently called the kinematic viscosity.

Two points should be noticed in the linear transport laws equations (1.1), (1.2), and (1.3). First, only the first derivative of some generalized "concentration" ITLITL appears on the right-hand side. This is because the transport is carried out by molecular processes, in which the length scales (say, the mean free path) are too small to feel the curvature of the ITLITL-profile. Second, the nonlinear terms involving higher powers of ∇ITLITL do not appear. Although this is only expected for small magnitudes of ∇ITLITL, experiments show that such linear relations are very accurate for most practical values of ∇ITLITL.

(Continues...)

Excerpted from Fluid Mechanics by Pijush K. Kundu Ira M. Cohen Copyright © 2008 by Elsevier Inc.. Excerpted by permission of Academic Press. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Fluid mechanics, the study of how fluids behave and interact under various forces and in various applied situations—whether in the liquid or gaseous state or both—is introduced and comprehensively covered in this widely adopted text. The leading advanced general text on fluid mechanics, Kundu & Cohen's Fluid Mechanics, 4e, now includes a free copy of the DVD "Multimedia Fluid Mechanics," 2e. With the inclusion of the DVD, students can gain additional insight about fluid flows through nearly 1,000 fluids video clips, can conduct flow simulations in any of more than 20 virtual labs and simulations, and can view dozens of other new interactive demonstrations and animations, thereby enhancing their fluid mechanics learning experience.

• Changes for the 4th edition from the 3rd edition:

  • Updates to several chapters and sections, including Boundary Layers, Turbulence, Geophysical Fluid Dynamics, Thermodynamics and Compressibility
  • Fully revised and updated chapter on computational fluid dynamics
  • New chapter on Biofluid Mechanics by Professor Portonovo Ayyaswamy, the Asa Whitney Professor of Dynamical Engineering at the University of Pennsylvania
  • New Visual Resources appendix provides a list of fluid mechanics films available for viewing online

  New enhanced version of the 4th edition:

  • Now includes free Multimedia Fluid Mechanics 2e DVD

Fluid mechanics, the study of how fluids behave and interact under various forces and in various applied situations-whether in the liquid or gaseous state or both-is introduced and comprehensively covered in this widely adopted text. Fully revised and updated with the addition of a new chapter on biofluid mechanics, Fluid Mechanics, Fourth Edition is suitable for both a first or second course in fluid mechanics at the graduate or advanced undergraduate level. The leading advanced general text on fluid mechanics, Fluid Mechanics, 4e guides students from the fundamentals to the analysis and application of fluid mechanics, including compressible flow and such diverse applications as hydraulics and aerodynamics. Updates to several chapters and sections, including Boundary Layers, Turbulence, Geophysical Fluid Dynamics, Thermodynamics and Compressibility. Fully revised and updated chapter on Computational Fluid Dynamics. New chapter on Biofluid Mechanics by Professor Portonovo Ayyaswamy, the Asa Whitney Professor of Dynamical Engineering at the University of Pennsylvania.New Visual Resources appendix provides a list of fluid mechanics films available for viewing online.Additional worked-out examples and end-of-chapter problems.Updated online Solutions Manual for adopting instructors. Fluid mechanics, the study of how fluids behave and interact under various forces and in various applied situations—whether in the liquid or gaseous state or both—is introduced and comprehensively covered in this widely adopted text. Fluid Mechanics, Fourth Edition is the leading advanced general text on fluid mechanics. Changes for the 4th edition from the 3rd edition: Updates to several chapters and sections, including Boundary Layers, Turbulence, Geophysical Fluid Dynamics, Thermodynamics and Compressibility Fully revised and updated chapter on computational fluid dynamics New chapter on Biofluid Mechanics by Professor Portonovo Ayyaswamy, the Asa Whitney Professor of Dynamical Engineering at the University of Pennsylvania "Fluid mechanics, the study of how fluids behave and interact under various forces and in various applied situations - whether in the liquid or gaseous state or both - is introduced and comprehensively covered in this widely adopted text. Fully revised and updated with the addition of a new chapter on biofluid mechanics, Fluid Mechanics, Fourth Edition is suitable for both a first or second course in fluid mechanics at the graduate or advanced undergraduate level. The leading advanced general text on fluid mechanics, Fluid Mechanics, Fourth Edition guides students from the fundamentals to the analysis and application of fluid mechanics, including compressible flow and such diverse applications as hydraulics and aerodynamics."--Jacket