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2000 | Buch

Bodies, Configurations, and Motions Kapitel lesen Erstes Kapitel lesen

An Introduction to the Mechanics of Fluids

verfasst von: C. Truesdell, K. R. Rajagopal

Verlag: Birkhäuser Boston

Buchreihe : Modern Birkhäuser Classics

Enthalten in: Professional Book Archive

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Über dieses Buch

“The authors have backgrounds which are ideally suited for writing this book. ...They have produced a compact, moderately general book which encompasses many fluid models of current interest…The book is written very clearly and contains a large number of exercises and their solutions. The level of mathematics is that commonly taught to undergraduates in mathematics departments. This is an excellent book which is highly recommended to students and researchers in fluid mechanics.” (Mathematical Reviews)

“The writing style is quintessential Truesdellania: purely mathematical, breathtaking, irrepressible, irreverent, uncompromising, taking no prisoners...The book is filled with historical nuggets…Its pure, exact mathematics will baptize, enlighten and exhilarate.” (Applied Mechanics Review)

“The most positive aspect of this book is its brevity; a large number of topics are covered within the space of a little more than 250 pages.” (Current Science)

“This advanced monograph presents one of the best new views on the subject for those who like relative simplicity and certain abstractness combined with mathematical rigor and elegance…The book should be useful for graduates and researchers not only in applied mathematics and mechanical engineering but also in advanced materials science and technology…Each public scientific library as well as hydrodynamics hand libraries should own this timeless book…Everyone who decides to buy this book can be sure to have bought a classic of science and the heritage of an outstanding scientist.” (Silikáty)

“All applied mathematicians, mechanical engineers, aerospace engineers, and engineering mechanics graduates and researchers will find the book an essential reading resource for fluids.” (Simulation News Europe)

Inhaltsverzeichnis

Frontmatter
1. Bodies, Configurations, and Motions
Abstract
A body ℬ is a set that has a topological structure and a measure structure. It is assumed to be a ρ-finite measure space with a nonnegative measure μ(Ƥ) defined over a ρ-ring of subsets Ƥ of ℬ called subparts of the body. The open sets of ℬ are assumed to be the ρ-ring of sets. The members of the smallest ρ-ring containing the open sets are called Borel sets of ℬ.
C. Truesdell, K. R. Rajagopal
2. Kinematics and Basic Laws
Abstract
Since the motion χκ is smooth and, Fis nonsingular, the polar decomposition theorem of Cauchy enables us to write it in the two forms
C. Truesdell, K. R. Rajagopal
3. Constitutive Equations, Reduced Constitutive Equations, and Internal Constraints
Abstract
The general principles of mechanics apply to all bodies and motions, and the diversity of materials in nature is represented in the theory by constitutive equations. A constitutive equation is a relation between forces and motions. In popular terms, forces applied to a body “cause” it to undergo a motion, and the motion “caused” differs according to the nature of the body. In continuum mechanics the forces of interest are contact forces, which are specified by the stress tensor T. Just as different figures are defined in geometry as idealizations of certain important natural objects, in continuum mechanics ideal materials are defined by particular relations between the stress tensor and the motion of the body. Some materials are important in themselves, but most of them are of more interest as members of a class than in detail. Thus a general theory of constitutive equations is needed. The material presented here draws heavily from the work of Noll.
C. Truesdell, K. R. Rajagopal
4. Simple Fluids
Abstract
There are various physical notions concerned with fluids. One is that a fluid is a substance that can flow. “Flow” itself is a vague term. One meaning of “flow“ is simply deformation under stress, which does not distinguish a fluid from any other nonrigid material. Another is that steady velocity results from constant stress, which seems to be special and to apply only with difficulty and to particular flows.
C. Truesdell, K. R. Rajagopal
5. Some Flows of Incompressible Fluids in General
C. Truesdell, K. R. Rajagopal
6. Some Flows of Particular Nonlinear Fluids
Abstract
The Principle of Local Action asserts that the stress at the body point Xis unaffected at the time t by the history of the motion at other body points except those in some arbitrarily small neighborhood of χ (X, t), but it allows influence to arbitrarily long-past time. Thus, in general, a material point may have an arbitrarily long memory. In viscometric flows (Chapter 5) and, more generally, in monotonous motions (4.2–3), any such memory is given scant opportunity to make itself known, and for this reason many special problems regarding such flows are amenable to an easy solution. There is a second way to find tractable problems: instead of specializing the motion, specialize the material. Because of the obvious difficulty introduced by long-range memory, it is natural to propose for study a class of materials in which the stress atXis affected by the history of the motion only within an arbitrarily short interval [t — δ, t] of preceeding time, where δ is some positive number. Materials of this kind have infinitesimal memory. The history of the motion before any given past time is irrelevant in determining the stress in such a material at the present time.
C. Truesdell, K. R. Rajagopal
7. Some Flows of Fluids of Grade 2
Abstract
The incompressible fluid of grade 2 is defined by the constitutive relation (6.1-17), in which μ, α1, and α2 are constants, and μ, the shear viscosity, is positive. The sign of the coefficient α1 has important effects on the nature of the solutions. If the constitutive relation is taken as defining a particular fluid, just as the Navier-Stokes fluid is almost always regarded, other restrictions (for example, those implied by thermodynamics) lead to the conclusion that α1 ≥ 0, α1 +α2 = 0. However, strong sentiments have been espoused for assuming that α1 > 0 when this constitutive relation is regarded as a second-order approximation in the sense of retarded motions (6.1-9). A critical discussion of the relevant issues can be found in the recent review article by Dunn and Rajagopal.1 In the following purely mechanical treatment we do not impose any of these adscititious inequalities. We instead emphasize the effects that the sign of α1 has upon the phenomena associated with fluids of grade 2.
We note that μ/ǀα1ǀ has the dimension of time. Thus the fluid of grade 2 may be expected to show evidence of having a time scale proper to itself. The special flows we study now will reveal some effects of the existence of this proper time. Of course these effects must vanish when α1 = α2 = 0, for then the fluid of grade 2 reduces to the Navier-Stokes fluid, with which no constitutive parameter bearing the dimension of time can be associated.
C. Truesdell, K. R. Rajagopal
8. Navier-Stokes Fluids
Abstract
Classical hydrodynamics and aerodynamics concern fluids having linear viscosity or none at all. We defined these fluids in Chapter 4; in Chapters 5 and 7 we referred to them many times as special instances. Thus, in one sense, we have studied them already, but now we consider some of their properties that seem, as yet at least, peculiar to them, not to be understood easily as being merely the most special examples of more general ideas and flows.
For convenience we recall here the basic equations and conditions set forth in Chapter 4. Using the equation numbers provided, students should review the appropriate parts of Chapter 4 before reading this chapter.
C. Truesdell, K. R. Rajagopal
9. Incompressible Euler Fluids
Abstract
While students would do well to reread Section 8.1, here for convenience some generalities regarding inviscid fluids are repeated. Many can be obtained formally by simply annulling μ and ν statements made in Section 8.1–8.4, but because Eulerian hydrodynamics is not only the prototype of continuum mechanics but also its most perfect example, we prefer to repeat its basic qualities here.
C. Truesdell, K. R. Rajagopal
10. Compressible Euler Fluids
Abstract
Thus far, we have mainly confined our attention to incompressible fluids. While the assumption of incompressibility is reasonable when we restrict our attention to liquids, it is inappropriate when dealing with gases. In this section, we shall discuss briefly some interesting features that are a consequence of compressibility. The main results presented in this section pertain to the Munk-Prim substitution principle and the uniqueness of the flow of a compressible fluid.
C. Truesdell, K. R. Rajagopal
11. Singular Surfaces and Waves
Abstract
Thus far we have considered “smooth” fields. When the hypothesis of smoothness is relaxed, any sort of singularity is possible. When we confine our attention to potential theory of classical hydrodynamics, the singularities of interest are point sources, dipoles, vortex lines, double layers, and the like. In gas dynamics, we are interested in another kind of singularity, the singular surface. We shall now study the kinematical properties of singular surfaces.
C. Truesdell, K. R. Rajagopal
Backmatter
Metadaten
Titel
An Introduction to the Mechanics of Fluids
verfasst von
C. Truesdell
K. R. Rajagopal
Copyright-Jahr
2000
Verlag
Birkhäuser Boston
Electronic ISBN
978-0-8176-4846-6
Print ISBN
978-0-8176-4845-9
DOI
https://doi.org/10.1007/978-0-8176-4846-6