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2002 | Buch

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Foundations of Fluid Dynamics

verfasst von: Professor Giovanni Gallavotti

Verlag: Springer Berlin Heidelberg

Buchreihe : Texts and Monographs in Physics

Enthalten in: Professional Book Archive

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Über dieses Buch

The imagination is struck by the substantial conceptual identity between the problems met in the theoretical study of physical phenomena. It is absolutely unexpected and surprising, whether one studies equilibrium statistical me chanics, or quantum field theory, or solid state physics, or celestial mechanics, harmonic analysis, elasticity, general relativity or fluid mechanics and chaos in turbulence. So when in 1988 I was made chair of Fluid Mechanics at the Universita La Sapienza, not out of recognition of work I did on the subject (there was none) but, rather, to avoid my teaching mechanics, from which I could have a strong cultural influence on mathematical physics in Rome, I was not excessively worried, although I was clearly in the wrong place. The subject is wide, hence in the last decade I could do nothing else but go through books and libraries looking for something that was within the range of the methods and experiences of my past work. The first great surprise was to realize that the mathematical theory of fluids is in an even more primitive state than I was aware of. Nevertheless it still seems to me that a detailed analysis of the mathematical problems is essential for anyone who wishes to do research into fluids. Therefore, I dedicated (Chap. 3) all the space necessary to a complete exposition of the theories of Leray, of Scheffer and of Caffarelli, Kohn and Nirenberg, taken directly from the original works.

Inhaltsverzeichnis

Frontmatter

1. Continua and Generalities About Their Equations

Abstract

A homogeneous continuum, chemically inert, in d dimensions is described by:

(a)

a region Ω in ambient space (Ω ⊂ ℝ^d), which is the occupied volume;

(b)

a function P → ρ(P) > 0, defined on Ω, giving the mass density;

(c)

a function P → T(P) defining the temperature;

(d)

a function P → s(P) defining the entropy density (per unit mass);

(e)

a function P → δ(P) defining the displacement with respect to a reference configuration;

(f)

a function P → u(P) defining the velocity field;

(g)

an equation of state relating T(P), s(P), ρ(P);

(h)

a stress tensor $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\tau } $, also denoted (τ _ij), giving the force per unit surface that the part of the continuum in contact with an ideal surface element dσ, with normal vector n, on the side of n, exercises on the part of the continuum in contact with dσ on the side opposite to n, via the formula

$$d\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\tau } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} d\sigma \;{(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\tau } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} )_i} = \sum\nolimits_{j = 1}^d {{\tau _{ij}}} {n_j}$$

(1.1.1)

(i)

a thermal conductivity tensor $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} $, giving the quantity of heat traversing the surface element dσ; in the direction of n per unit time via the formula

$$dQ = - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\partial } T\;d\sigma $$

(1.1.2)

(l)

a volume force density P → g(P);

(m)

a relation expressing the stress and conductivity tensors as functions of the observables δ, u, ρ, T, s.

Giovanni Gallavotti

2. Empirical Algorithms

Abstract

Imagine an incompressible Euler fluid in a fixed volume Ω with a (C^∞)-regular boundary. The equations describing it are

$$\begin{gathered} (1)\underline \partial \cdot \underline u = 0\Omega in \hfill \\ (2){\underline \partial _t}\underline u + u \cdot \partial u = - {\partial ^{ - 1}}\rho - g\Omega in \hfill \\ (3)\underline u \cdot \underline n = 0\Omega in \hfill \\ (4)\underline u \left( {\xi ,0} \right) \equiv {\underline u _0}\left( {\xi ,0} \right),t = 0 \hfill \\ \end{gathered} $$

(2.1.1)

where n denotes the external normal to ∂Ω and the boundary condition (3) expresses the condition that the fluid “glides” (without friction) on the boundary of Ω.

Giovanni Gallavotti

3. Analytical Theories and Mathematical Aspects

Abstract

One of the most immediate and elementary applications of the spectral method of Sect. 2.2 is the local existence, regularity and uniqueness theory of the solutions of the Euler and Navier—Stokes equations in arbitrary dimension.

Giovanni Gallavotti

4. Incipient Turbulence and Chaos

Abstract

Analysing the fundamental problems of the NS equation has, in particular, brought up clearly the lack of an adequate algorithm, i.e. convergent and constructive, for its solution. Furthermore even if we knew that the fluid equations had unique and regular solutions, for regular initial data (for the NS equation this is true if d = 2 and likely if d = 3, but false if d >4) this would not help much in the understanding of the physical properties of such solutions at large times.

Giovanni Gallavotti

5. Ordering Chaos

Abstract

After the discussions of the previous chapters it becomes imperative to find quantitative methods of study, or even simply of description, of the various phenomena that one expects to observe in experiments on fluids.

Giovanni Gallavotti

6. Developed Turbulence

Abstract

From a qualitative viewpoint the onset of turbulence, i.e. the birth of chaos, is rather well understood, as analysed in Chap. 4.

Giovanni Gallavotti

7. Statistical Properties of Turbulence

Abstract

It is now convenient to reexamine some questions of a fundamental nature with the purpose of analysing the possible consequences of Ruelle’s principle introduced in Sect. 5.7. We shall make frequent reference to the general description of motions given in Chap. 5 in a context in which we imagine that the motions are attracted by some attracting set in phase space, which will have zero volume when energy dissipation occurs in the system. The main purpose of this section and of the following ones is to analyse the consequences of Ruelle’s principle, see Sect. 5.7, with particular attention to fluid motions.

Giovanni Gallavotti

Backmatter

Titel: Foundations of Fluid Dynamics
verfasst von: Professor Giovanni Gallavotti
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-04670-8
Print ISBN: 978-3-642-07468-4
DOI: https://doi.org/10.1007/978-3-662-04670-8

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Continua and Generalities About Their Equations

2. Empirical Algorithms

3. Analytical Theories and Mathematical Aspects

4. Incipient Turbulence and Chaos

5. Ordering Chaos

6. Developed Turbulence

7. Statistical Properties of Turbulence

Backmatter