Rigorous Mathematical Results
Obviously, there is always considerable interest in rigorous formulations of initial-boundary value (I-BV) problems for various systems of partial differ- ential equations that arise in fluid dynamics. This interest stems, primarily, from efforts to creat
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Springer-Verlag Berlin Heidelberg GmbH
Radyadour Kh. Zeytounian
Theory and Applications of Nonviscous Fluid Flows With 38 Figures
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Radyadour Kh. Zeytounian 12, rue Saint-Fiacre 75002 Paris, France
Honorary professor of the Université des Sciences et Technologies de Lille; Villeneuve d’Ascq, France
Library of Congress Cataloging-in-Publication Data Zeytounian, R. Kh. (Radyadour Kh.), 1928- Theory and applications of nonviscous fluid flows / Radyadour K. Zeytounian. p. cm. Includes biblographical references and index. ISBN 3540414126 (alk. paper) 1. Fluid dynamics. 2. Newtonian fluids. I. Title. QA911.Z495 2001 532’.0533--dc21 2001041106
ISBN 978-3-642-62551-0 ISBN 978-3-642-56215-0 (eBook) DOI 10.1007/978-3-642-56215-0 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. http://www.springer.de c 6SULQJHU9HUODJ %HUOLQ +HLGHOEHUJ
2ULJLQDOO\SXEOLVKHGE\6SULQJHU9HUODJ%HUOLQ+HLGHOEHUJ1HZ 0 is a scalar. The Navier–Stokes (N–S) and Navier–Stokes–Fourier (N–S–F), compressible, viscous equations are more complicated and have no obvious scaling. They are given by correcting the Euler compressible equations with viscous and thermal diffusivity terms, described by second-order derivatives, and also by viscous dissipation – a quadratic function, of conserved quantities such as energy, momentum, and mass. In the incompressible and viscous regime, we obtain the Navier equations: ∇ · u=0, 1 du + ∇p + g = ν0 ∇2 u , dt ρ0
(2) (3)
where d/dt = ∂/∂t + u · ∇ is the derivative along the trajectories, u is the velocity field, ∇ is the gradient operator, p is the pressure, ρ0 is the constant density, g is the acceleration of gravity, and the constant ν0 > 0 is
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Introduction
the kinematic viscosity. When ν0 ↓ 0 (which is a singular limit!), from (2), (3), we derive the incompressible, inviscid (nonviscous), Euler equations: ∇ · u = 0,
du 1 + ∇p + g = 0 . dt ρ0
(4)
Because Euler, Navier, N–S and N–S–F equations involve macroscopic quantities, derivation of these equations from microscopic Hamiltonian dynamics is understood in the sense of the law of large numbers, as the number of particles tends to infinity. In that case, we fix the space and time scales by choosing the typical interparticle distance as the unit length scale and the mean free time of particles as unit time scale. Suppose the range of molecular interactions is of the order of the typical interparticle distance. Then each particle typically interacts with at most an finite number of nearby particles. Let ε (the Kn
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