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2017 | Book

Singular Limits in Thermodynamics of Viscous Fluids

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About this book

This book is about singular limits of systems of partial differential equations governing the motion of thermally conducting compressible viscous fluids.

"The main aim is to provide mathematically rigorous arguments how to get from the compressible Navier-Stokes-Fourier system several less complex systems of partial differential equations used e.g. in meteorology or astrophysics. However, the book contains also a detailed introduction to the modelling in mechanics and thermodynamics of fluids from the viewpoint of continuum physics. The book is very interesting and important. It can be recommended not only to specialists in the field, but it can also be used for doctoral students and young researches who want to start to work in the mathematical theory of compressible fluids and their asymptotic limits."
Milan Pokorný (zbMATH)

"This book is of the highest quality from every point of view. It presents, in a unified way, recent research material of fundament

al importance. It is self-contained, thanks to Chapter 3 (existence theory) and to the appendices. It is extremely well organized, and very well written. It is a landmark for researchers in mathematical fluid dynamics, especially those interested in the physical meaning of the equations and statements."

Denis Serre (MathSciNet)

Table of Contents

Frontmatter
Chapter 1. Fluid Flow Modeling
Abstract
Physics distinguishes four basic forms of matter: solids, liquids, gases, and plasmas. The last three forms fall in the category of fluids. Fluid is a material that can flow, meaning fluids cannot sustain stress in the equilibrium state. The mathematical theory of continuum fluid mechanics is based on fundamental physical principles that can be expressed in terms of balance laws. These may be written by means of either Lagrangian or Eulerian reference system. In the Lagrangian coordinates, the description is associated to the particles moving in space and time. The Eulerian reference system is based on a fixed frame attached to the underlying physical space.
Eduard Feireisl, Antonín Novotný
Chapter 2. Weak Solutions, A Priori Estimates
Abstract
The fundamental laws of continuum mechanics that can be interpreted as infinite families of integral identities equivalent to systems of partial differential equations give rise to the concept of weak (or variational) solutions that can be vastly extended to extremely divers physical systems of various sorts. We introduce the concept of weak solution to the Navier-Stokes-Fourier system based postulating, besides the principles of mass and momentum conservation, a variational form of the total energy balance and the entropy balance inequality. Basic a priori bounds are derived.
Eduard Feireisl, Antonín Novotný
Chapter 3. Existence Theory

We give a complete proof of existence of weak solutions to the Navier–Stokes–Fourier system. The proof is very technical and rather involved combining various techniques of nonlinear analysis and the theory of partial differential equations. Our goal was to provide a concise but at the same time self–contained treatment of the problem without any restriction on the size of the initial data and the length of the existence interval.

Eduard Feireisl, Antonín Novotný
Chapter 4. Asymptotic Analysis: An Introduction
Abstract
The extreme generality of the full Navier-Stokes-Fourier system whereby the equations describe the entire spectrum of possible motions—ranging from sound waves, cyclone waves in the atmosphere, to models of gaseous stars in astrophysics—constitutes a serious defect of the equations from the point of view of applications. Eliminating unwanted or unimportant modes of motion, and building in the essential balances between flow fields, allow the investigator to better focus on a particular class of phenomena and to potentially achieve a deeper understanding of the problem. Scaling and asymptotic analysis play an important role in this approach. By scaling the equations, meaning by choosing appropriately the system of the reference units, the parameters determining the behavior of the system become explicit. Asymptotic analysis provides a useful tool in the situations when certain of these parameters called characteristic numbers vanish or become infinite.
Eduard Feireisl, Antonín Novotný
Chapter 5. Singular Limits: Low Stratification
Abstract
We develop the general ideas concerning singular limits in fluid mechanics focusing on the incompressible (low Mach number) limit with spatially homogeneous (constant) density profile. We show that solutions of the Navier-Stokes-Fourier system approach in the regime the Oberbeck- Boussinesq approximation.
Eduard Feireisl, Antonín Novotný
Chapter 6. Stratified Fluids
Abstract
We develop methods to handle the case of strongly stratified fluids. A new aspect with respect to the “standard” low Mach number limits arises, namely the thermodynamic state variables–the density and the temperature–undergo a scaling procedure similar to that of kinematic quantities like velocity, length, and time. In particular, both thermal and caloric equations of state modify their form reflecting substantial changes of the material properties of the fluid.
Eduard Feireisl, Antonín Novotný
Chapter 7. Interaction of Acoustic Waves with Boundary
Abstract
One of the most delicate issues in the analysis of singular limits for the Navier-Stokes-Fourier system in the low Mach number regime is the influence of acoustic waves. If the physical domain is bounded and the complete slip boundary conditions imposed, the acoustic waves, being reflected by the boundary, inevitably develop high frequency oscillations resulting in the weak convergence of the velocity field, in particular, its gradient part converges to zero only in the sense of integral means. This rather unpleasant phenomenon creates additional problems when handling the convective term in the momentum equation. Here, we focus on the mechanisms by which the acoustic energy may be dissipated, and the ways how the dissipation may be used in order to show strong (pointwise) convergence of the velocities in the incompressible limit.
Eduard Feireisl, Antonín Novotný
Chapter 8. Problems on Large Domains
Abstract
Many theoretical problems in continuum fluid mechanics are formulated on unbounded physical domains, most frequently on the whole Euclidean space. Although, arguably, any physical but also numerical space is necessarily bounded, the concept of unbounded domain offers a useful approximation in the situations when the influence of the boundary or at least its part on the behavior of the system can be neglected. We examine the incompressible limit of the Navier–Stokes–Fourier System in the situation when the spatial domain is large with respect to the characteristic speed of sound in the fluid. Remarkably, although very large, our physical space is still bounded exactly in the spirit of the idea that the notions of “large” and “small” depend on the chosen scale.
Eduard Feireisl, Antonín Novotný
Chapter 9. Vanishing Dissipation Limits
Abstract
The behavior of fluids in the vanishing dissipation regime, meaning when both the Reynolds number and the Péclet number are large, plays an important role in the study of turbulence. We examine the situation when the Mach number is small while the Reynolds and Péclet numbers are large.
Eduard Feireisl, Antonín Novotný
Chapter 10. Acoustic Analogies
Abstract
We interpret certain results on the singular limits of the Navier-Stokes-Fourier system in terms of the acoustic analogies. An acoustic analogy is represented by a non-homogeneous wave equation supplemented with source terms obtained simply by regrouping the original (primitive) system. In the low Mach number regime, the source terms may be evaluated on the basis of the limit (incompressible) system. This is the principal idea of the so-called hybrid method used in numerical analysis.
Eduard Feireisl, Antonín Novotný
Chapter 11. Appendix
Abstract
Appendix collects a number of standard results for reader’s convenience. Nowadays classical statements are appended with the relevant reference material, while complete proofs are provided in the cases when a compilation of several different techniques is necessary. A significant part of the theory presented below is related to general problems in mathematical fluid mechanics and may be of independent interest.
Eduard Feireisl, Antonín Novotný
Chapter 12. Bibliographical Remarks
Abstract
The material collected in Chapter 1 is standard. We refer to the classical monographs by Batchelor [20] or Lamb [180] for the full account on the mathematical theory of continuum fluid mechanics. A more recent treatment may be found in Truesdell and Noll [259] or Truesdell and Rajagopal [260]. An excellent introduction to the mathematical theory of waves in fluids is contained in Lighthill’s book [188].
Eduard Feireisl, Antonín Novotný
Backmatter
Metadata
Title
Singular Limits in Thermodynamics of Viscous Fluids
Authors
Dr. Eduard Feireisl
Dr. Antonín Novotný
Copyright Year
2017
Electronic ISBN
978-3-319-63781-5
Print ISBN
978-3-319-63780-8
DOI
https://doi.org/10.1007/978-3-319-63781-5

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