Asymptotic Stability of Steady Compressible Fluids
This volume introduces a systematic approach to the solution of some mathematical problems that arise in the study of the hyperbolic-parabolic systems of equations that govern the motions of thermodynamic fluids. It is intended for a wide audience of theo
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Topics in Fluid Mechanics
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1.1
Introduction
The aim of this chapter is, after some mathematical preliminaries, to introduce the physical equations governing steady and unsteady motions of barotropic and polytropic fluids. The field of macroscopic thermodynamics provides us with a general framework for the description of irreversible continuum processes. Certain areas of macroscopic physics have connections with fluid dynamics, and the first part of the chapter is devoted to a systematic development of this theory, referencing [143]. More specifically, we will be treating state parameters as field variables, thus formulating the basic equations of continuous thermodynamics in the form of local equations. This will allow us to formulate correct well-posed problems. In this chapter we will distinguish three model problems: (a) Barotropic viscous fluid , fluid filling a domain Ω with rigid boundaries (b) Isothermal viscous fluid , fluid filling a domain Ω with deformable boundaries (c) Polytropic viscous fluid , fluid filling a domain Ω with rigid, perfectly heat-conducting boundaries It is clear that in cases (a), (b) and (c), the unknown functions correspond to different physical variables. In addition, as observed in the preface, each M. Padula, Asymptotic Stability of Steady Compressible Fluids, Lecture Notes in Mathematics 2024, DOI 10.1007/978-3-642-21137-9 1, © Springer-Verlag Berlin Heidelberg 2011
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Topics in Fluid Mechanics
unknown function satisfies a different PDE. The character of these PDEs changes by changing the physical model, thus steady or unsteady problems are governed by different mathematical problems (see cases (i), (ii), (iii) defined in the preface). In all cases that we are dealing with, problems of unsteady fluid flow are all described by a coupled parabolic-hyperbolic system. Chapter 1 will proceed as follows: Section 1.2 Geometric and analytical tools are introduced. Section 1.3 Kinematical tools are introduced. Section 1.4 Systems of equations governing motions of general continua are written in local form. Section 1.5 Equations of thermodynamics are introduced, and constitutive equations for a fluid are given. Section 1.6 General physical boundary conditions are introduced for different domains and material boundaries. Section 1.7 The exact mathematical position is given for three model problems.
1.2
Mathematical Notations
In this section we will introduce some notations concerning the geometry of the region where the motion occurs, and the functional spaces, with relative norms, where stability is studied. Finally, some elementary topics in kinematics are covered.
1.2.1
Geometrical Notations
The domain Let a fluid fill the region Ωt at time t, with boundary ∂Ωt . If the domain is fixed we omit the subscript t. In subsequent sections we will use the symbol Ω to analyze motions occurring only in one of the following regions Ω:
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