Skip to main content
main-content

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
There are two main descriptions of fluid flows, namely, (1) from the Boltzmann equation and (2) from the classical continuum theory. The difference comes from the scale at which fluid flows are observed. Therefore they involve the observer or the experimental and numerical devices that may be used. Naturally, at different scales ranging from microscopic (molecular dynamics) to macroscopic (which is the real scale for applications), the fluid flow equations are different. There is also a deeper description, and one of the open problems is the derivation of the fluid flow equations from microscopic Hamiltonian dynamics.
Radyadour Kh. Zeytounian

1. Fluid Dynamic Limits of the Boltzmann Equation

Abstract
In 1866, James Clerk Maxwell (1831–1879) developed a fundamental theoretical basis for the kinetics theory of gases. Maxwell’s theory is based on the idea of Daniel Bernoulli (1738), which gave birth to the kinetic theory of gases, that gases are formed of electric molecules rushing hither and thither at high speeds, colliding and rebounding according to the laws of elementary mechanics (see, Cercignani, Illner, and Pulvirenti 1994, pp. 8–12). In fact, Maxwell developed, first, a theory of transport processes and gave a heuristic derivation of the velocity distribution function that bears his name. Next, he developed a much more accurate model (Maxwell 1867), based on transfer equations, in fact, a model, according to which the molecules interact with a force inversely proportional to the fifth power of the distance between them (now commonly called Maxwellian molecules). With these transfer equations, Maxwell came very close to an evolution equation for the distribution, but this step (1872) must be credited to Ludwig Boltzmann (1844–1906). The equation under consideration is usually called the Boltzmann equation.
Radyadour Kh. Zeytounian

2. From Classical Continuum Theory to Euler Equations via N-S-F Equations

Abstract
In classical continuum theory, we relate field variables by specific axioms called constitutive relations. For an elementary introduction to the basic concepts and assumptions of continuum mechanics, the reader may consult Truesdell (1977). Many materials are homogeneous in the sense that each part of the material has the same response to a given set of stimuli as all of the other parts. An example of such a material is pure water. Formulation of equations that describe the behavior of homogeneous materials is well understood and is described in numerous standard textbooks (see, for instance, Gurtin, 1981). We expect the reader of this book to have sufficient background to follow our use of classical results in continuum mechanics. The short introduction to the subject we give here is intended only to fix notation and our basic ideas. Our intent is not to condense all of the knowledge about continuum mechanics into a few pages. Rather, we present the material we will use in later chapter.
Radyadour Kh. Zeytounian

3. A Short Presentation of Asymptotic Methods and Modelling

Abstract
The method of perturbation expansions is a well-established analytical tool that has found applications in many areas of fluid dynamics. The subject is covered in detail in several currently available books, and here we note only four: Van Dyke (1975), Lagerstrom (1988), Zeytounian (1994b), and Kevorkian and Cole (1996). In the recent book by Kevorkian and Cole (1996), the reader can find a comprehensive survey of perturbation methods currently used in various engineering applications, when the problems are governed by partial differential equations. As an example of asymptotic modelling of fluid flows, see the special issue of the Journal of Theoretical and Applied Mechanics (1986b), edited by Guiraud and Zeytounian and also Zeytounian (2001). For asymptotic modelling of atmospheric flows, see Zeytounian (1990).
Radyadour Kh. Zeytounian

4. Various Forms of Euler Equations and Some Hydro-Aerodynamics Problems

Abstract
Fluid dynamics was first envisaged as a systematic mathematical-physical science in Johann Bernoulli’s “Hydraulics”, in Daniel Bernoulli’s “Hydrodynamica”, and also in d’Alembert’s “Traité de l’;équilibre et du mouvement des fluides.”
Radyadour Kh. Zeytounian

5. Atmospheric Flow Equations and Lee Waves

Abstract
In what follows, we will restrict our discussion to atmospheric flows for which the horizontal scale L 0 is much smaller than the mean radius a 0 of the earth(a 0 ≅ 6367km). Based on this hypothesis, because the ratio δ = LO/aO ≪ 1, one can describe with a very good approximation the “regional” atmospheric flows in a system of Cartesian coordinates, associated with the plane normal to the gravitational acceleration 9 (of magnitude 9) resulting from the Newtonian gravitational pull (the true gravitational acceleration from the pull of the earth on the surface) and the centrifugical force per unit mass, due to the earth’s rotation.
Radyadour Kh. Zeytounian

6. Low Mach Number Flow and Acoustics Equations

Abstract
Low Mach number flow theory is a singular asymptotic theory. A typical example is that considered in Sect. 4.7.2 relative to the degeneracy of the unsteady-state Steichen, hyperbolic equation (4.181a), in an elliptical Laplace equation (4.182). As a matter of fact, this degeneracy is a consequence of filtering acoustics waves that are present in (4.181a) but absent in (4.182). In this chapter, we derive, first, the incompressible Euler equations as an outer approximation (see Sect. 6.1) and then, in the case of the external aerodynamics, the linear acoustics equations — as associated inner equations valid near time zero (see the Sect. 6.2.1). In Sect. 6.2.2, we also give some brief information concerning the very interesting case of a slightly compressible inviscid fluid flow in a bounded deformable (in time) container (internal aerodynamics). Finally, in Sect. 6.2.3, the singular nature of the far field is investigated.
Radyadour Kh. Zeytounian

7. Turbo-Machinery Fluid Flow

Abstract
Fluid flow in turbines and compressors has long defied analysis using the level of sophistication given by partial differential equations and complex configuration of fluid flow. Yet, one may wonder, whether good mathematical models exist for describing such complex fluid flow (see, for instance, the figure at the end of this short introduction)
Radyadour Kh. Zeytounian

8. Vortex Sheets and Shock Layer Phenomena

Abstract
Often, inviscid and heat nonconducting fluid flow is not continuous as assumed before, and the Euler differential equations of motion are not valid on surfaces at which the velocity is discontinuous. So new equations relating the variables on different sides of such surfaces are required.
Radyadour Kh. Zeytounian

9. Rigorous Mathematical Results

Abstract
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 create useful computational models of various processes for the prediction and the detailed study of various flow phenomena. As a matter of fact, the main mathematical problem is to discover and specify the cir- cumstances that give rise to solutions that persist forever. Only after having done that can we expect to construct proofs that such solutions exist, are unique, and regular.
Radyadour Kh. Zeytounian

Backmatter

Weitere Informationen