Asymptotic modeling in Newtonian fluid mechanics
The aim of this book is to obtain, via the asymptotic techniques (by taking limit values of some dimensionless parameters), rational simplified models derived from the Navier-Stokes model for compressible, viscous, heat conducting fluids. The first chapter is dedicated to the exact Navier-Stokes model and to some variants of this model useful for the oceanic and atmospheric flows. The Navier-Stokes equations in dimensionless variables and the fundamental dimensionless parameters (Reynolds, Mach, Strouhal, Froude, Prandtl, Rossby, etc.) are presented in the second chapter. The concept of asymptotic modelling is introduced in chapter III. Using heuristic methods (different from the asymptotic formalism), one presents in chapter IV some simplified forms of the Navier-Stokes equations (Euler, Navier, Prandtl, Stokes and Oseen equations). The instruments of asymptotic modelling, especially the method of matched asymptotic expansions and the method of multiple scales, are introduced in chapter V. In chapter VI, one exposes the fundamental notions of the asymptotic modelling. Starting from the Navier model, one examines in chapter VII the models related to large Reynolds numbers (Euler and Prandtl) and to small ones (Stokes and Oseen). In chapter VIII one develops, according to the values of Mach number, various models for compressible, heat conducting fluids. We mention the acoustic model, the subsonic, transonic, supersonic and hypersonic flows of ideal fluids, and the subsonic flow of slightly viscous fluids. Chapter IX is dedicated to Boussinesq and quasi-geostrophic models for atmospheric flow and to the applications of the method of multiple scales to oceanic flow. In chapter X, the homogenization method is used for studying some local models in aerodynamics (turbulent motion, evolution of acoustic waves, stationary flow in a turbine with many blades). Chapter XI is dedicated to the problem of water waves. Using the model of shallow water, one presents the equations of Boussinesq, Korteweg-de Vries and Schrödinger. Although the author presented his work as a textbook for students in fluid mechanics, it is actually one of the first books presenting a vast panorama of the asymptotic methods (concerning both the concepts and the effective modelling of concrete problems) used for investigating the flows of Newtonian fluids.
scientific article; zbMATH DE number 590350