Dissipative Structures and Weak Turbulence

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• Publisher: Academic Press
• Year: 2014
• Print ISBN: 9780124692602
• eBook ISBN: 9780080924458
• Topic: Scienze fisiche
• Subtopic: Meccanica dei fluidi

Dissipative Structure and Weak Turbulence provides an understanding of the emergence and evolution of structures in macroscopic systems. This book discusses the emergence of dissipative structures. Organized into 10 chapters, this book begins with an overview of the stability of a fluid layer with potentially unstable density stratification in the field of gravity. This text then explains the theoretical description of the dynamics of a given system at a formal level. Other chapters consider several examples of how such simplified models can be derived, complicating the picture progressively to account for other phenomena. This book discusses as well the theory and experiments on plain Rayleigh–Bénard convection by setting first the theoretical frame and deriving the analytical solution of the marginal stability problem. The final chapter deals with building a bridge between chaos as studied in weakly confined systems and more advanced turbulence in the most conventional sense. This book is a valuable resource for physicists.

Chapter 1

Outlook

Publisher Summary

This chapter provides an overview of equilibrium. The state of macroscopic systems can be identified by a small number of parameters entering thermodynamic functions. For example, the description of a closed system in contact with a thermal bath is contained in a free energy, F = U − TS with U the internal energy, S the entropy, T the temperature, and the equilibrium state corresponds to a minimum of this function. This extremum property can be understood as the result of a competition between the organization induced by mechanical interactions and the disorder originating from the degeneracy among macroscopic states with the same energy. The structure of the equilibrium state is then controlled by the temperature and can be characterized by uniform macroscopic parameters, whereas the ultimate fate of fluctuations around this state amounts to an exponential relaxation well-described by the linear theory of irreversible processes.

The state of macroscopic systems can be identified by a small number of parameters (volume, number of particles, …) entering thermodynamic functions (energy, entropy, …). For example, the description of a closed system in contact with a thermal bath is contained in a free energy, F = U – TS with U the internal energy, S the entropy, T the temperature, and the equilibrium state corresponds to a minimum of this function. This extremum property can be understood as the result of a competition between the organization induced by mechanical interactions (the energy term) and the disorder originating from the degeneracy between macroscopic states with the same energy (the entropy term). The structure of the equilibrium state is then controlled by the temperature and can be characterized by uniform macroscopic parameters, whereas the ultimate fate of fluctuations around this state amounts to an exponential relaxation well described by the linear theory of irreversible processes.

Global thermodynamic equilibrium is an exceptional situation; most often systems are out of equilibrium and evolve spontaneously to recover the lost equilibrium. In isolated systems displaying phase coexistence, metastability, and hysteresis, this immediately raises the problem of the nucleation and growth of the most stable phase inside the metastable phase and, at later stages, the problem of the propagation and stability of the front separating the two phases. In spite of its intrinsic interest, we will not consider such irreversible macroscopic evolution of heterogeneous systems in the following but concentrate our attention on homogeneous systems.

Out-of-equilibrium situations in isolated systems are only transient. In contrast, a system allowed to exchange matter or energy with the exterior world can be maintained permanently far from equilibrium when submitted to a gradient of intensive thermodynamic quantities. For example, a macroscopic motion is driven by a pressure gradient, a heat flux is the response to a temperature gradient, etc. In principle we need a nonlinear theory to determine the response of the system to applied stresses of arbitrary strength. In practice, however, the stresses that we are able to apply to a continuous medium are usually very weak when compared with microscopic interactions. We can therefore safely make an assumption of local equilibrium and define a mesoscopic scale over which this assumption holds. The corresponding size of the “infinitesimal” elements of the continuous medium must remain large when compared to the molecular dimensions but very small at a macrosopic scale.

Though the local equilibrium assumption is valid at this scale and molecular transport well described by a linear response theory, the system can be driven far from equilibrium on a global scale. Although solid media give many examples of strong nonlinearities (in electronics, optics, etc.), as far as physical applications are concerned, we will deal mostly with fluids. In fluid systems, the mesoscopic scale is that of the so-called fluid particle and macroscopic motion is governed by hydrodynamic equations as introduced briefly in Appendix 1. Most of the properties to be discussed arise from the possibility of advection of some physical quantity: momentum, heat, etc. In the hydrodynamic equations this is accounted for by the term v ·

() where v is the macroscopic velocity field and

() stands for the local gradient of the quantity transported.

Extremely close to equilibrium, the quadratic convective term above can be neglected and the evolution equations are linear. As a result, the solution is unique and completely controlled by dissipative processes. It derives continuously from the equilibrium state and displays the same space-time symmetries as the driving stresses, stationary or periodic in time, uniform or periodic in space, etc. The solution is said to belong to the thermodynamic branch. For fluid flows, this situation corresponds to the Stokes approximation. The corresponding velocity field, called laminar, is entirely predictable and has the same regularity properties as the applied external forces.

Farther from equilibrium, nonlinearities are no longer negligible. In principle, one should be able to follow the solution belonging to the thermodynamic branch by extrapolat...

Table of contents

• Cover image
• Title page
• Table of Contents
• PERSPECTIVES IN PHYSICS
• Copyright
• Foreword
• Chapter 1: Outlook
• Chapter 2: Evolution and Stability, Basic Concepts
• Chapter 3: Instability Mechanisms
• Chapter 4: Thermal Convection
• Chapter 5: Low-Dimensional Dynamical Systems
• Chapter 6: Beyond Periodic Behavior
• Chapter 7: Characterization of Temporal Chaos
• Chapter 8: Basics of Pattern Formation in Weakly Confined Systems
• Chapter 9: Applications of the Envelope Formalism
• Chapter 10: Dynamics of Textures and Turbulence
• Appendix 1: Macroscopic Dynamics
• Appendix 2: Differential Calculus
• Appendix 3: Numerical Simulations
• Subject Index
• Perspectives in Physics