Hydrodynamic Fluctuations in Fluids and Fluid Mixtures

Jose M. Ortiz de Zarate, Jan V. Sengers

• Publisher: Elsevier Science
• Year: 2006
• ISBN:
  9780080459431
• Topic: Physical Sciences
• Subtopic: Fluid Mechanics

This book deals with density, temperature, velocity and concentration fluctuations in fluids and fluid mixtures. The book first reviews thermal fluctuations in equilibrium fluids on the basis of fluctuating hydrodynamics. It then shows how the method of fluctuating hydrodynamics can be extended to deal with hydrodynamic fluctuations when the system is in a stationary nonequilibrium state. In contrast to equilibrium fluids where the fluctuations are generally short ranged unless the system is close to a critical point, fluctuations in nonequilibrium fluids are always long-ranged encompassing the entire system. The book provides the first comprehensive treatment of fluctuations in fluids and fluid mixtures brought out of equilibrium by the imposition of a temperature and concentration gradient but that are still in a macroscopically quiescent state. By incorporating appropriate boundary conditions in the case of fluid layers, it is shown how fluctuating hydrodynamics affects the fluctuations close to the onset of convection. Experimental techniques of light scattering and shadowgraphy for measuring nonequilibrium fluctuations are elucidated and the experimental results thus far reported in the literature are reviewed.· Systematic exposition of fluctuating hydrodynamics and its applications· First book on nonequilibrium fluctuations in fluids· Fluctuating Boussinesq equations and nonequilibrium fluids· Fluid layers and onset of convection· Rayleigh scattering and Brillouin scattering in fluids· Shadowgraph technique for measuring fluctuations· Fluctuations near hydrodynamic instabilities

Chapter 1

Introduction

Publisher Summary

This chapter presents a systematic treatment of hydrodynamic fluctuations in fluids and fluid mixtures. The theory of fluctuations in fluids that are in thermodynamic equilibrium is well developed. Specifically, the intensity of density fluctuations is proportional to the isothermal compressibility and the intensity of concentration fluctuations in mixtures is proportional to the osmotic compressibility. Fluctuating hydrodynamics is a stochastic fluid dynamics approach in which it is assumed that the fluctuations can be described by the usual hydrodynamic equations, but supplemented with random noise terms whose correlation functions are determined by a fluctuation-dissipation theorem. It is found that hydrodynamic correlation functions in fluids in nonequilibrium states always extend over a spatial range that is longer than what one would expect for the corresponding local-equilibrium expressions for these correlation functions, even when the system is far away from any hydrodynamic instability. A review of some of the basic concepts of nonequilibrium thermodynamics and a derivation of the hydrodynamic equations for fluids and fluid mixtures are also presented.

The purpose of this book is to present a systematic treatment of hydrodynamic fluctuations in fluids and fluid mixtures. The theory of fluctuations in fluids that are in thermodynamic equilibrium is well developed (Boon and Yip, 1980). Specifically, the intensity of density fluctuations is proportional to the isothermal compressibility and the intensity of concentration fluctuations in mixtures is proportional to the osmotic compressibility. Moreover, the decay of thermally excited fluctuations is governed by Onsager’s regression hypothesis which says that the decay rates of the fluctuations is determined by the transport coefficients that appear in the linear relations between fluxes and gradients in nonequilibrium thermodynamics. A theoretical frame work for dealing with fluctuations in fluids in thermodynamic equilibrium states is provided by Landau’s fluctuating hydrodynamics (Landau and Lifshitz, 1958, 1959; Fox and Uhlenbeck, 1970a). Such fluctuations can be investigated experimentally by light scattering and neutron scattering (Berne and Pecora, 1976).

An interesting issue is the nature of thermal fluctuations in fluids and fluid mixtures that are in thermodynamic nonequilibrium states. While it is well known that nonequilibrium fluids can exhibit large fluctuations associated with convection patterns or turbulence, a rather new subject is that of fluctuations in fluids that are subjected to a temperature gradient or to shear in the absence of convective patterns or turbulent flows. About 50 years ago a microscopic picture of nonequilibrium phenomena in fluids was proposed by Bogoliubov (1946, 1962). It was based on a postulate that a fluid away from equilibrium would proceed to a thermodynamic equilibrium state in two distinct stages: first a microscopic kinetic stage with a time scale of the order of the time between molecular collisions, which for dense fluids or liquids is of the same order as the duration of the molecular collisions, after which local equilibrium is established; second a macroscopic hydrodynamic stage during which the fluid evolves in accordance with the hydrodynamic equations. Implicit in this postulate was the idea that no long-ranged dynamic correlations would be present in a fluid of molecules with short-ranged forces, unless the system would be near an incipient thermodynamic or hydrodynamic instability.

The subsequent history of nonequilibrium statistical physics has revealed a basic flaw in this picture. In evaluating the randomizing nature of molecular collisions one may distinguish between quantities like mass, momentum and energy, that are conserved in molecular collisions, and non-conserved physical quantities. However, it turns out that the slow hydrodynamic modes associated with the conserved quantities and the fast modes associated with non-conserved quantities are not independent, but can interact to cause a coupling between modes resulting in long-ranged (mesoscopic) dynamic correlations. The classical picture of short-ranged dynamic correlations first appeared to be inadequate in fluids near the critical point when experiments revealed a divergent thermal conductivity which could not be explained by the Van Hove theory of critical slowing down of the fluctuations that was based on strictly thermodynamic considerations (Michels et al., 1962; Sengers, 1966; Sengers and Keyes, 1971); this observation led to the development of the mode-coupling theory of critical dynamics (Fixman, 1967; Kadanoff and Swift, 1968; Kawasaki, 1970). Some time later it turned out that the same mode-coupling theory could also account for the presence of long-time tails in the Green-Kubo correlation functions for the transport coefficients that were originally noticed in computer simulations of molecular dynamics (Pomeau and Résibois, 1975; Dorfman, 1975; Ernst et al., 1976a). The presence of mesoscopic dynamic correlations also caused the appearance of a divergence in the virial expansion for the transport coefficients of moderately dense gases (Dorfman and Cohen, 1967; Brush, 1972).

Around 1980 it became evident that the mode-coupling theory would also predict the existence of long-ranged fluctuations in fluids that are kept in stationary nonequilibrium states (Kirkpatrick et al., 1982a,b; Fox, 1982; Tremblay, 1984). Specifically, when a fluid is subjected to a stationary temperature gradient, the temperature gradient causes a coupling between the component of the velocity fluctuations parallel to the gradient and the temperature fluctuations, leading to an algebraic divergence of the fluctuations in the limit of small wave numbers (Kirkpatrick et al., 1982b; Ronis and Procaccia, 1982). An algebraic dependence of the nonequilibrium fluctuations as a function of the wave number is now believed to be a general feature of fluctuations in fluids in stationary nonequilibrium states (Grinstein, 1991; Dorfman et al., 1994) and is a manifestation of a general principle of generic scale invariance in nonequilibrium statistical mechanics (Kirkpatrick et al., 2002; Belitz et al., 2005). The ultimate divergence of the intensity of nonequilibrium fluctuations for small wave numbers, i.e., for large wavelengths, in the presence of temperature or concentration gradients will be prevented by gravity and finite-size effects. This is the reason that gravity and finite-size effects can play an important role in the wave-number dependence of fluctuations in fluids in nonequilibrium states.

In this book we shall elucidate how fluctuating hydrodynamics can be used to deal with both equilibrium and nonequilibrium fluctuations in fluids and fluids mixtures. Fluctuating hydrodynamics is a stochastic fluid dynamics approach in which it is assumed that the fluctuations can be described by the usual hydrodynamic equations but supplemented with random noise terms whose correlation functions are determined by a fluctuation-dissipation theorem. This approach can be extended to fluctuations in fluids in nonequilibrium states by assuming that the noise correlations (in contrast to the correlation functions of the hydrodynamic variables) satisfy local thermal equilibrium, that is, they are assumed to be given by the same fluctuation-dissipation theorem as for fluids in thermodynamic equilibrium, but with local values of the temperature, density and concentration.

We shall find that hydrodynamic correlation functions in fluids in nonequilibrium states always extend over a spatial range that is longer than what one would expect for the corresponding local-equilibrium expressions for these correlation functions, even when the system is far away from any hydrodynamic instability. The long-ranged nature of the nonequilibrium fluctuations causes a volume element inside the fluid to receive information from neighboring volume elements. Thus nonequilibrium fluctuations even far away from any hydrodynamic instability already foreshadow to some extent the ultimate appearance of complex spatio-temporal behavior of fluids far from equilibrium like the onset of convection or the onset of turbulence. In this book we shall demonstrate these features by presenting a detailed analysis of fluctuations in fluid layers subjected to a temperature gradient. However, the fluctuating-dynamics approach elucidated in this book can be used to deal with a wide variety of nonequilibrium fluctuations in fluids and fluid mixtures.

We shall proceed as follows. In Chapter 2 we present a review of some of the basic concepts of nonequilibrium thermodynamics and a derivation of the hydrodynamic equations for fluids and fluid mixtures. Chapter 3 presents the theory of fluctuations in fluids that are in thermodynamic equilibrium. The major part of the book, namely Chapters 4 through 10 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10, deal with thermal fluctuations in fluids and fluid mixtures in the presence of a stationary temperature gradient, to which we refer as the Rayleigh-Bénard problem. The Rayleigh-Bénard problem has been historically the most thoroughly studied nonequilibrium system, both from a theoretical and an experimental point of view. It may be considered as a kind of paradigm for nonequilibrium systems. Ideas and insights obtained from a careful study of the Rayleigh-Bénard problem can be extended and used to interpret other nonequilibrium systems. We start our study of the Rayleigh-Bénard problem in Chapters 4 and 5, where we consider “bulk” thermal nonequilibrium fluctuations in fluids and fluid mixtures, namely, fluctuations with wavelengths much smaller than any finite height of the fluid layer. In Chapter 6 we develop a general procedure for incorporating finite-size effects in fluctuating hydrodynamics. In Chapter 7 we apply the procedure to a study of thermal nonequilibrium fluctuations in one-component fluid layers. In Chapter 8 we consider the nature of such nonequilibrium fluctuations close to the onset of Rayleigh-Bénard convection. Chapter 9 deals with thermal nonequilibrium fluctuations in binary-fluid layers. In this book we consider fluctuations at hydrodynamic space and time scales that are experimentally accessible with light scattering or shadowgraph techniques. In Chapter 10 we present a review of the experimental attempts to measure nonequilibrium fluctuations. In Chapter 11 we give a brief discussion of some other types of nonequilibrium fluctuations, namely nonequilibrium fluctuations in fluids subjected to shear, nonequilibrium interface fluctuations, nonequilibrium fluctuations in liquid crystals and in mixtures with chemical reactions. This chapter confirms the general theme of the book about the long-ranged nature of nonequilibrium fluctuations as summarized in the Epilogue.

One of the problems we encountered when preparing this volume is a lack of uniform notation among the various communities (such as statistical physicists, physical chemists, fluid dynamists, mechanical and chemical engineers) to whom this book is addressed. Even more striking is the presence in the literature of different definitions for some of the thermophysical properties and dimensionless numbers that enter in the description of fluids. In this book, we tried to follow the definitions and nomenclature recommendations of both IUPAP (Cohen and Giacomo, 1987) and IUPAC (Mills et al., 1988). Regarding the description of thermal diffusion, where the coexistence of various definitions and sign conventions is particularly confounding, we have adopted the recommendations contained in the book edited by Köhler and Wiegand (2002). For the benefit of the reader, we have included at the end of the book a List of symbols and corresponding SI units, where the nomenclature used is summarized, and reference is made to the page number where the property is first introduced. In addition, we have prepared a List of abbreviations which the reader can find at the end of the volume, prior to the Subject index.

Chapter 2

Nonequilibrium thermodynamics

Publisher Summary

This chapter...