de Zarate, Jose M Ortiz

Sengers, Jan V

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Hydrodynamic Fluctuationsin Fluids and Fluid MixturesbyJosd M. Ortiz de ZärateDepartamento de Fisica Aplicada IUniversidad Complutense, Madrid, SpainJan V. SengersInstitute for Physical Science and Technologyand Burgers Program for Fluid DynamicsUniversity of Maryland, College ParkMaryland, USAELSEVIERAmsterdam - Boston - Heidelberg - London - New York - OxfordParis - San Diego - San Francisco - Singapore - Sydney - TokyoContents123IntroductionNonequilibrium thermodynamics2.1	 Local thermodynamic properties	 	2.1.1	 Equilibrium thermodynamics 	2.1.2	 Continuum hypothesis and the assumption of localequilibrium 	2.2	 Balance laws	 	2.2.1	 Mass balance 	2.2.2	 Pressure tensor 	2.2.3	 Momentuni balance	 	2.2.4	 Energy balance 	2.2.5	 Additional remarks 	2.3	 Entropy balance, dissipative fluxes and thermodynamic forces2.3.1	 Phenomenological relations 	 	2.3.2	 Onsager's reciprocal relations 	2.4	 Hydrodynamic equations 	2.4.1	 Hydrodynamic equations for a one-componentfluid 	2.4.2	 Hydrodynamic equations for a binary mixture . .2.5	 Boundary conditions 	2.5.1	 Conditions at the interface between two continuousmedia 	2.5.2	 A flat horizontal fluid boundary 	 	Fluctuations in fluids in thermodynamic equilibrium3.1	 Fluctuating hydrodynamics 	3.2	 Fluctuation-dissipation theorem for fluids and fluid mixtures3.2.1	 Fluctuation-dissipation theorem for a one-componentfluid 	3.2.2	 Fluctuation-dissipation theorem for a binary mixture3.3	 Hydrodynamic fluctuations in a one-component fluid . .. .155578911151619202223242428333336394043434546viiviii	 CONTENTS3.3.1 Structure factor 		513.3.2 Equal-time correlations 		543.4 Equilibrium correlation functions and entropy probabilityfunctional 		563.5 Extension of fluctuating hydrodynamics to nonequilibriumsteady states 		604 Thermal nonequilibrium fluctuations in one-componentfluids	 634.1 Boussinesq approximation 	  644.2 Bulk structure factor in the presence of a stationary temper-ature gradient 	 	 704.3 Nonequilibrium effects an the Brillouin doublet 	 	 764.4 Nonequilibrium fluctuations in heat conduction 	  825 Thermal nonequilibrium fluctuations in fluid mixtures 	 895.1 Linearized fluctuating Boussinesq equations for a binary liquid 915.1.1 The linear stability problem 	 	 945.2 Structure factor in a large-Lewis-number approximation . 	 	 975.3 Comprehensive structure factor of a nonequilibrium binarymixture 	  1025.4 Nonequilibrium fluctuations in isothermal free-diffusion pro-cesses 	  1106 Finite-size effects in hydrodynamic fluctuations 	 1136.1 The hydrodynamic operator 	 1156.1.1 Structure factors 	 1226.2 Hydrodynamic modes and decay rates for two free boundaries 1256.3 Hydrodynamic modes and decay rates for two rigid boundaries 1296.3.1 Decay rates for rigid boundaries in the limit of smallqll 	 	 1316.3.2 Decay rates for rigid boundaries in the limit of largeqll 134 6.4 The slowest decay rate 	 1367 Thermal nonequilibrium fluctuations in one-component-fluid layers	 1417.1 A fluid confined between two free boundaries 	 1427.2 A fluid confined between two rigid boundaries 	 1477.3 Limiting behavior of the structure factor for rigid boundariesat small and large wave numbers 	 1527.4 A Galerkin approximation for two rigid boundaries 	  1557.5 Correlations in real space 	 1627.6 Contribution of nonequilibrium fluctuations to heat transfer 166CONTENTS	 ix8 Thermal fluctuations close to the Rayleigh-Benard insta-bility	 1678.1 Critical slowing down of nonequilibrium fluctuations	 1688.2 The most-unstable-mode approximation 	 1698.3 The Swift-Hohenberg approximation 	 1728.4 Power of thermal fluctuations 	 1758.5 Wave number of maximum enhancement of fluctuations 	  1798.6 Wave number of fluctuations with maximum growth rate 	  1819 Thermal nonequilibrium fluctuations in binary-fluid layers 1859.1 The hydrodynamic Operator for binary Boussinesq 	  1879.2 A Galerkin approximation for rigid and impermeable bound-ary conditions 	 1899.3 Evaluation of nonequilibrium structure factors 	 1949.4 Structure factor in thermal equilibrium	 1979.5 Nonequilibrium fluctuations for positive separation ratio	 1989.5.1 Temperature fluctuations 	 1999.5.2 Concentration fluctuations 	 2049.6 Large-Lewis-number approximation for positive separationratio 	 2079.7 Concluding remarks 	  21110 Experiments on nonequilibrium fluctuations in the Rayleigh-Benard problem	 21310.1 Experimental techniques 	 21410.1.1 Small-angle light scattering 	 21410.1.2 Quantitative shadowgraph analysis 	 22210.1.3 Dependence of shadowgraph images on collecting times 22810.2 Nonequilibrium light-scattering experiments in one-componentfluids 	  22910.2.1 Propagating modes 	 22910.2.2 Brillouin doublet 	  23010.2.3 Rayleigh scattering 	 23210.3 Shadowgraph experiments in one-component fluids 	 23510.3.1 Divergence of critical fluctuations near the onset ofconvection 	  23510.3.2 Critical slowing down of fluctuations near the onsetof convection 	  23710.4 Light scattering in binary systems 	  23910.4.1 Nonequilibrium fluctuations in liquid mixtures . 	  24010.4.2 Concentration fluctuations in dilute polymer solutions 24310.5 Shadowgraphy in binary mixtures 	  246CONTENTS11 Other nonequilibrium fluctuations	 24711.1 Nonequilibrium fluctuations in isothermal one-componentfiuids under shear 	  24811.2 Nonequilibrium interface fluctuations 	 25211,3 Nonequilibrium fluctuations in nematic liquid crystals	25311.4 Concentration fluctuations in reaction-diffusion problems . 25611.4.1 Equilibrium concentration fluctuations in reaction-diffusion 	 25711.4.2 Nonequilibrium concentration fluctuations in reaction-diffusion 	 26312 Epilogue	 269Bibliography	 273List of symbols and corresponding SI units	 297List of abbreviations	 303Subject index	 305

Hydrodynamic Fluctuations in Fluids and Fluid Mixtures

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