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International audience; We present the complete theory for the decay rates of non-equilibrium fluctuations in a ternary liquid mixture subjected to a stationary temperature gradient, when the quiescent non-convective state is stable. In the most general case, within Boussinesq approximation, four fluctuating modes exist. Depending on the parameter values, propagative modes may be present, and we discuss numerically some cases where that is so. We complete the work with a discussion of symmetry upon changes in concentration representation, as well as examination of some limiting cases with practical relevance for which analytical progress is possible. We make contact with previous publications, which were based on some kind of approximation.

Presented in this paper is an investigation on wire-length dependence of transient hot-wire calorimetry, a technique generally used to measure the thermal conductivity of materials. In particular, a measurement protocol that includes the use of a calibration fluid that was designed for wires of reduced lengths; so as to reliably apply the transient hot-wire method to materials exhibiting a solid-liquid phase transition as a function of temperature. An experimental study of the onset of convection was also carried out, including a comparison between measurements performed in the liquid and solid phases. For the purpose of validation, presented are experimental measurements of the thermal conductivity of water Ih-ice at 0.1 MPa, within a temperature range of 259–266 K, and that of n-eicosane within the range of 259–348 K, which encompasses its melting point.

Definition of frame-invariant Soret coefficients for ternary mixtures The definition of the Soret coefficient of a binary mixture includes a concentration prefactor, x(1-x) when mol fraction x is used, or w(1-w) when mass fraction w is used. In this presentation the physical reasons behind this choice are reviewed, emphasizing that the use of these prefactors makes the Soret coefficient invariant upon change in the reference frame, either mass or molar. Then, it will be shown how this invariance property can be extended to ternary mixtures by using an appropriate concentration prefactor in matrix form. The presentation will be completed with some considerations of general non-isothermal diffusion fluxes, binary limits of the concentration triangle, selection of the dependent concentration in a ternary mixture, and, finally, extension to multi-component mixtures.References[1] J. M. Ortiz de Zarate. Definition of frame-invariant thermodiffusion and Soret coefficients for ternary mixtures. Eur. Phys. J. E 42(2019), 43

Thermodiffusion, which leads to a component separation in a mixture due to the thermal gradient, still does not have an unambiguous microscopic picture. Therefore, experimental studies, especially in convection free environment, are important. As part of the 4th campaign on the DCMIX project, thermodiffusion experiments on three compositions of the toluene-methanol-cyclohexane ternary mixture, on a mixture of fullerene-tetralin-toluene and on a mixture of polystyrene-toluene-cyclohexane have been performed in microgravity conditions on board the International Space Station. A binary mixture of polystyrene-toluene has been filled into the companion cell for the campaign. The Selectable Optical Diagnostics Instrument (SODI), wich is a two-wavelength Mach-Zehnder interferometer for the ternary mixtures, plus a monochromatic Mach-Zehnder interferometer for the binary mixture, has been used in order to obtain the temperature and the concentration fields in the cells. Precisely, it is a 5-steps phase-shifting interferometry technique which is implemented with SODI, producing by means of laser illumination a set of 5 phase-shifted images of π/2 between them as function of the time. A total of 58 runs of various durations and at different mean temperatures have been conducted. Here, we evaluate the contrast of the interferograms, the quality in the phase stepping, the stability of the thermal regulation of the experiments and the level of environmental disturbances on board the space station during the campaign., SCOPUS: ar.j, info:eu-repo/semantics/published

The GRADFLEX experiment of ESA has shown that under microgravity conditions a stationary thermodiffusion process is accompanied by giant non-equilibrium fluctuations with size as large as the size of the sample. In the presence of small concentration gradients, the features of the non-equilibrium fluctuations can be described by means of linearized hydrodynamics. However, the linear models are not suitable to describe most cases of applicative interest, such as fluctuations induced by large gradients and under non-stationary conditions. Moreover, presently the investigation of non-equilibrium fluctuations has mainly involved single component fluids and binary mixtures, but recently transport processes in ternary mixtures have attracted increasing interest due to the experiments performed on the International Space Station in the framework of the DCMIX project of ESA. The Giant Fluctuations (NEUF-DIX) project of ESA will investigate non-equilibrium fluctuations during diffusive processes occurring in complex multi-component mixtures, where one of the components is a macromolecule, such as a polymer, a colloid or a protein. Important objectives will be the exploration of the features of the fluctuations under non-ideal conditions, such as large gradients, transient processes, and concentrated samples, and the understanding of how the fluctuations affect the interactions between macromolecules. The project involves the development of a dedicated facility, consisting of an array of shadowgraph optical instruments working in parallel, each one equipped with a thermal gradient cell. Here we outline the design concept of the facility and the results of performance tests performed on a breadboard to evaluate the suitability of the designed instrument to carry out scientific measurements of non-equilibrium concentration fluctuations in space.

Extension of a description of mass diffusion in binary fluids based on Fick's law to multicomponent fluids requires introduction of diffusion matrices. A problem is that Fick diffusion matrices commonly adopted for multicomponent fluids depend on the velocity frame of reference. In this paper we show how one can define Fick diffusion matrices for multicomponent fluids that are frame invariant.

The accurate knowledge of pre-exploitation fluid compositional profile is one of the necessary pre-requisites for a successful field plan development of a petroleum reservoir by the oil and gas industry. Thermodiffusion, leading to a partial diffusive separation of species in a mixture subject to thermal gradient, is thought to play an important role in oil and gas reservoir due to the geothermal gradient. Although major improvements in measuring, simulating and modelling thermodiffusion coefficients have been achieved in the last decades, the improvements are mostly limited to binary liquid mixtures at atmospheric pressure. Thus, the need for accurate data, that would prove invaluable as benchmark reference data for validating models and simulations, was one of the main drivers behind the project “Soret Coefficient measurements of Crude Oil” (SCCO) which used a microgravity set-up implemented in the SJ-10 satellite. This unique project, resulting from a partnership between European Space Agency and China’s National Space Science Center enhanced by collaboration among academics from France, Spain, United Kingdom, China and industrialists from France and China, aimed to measure the thermodiffusion coefficients of multicomponent oil and gas mixtures under high pressures. Within this framework, some results on thermodiffusion of one ternary oil mixture and one quaternary gas condensate have been obtained in microgravity and have been qualitatively confirmed by molecular simulations. More precisely, these microgravity results have confirmed on multicomponent mixtures that thermodiffusion leads to a relative migration of the lightest hydrocarbon to the hot region. These results support the idea that, in oil and gas reservoirs, thermodiffusion is not negligible being able to counteract the influence of gravity-driven segregation on the vertical distribution of species.

The definitions of thermodiffusion and Soret coefficients for a binary mixture include a concentration prefactor x(1 - x), when mole fraction x is used, or w(1 - w), when mass fraction w is used. In this paper the physical reasons behind this choice are reviewed, emphasizing that the use of these prefactors makes the thermodiffusion and the Soret coefficients invariant upon changing in the concentration representation, using either mole fraction or mass faction. Then, it is shown how this invariance property can be extended to ternary mixtures by using appropriate concentration prefactors in matrix form. The paper is completed with some considerations about alternative definitions of thermodiffusion coefficients, binary limits of the concentration triangle, selection of the dependent concentration in a ternary mixture, use of molar concentrations and, finally, extension to multi-component mixtures.

8 pags., 4 figs., Nonequilibrium temperature and concentration fluctuations inside a binary liquid mixture under the action of a temperature gradient relax back to equilibrium either due to conduction and diffusion at large wave numbers, or due to the quenching determined by gravity at small wave numbers. We investigate the dynamics of nonequilibrium fluctuations in a binary liquid mixture of polystyrene and toluene heated from above under stationary conditions in a thermodiffusion experiment. We show that the strong gravitational stabilization at small wave numbers determines the appearance of propagating modes of nonequilibrium fluctuations as detected through the structure function of shadowgraph images. The propagating modes are the combined effect of temperature and velocity nonequilibrium fluctuations induced by the buoyancy force. The experimental results are in good agreement with a fluctuating hydrodynamics theroretical model including the coupling of fluctuations of velocity, temperature and concentration., We gratefully acknowledge support from the European Space Agency (ESA) and the Centre National d’Etudes Spatiales (CNES) of France, specifically for supporting the acquisition of the fast camera that made it possible to observe the propagating mode described in this paper. F.C., L.G.F., and H.B. would like to thank E2S-UPPA for financial support. The research at the Complutense University was supported by Grant No. ESP2017-83544-C3-2-P of the Spanish Agencia Estatal de Investigación.

ACL; International audience; In this work we propose a methodology, based on molecular dynamics simulations, to quantify the influence of segregation and thermodiffusion on the initial state distribution of the fluid species in hydrocarbon reservoirs. This convection-free approach has been applied to a synthetic oil composed of three normal alkanes and to a real acid gas. It has been found that the thermodiffusion effect induced by the geothermal gradient is similar (but opposite in sign) to that due to segregation for both mixtures. In addition, because of the combined effect of thermal expansion and thermodiffusion, it has been observed that the density gradient can be reversed, in the presence of a geothermal gradient. These numerical results emphasize the need of improving our quantification of thermodiffusion in multicomponent mixtures. The SCCO-SJ10 experiments will be a crucial step towards this goal. © 2015, Springer Science+Business Media Dordrecht.

The thermal properties including the thermal conductivity ( λ ), the specific heat ( c p ), the melting and crystallization temperatures together with their corresponding transition heat enthalpies of the odd-numbered linear alkane hydrocarbons n-pentadecane, n-heptadecane and n-nonadecane were measured in both the solid and liquid phases using the transient hot wire and the differential scanning calorimetry (DSC) techniques. The obtained results were compared with available literature data. Discussions were based also on the X-ray diffraction spectra of the three n-alkanes carried out at different temperatures and including both solid and liquid phases. A discontinuity of λ is observed at the solid - liquid phase transition of the three n-alkane hydrocarbons.

International audience

ACL; International audience; Abstract.: We discuss the gravity effects on the dynamics of composition fluctuations in a ternary mixture around the non-equilibrium quiescent state induced by thermodiffusion when subjected to a stationary temperature gradient. We found that the autocorrelation matrix of concentration fluctuations can be expressed as the sum of two exponentially decaying concentration modes. Without accounting for confinement, we obtained exact analytical expressions for the two decay rates which, as a consequence of gravity, display a wave-number-dependent mixing. The stability of the quiescent solution is also examined, as a function of the two solutal Rayleigh numbers used to express the decay rates. After having discussed the dynamics of the two concentration modes, we calculate the corresponding amplitudes. Consequences for optical experiments are discussed. Graphical abstract: [Figure not available: see fulltext.] © 2017, The Author(s).

Experimental measurements are presented of the thermal conductivity ( λ ) and the density ( ρ ) of several clay pastes used in pelotherapy (peloids). Up to 22 peloids were tested. They were prepared by adding various amounts of water to seven mineralogically different powder clays or clay minerals. λ of each clay paste was measured at six different temperatures ranging between 293 K and 318 K, while ρ was measured at ambient temperature. A clear correlation was established between the density and the thermal conductivity of the peloids.

We use fluctuating hydrodynamics to derive expressions for the spatial and temporal spectra of concentration fluctuations in a ternary liquid mixture in equilibrium. Our results agree with those found by other investigators who have used a procedure in which arbitrary initial values are adopted. Our method of using fluctuating hydrodynamics has the advantage that it can also be extended to deal with fluctuations in systems out of equilibrium.

ACL; International audience; Abstract.: In a recent letter (C. Giraudet et al., EPL 111, 60013 (2015)) we reported preliminary data showing evidence of a slowing-down of non-equilibrium fluctuations of the concentration in thermodiffusion experiments on a binary mixture of miscible fluids. The reason for this slowing-down was attributed to the effect of confinement. Such tentative explanation is here experimentally corroborated by new measurements and theoretically substantiated by studying analytically and numerically the relevant fluctuating hydrodynamics equations. In the new experiments presented here, the magnitude of the temperature gradient is changed, confirming that the system is controlled solely by the solutal Rayleigh number, and that the slowing-down is dominated by a combined effect of the driving force of buoyancy, the dissipating force of diffusion and the confinement provided by the vertical extension of the sample cell. Moreover, a compact phenomenological interpolating formula is proposed for easy analysis of experimental results. Graphical abstract: [Figure not available: see fulltext.] © 2016, EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.

International audience; Dynamic analysis of the light scattered by non-equilibrium fluctuations in a thermodiffusion experiment has been performed on a sample of polystyrene-toluene-n-hexane, at 0.9-49.55-49.55% mass fraction. Time decays of the non-equilibrium fluctuations have been obtained revealing the accurate de-tectability of three modes. The slowest mode has been attributed to the mass diffusion of the polymer into the binary solvent; the intermediate one to mass diffusion of the two molecular components of the solvent; finally, the fastest one has been attributed to the thermal diffusivity of the overall mixture. The two eigenvalues of the mass diffusion matrix have been evaluated with accuracy in the order of 1%. Neglecting cross-diffusion effects we obtain a simplified expression for the relative amplitude of the two mass diffusion modes, allowing a parameterized determination of polystyrene and toluene Soret coefficients in the ternary mixture. We suggest that a two wavelength shadowgraph experiment is needed for a complete determination of all the coefficients.

ACL; International audience; Abstract.: Diffusion and thermal diffusion processes in a liquid mixture are accompanied by long-range non-equilibrium fluctuations, whose amplitude is orders of magnitude larger than that of equilibrium fluctuations. The mean-square amplitude of the non-equilibrium fluctuations presents a scale-free power law behavior q-4 as a function of the wave vector q, but the divergence of the amplitude of the fluctuations at small wave vectors is prevented by the presence of gravity. In microgravity conditions the non-equilibrium fluctuations are fully developed and span all the available length scales up to the macroscopic size of the systems in the direction parallel to the applied gradient. Available theoretical models are based on linearized hydrodynamics and provide an adequate description of the statics and dynamics of the fluctuations in the presence of small temperature/concentration gradients and under stationary or quasi-stationary conditions. We describe a project aimed at the investigation of Non-EquilibriUm Fluctuations during DIffusion in compleX liquids (NEUF-DIX). The focus of the project is on the investigation in micro-gravity conditions of the non-equilibrium fluctuations in complex liquids, trying to tackle several challenging problems that emerged during the latest years, such as the theoretical predictions of Casimir-like forces induced by non-equilibrium fluctuations; the understanding of the non-equilibrium fluctuations in multi-component mixtures including a polymer, both in relation to the transport coefficients and to their behavior close to a glass transition; the understanding of the non-equilibrium fluctuations in concentrated colloidal suspensions, a problem closely related with the detection of Casimir forces; and the investigation of the development of fluctuations during transient diffusion. We envision to parallel these experiments with state-of-the-art multi-scale simulations. Graphical abstract: [Figure not available: see fulltext.] © 2016, EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.

ACL; International audience; From the benchmark values of the diffusion and thermodiffusion coefficients of the tetralin, isobutylbenzene and n-dodecane ternary mixture, and the published optical contrast factors, we have evaluated the theoretical amplitudes of the two composition modes of the refractive-index fluctuations. Shadowgraph experiments have been performed on ground, where the current theory is expected to be correct only for large wave vectors. Two decay times have been observed experimentally. The fastest one being related to the thermal diffusivity of the mixture, while the slower one to mass diffusion. Hence, it has not been possible to distinguish the two eigenvalues of the mass diffusion matrix, a problem also encountered in traditional light-scattering with ternary mixtures of similar-size molecules. Thus, to compare the measured Intermediate Scattering Function with theory, we fix the amplitudes and decay rates to the benchmark values, use the wave number as a fitting parameter, and compare it to the experimental wave number. The good agreement between theory and experiments for the larger wave numbers validates the theory developed for the microgravity conditions. © 2016, Springer Science+Business Media Dordrecht.

Recently, we have started to develop mesoscopic nonequilibrium thermodynamics for a reaction far from equilibrium in the presence of a temperature gradient, so that the interplay between the chemical reaction, diffusion and thermal diffusion can be described within the same theoretical framework. In this article we show that the spatial symmetry properties of the deterministic solution, which were so characteristic for conditions close to equilibrium, are no longer valid far from equilibrium. This is expected to have some conceptual consequences for the spatial spectrum of the fluctuations of temperature and concentrations around their local equilibrium values.

We measured, using the multi-current hot-wire method, the thermal conductivity (λ) of two different clay pastes suitable to be used as peloids. The same wire and system was used to measure the λ of water, and the results were compared with literature data. Special attention was paid to statistical data analysis. The obtained λ values for the clay pastes are higher than the λ of water at the same temperature.

In this article we describe and build a model of a historical weighing device proposed by the Spanish engineer Dario Bacas in the second half of the 19th century. The balance was named 'goniobarimeter' by its inventor, and the weighing principle is based on a curious, and not very well known, property of the cycloid. The simplicity of the design makes it especially suitable for a hands-on educational project with pre-university physics students.

The Jarzynski theorem is perhaps the most recently discovered simple general formula in elementary statistical physics. In this paper, written with a pedagogical aim, we illustrate the physical concepts under the Jarzynski and related results by a detailed calculation with a representative example. The physics of the model is sufficiently transparent so that it becomes suitable to incorporate in a general course of statistical physics.

Fluctuations in a fluid are strongly affected by the presence of a macroscopic gradient making them long-ranged and enhancing their amplitude. While small-scale fluctuations exhibit diffusive lifetimes, larger-scale fluctuations live shorter because of gravity, as theoretically and experimentally well-known. We explore here fluctuations of even larger size, comparable to the extent of the system in the direction of the gradient, and find experimental evidence of a dramatic slowing-down in their dynamics. We recover diffusive behaviour for these strongly-confined fluctuations, but with a diffusion coefficient that depends on the solutal Rayleigh number. Results from dynamic shadowgraph experiments are complemented by theoretical calculations and numerical simulations based on fluctuating hydrodynamics, and excellent agreement is found. The study of the dynamics of non-equilibrium fluctuations allows to probe and measure the competition of physical processes such as diffusion, buoyancy and confinement., Comment: Includes see Supplementary Material. Submitted to PRL

This chapter deals with thermal fluctuations in fluids and fluid mixtures in non-equilibrium steady states without any macroscopic fluid motion. Thermal fluctuations in fluids in the presence of a temperature and/or concentration gradient exhibit a pronounced non-equilibrium enhancement and are long ranged extending over the entire system. As a consequence, non-equilibrium fluctuations are profoundly affected by gravity and by finite-size effects. In addition, they induce large non-equilibrium Casimir forces. These phenomena change the traditional picture of thermal fluctuations in fluids and fluid mixtures.

Non-equilibrium thermodynamics is not only a theory of transport, fluxes and associated thermodynamic forces, but also yields a theory of thermodynamic fluctuations. In this chapter we present a brief summary of fluctuating hydrodynamics, which is the extension of linear non-equilibrium thermodynamics to deal with thermal fluctuation phenomena. At the core of fluctuating hydrodynamics is the fluctuation–dissipation theorem.

Thermodynamic fluctuations in systems that are in nonequilibrium steady states are always spatially long ranged, in contrast to fluctuations in thermodynamic equilibrium. In the present paper we consider a fluid subjected to a stationary temperature gradient. Two different physical mechanisms have been identified by which the temperature gradient causes long-ranged fluctuations. One cause is the presence of couplings between fluctuating fields. Secondly, spatial variation of the strength of random forces, resulting from the local version of the fluctuation-dissipation theorem, has also been shown to generate long-ranged fluctuations. We evaluate the contributions to the long-ranged temperature fluctuations due to both mechanisms. While the inhomogeneously correlated Langevin noise does lead to long-ranged fluctuations, in practice, they turn out to be negligible as compared to nonequilibrium temperature fluctuations resulting from the coupling between temperature and velocity fluctuations.

International audience; We present, based on fluctuating hydrodynamics, the theory of concentration fluctuations in a ternary mixture subjected to a stationary temperature gradient, so that composition gradients are present due to thermal diffusion (Soret effect). We neglect gravity and confinement (boundary conditions) but consider a completely generic diffusion matrix, including cross-diffusion effects. We find, as in the case of binary mixtures, an important non-equilibrium enhancement of the concentration fluctuations, which is proportional to the square of the gradient and inversely proportional to the fourth power of the fluctuations wave number, q−4. The results of this paper are expected to be asymptotically correct for fluctuations of large q, while for shorter q gravity and confinement effects need to be incorporated. Comparison with previous work in the topic is included. Graphical abstract: [Figure not available: see fulltext.]

In the present paper the feasibility of using direct contact membrane distillation for the concentration of glycol from used coolant liquids is analyzed. Membrane distillation experiments were performed with water–ethylene glycol mixtures, using a tangential flow cell, various membranes and different temperatures. The experiments show that the membranes reject almost completely the glycol and effective concentration is achieved. This result is expected, since the vapor pressure of glycol is negligible, compared to that of water, at the temperatures employed in the experiments. On the other hand, the effects of both temperature and concentration polarization were very important, and a specific computer code was developed in order to take them into account. From the computer code, values are obtained for the permeability of the membrane, and for the heat and mass transfer coefficients of the boundary layers. The values obtained agree with those of other literature sources and with the theoretical expectations.

We present a study of the spatial correlation functions of a one-dimensional reaction-diffusion system in both equilibrium and out of equilibrium. For the numerical simulations we have employed the Gillespie algorithm dividing the system in cells to treat diffusion as a chemical process between adjacent cells. We find that the spatial correlations are spatially short ranged in equilibrium but become long ranged in nonequilibrium. These results are in good agreement with theoretical predictions from fluctuating hydrodynamics for a one-dimensional system and periodic boundary conditions., Comment: 11 pages, 4 figures

We use fluctuating hydrodynamics to evaluate the enhancement of thermally excited fluctuations in laminar fluid flow using plane Couette flow as a representative example. In a previous publication [J. Stat. Phys. vol. 144 (2011) p. 774] we derived the energy amplification arising from thermally excited wall-normal fluctuations by solving a fluctuating Orr-Sommerfeld equation. In the present paper we derive the energy amplification arising from wall-normal vorticity fluctuation by solving a fluctuating Squire equation. The thermally excited wall-normal vorticity fluctuations turn out to yield the dominant contribution to the energy amplification. In addition, we show that thermally excited streaks, even in the absence of any externally imposed perturbations, are present in laminar fluid flow., Comment: 21 pages, 4 figures, Journal of Statistical Physics, accepted

We employed an easy and direct method to measure the thermal conductivity of epoxy in the liquid (nanofluid) and solid (nanocomposite) states using both rodlike and platelet-like carbon-based nanostructures. Comparing the experimental results with the theoretical model, an anomalous enhancement was obtained with multiwall carbon nanotubes, probably due to their layered structure and lowest surface resistance. Puzzling results for functionalized graphene sheet nanocomposites suggest that phonon coupling of the vibrational modes of the graphene and of the polymeric matrix plays a dominant role on the thermal conductivities of the liquid and solid states. PACS: 74.25.fc; 81.05.Qk; 81.07.Pr., The work was supported by the Spanish Ministry of Science and Innovation (MICINN) under project MAT 2010-18749. MMG thanks the CSIC for a JAE-Pre grant.

Non-isothermal transport of water through porous hydrophobic membranes is interpreted on the basis of a modified membrane distillation model. In previous papers, it was suggested that two opposing fluxes can take place through the partially wet membrane. The first is a vapour flux, from hot to cold through dry pores, and the second a liquid flux, from cold to hot through liquid-filled pores. We now present some results that confirm the above hypothesis. Temperature- and/or pressure-driven water transport through two PTFE membranes has been studied. A model is developed which characterizes the temperature- and pressure-driven steady states in accordance with the membrane distillation theory, explains the decrease of the fluxes and the steady-state pressure differences by progressive membrane wetting, and defines and evaluates a new parameter, the percentage of pores filled with liquid water, which describes that membrane wetting.

In this paper we investigate intrinsic thermally excited nonequilibrium velocity fluctuations in laminar planar Couette flow. For this purpose we have complemented the solution of the stochastic Orr-Sommerfeld equation for the intensity of the fluctuations of the wall-normal velocity, presented in a previous publication, with a solution of the stochastic Squire equation for the intensity of the fluctuations of the wall-normal vorticity. We have obtained exact solutions of these equations without boundary conditions and solutions in a Galerkin approximation when appropriate boundary conditions are included. These results enable us to make a quantitative assessment of the intensity of these nonequilibrium fluctuations, as well as of the related energy amplification, which are always present, even in the absence of any externally imposed noise.

We present an analysis of the transverse-velocity fluctuations in an isothermal liquid layer with a uniform shear rate between two parallel horizontal boundaries as a function of the wave number and the Reynolds number. The results were obtained by solving a stochastic version of the Orr-Sommerfeld equation subject to no-slip boundary conditions in a second-order Galerkin approximation. We find that the spatial Fourier transform of the transverse-velocity fluctuations exhibits a maximum as a function of the (horizontal) wave number ${q}_{\ensuremath{\parallel}}$. This maximum is associated with a crossover from a ${q}_{\ensuremath{\parallel}}^{\ensuremath{-}4}$ dependence for larger ${q}_{\ensuremath{\parallel}}$ to a ${q}_{\ensuremath{\parallel}}^{2}$ dependence for small ${q}_{\ensuremath{\parallel}}$. The ${q}_{\ensuremath{\parallel}}^{\ensuremath{-}4}$ dependence at larger wave numbers is independent of the boundary conditions, but the small-${q}_{\ensuremath{\parallel}}$ behavior is strongly affected by the boundary conditions. The nonequilibrium enhancement of the intensity of the transverse-velocity fluctuations remains finite for all values of the Reynolds number, but increases approximately with the square of the Reynolds number. The relation between our results and those obtained by previous authors in the absence of boundary conditions is elucidated.

The water flux induced by transmembrane temperature gradients through five PTFE membranes has been studied. Two sets of experiments have been carried out. In the first set the temperature difference between the two bulk phases was fixed at 10 K while the mean temperature and the stirring rate were varied independently. The mean temperature values were varied between 30 and 55 °C and the stirring-rate values were varied between 0 and 360 r.p.m. In the second set the mean temperature was fixed at 45 °C and the temperature difference and the stirring rate were varied independently. The temperature difference was varied between 5 and 25 K and the stirring rate was varied as above. The results permit us to obtain a ‘correction factor’ for the temperature polarization in non-isothermal water transport. In the literature this correction factor is supposed to be independent of mean temperature and temperature difference. In the present paper this assumption has been experimentally checked. The experiments state that the correction factor is virtually independent of mean temperature but grows with temperature difference.

In addition to the fast correlation for local stochastic motion, the velocity correlation function in a fluid enclosed within the pore boundaries features a slow long time-tail decay. At late times, the flow approaches that of an incompressible fluid. Here, we consider the motion of a viscous fluid, at constant temperature, in a rectangular semipermeable channel. The fluid is driven through the rectangular capillary by a uniform main pressure gradient. Tiny pressure gradients are allowed perpendicular to the main flux. We solve numerically the three-dimensional Navier-Stokes equations for the velocity field to obtain the steady solution. We then set and solve the Langevin equation for the fluid velocity. We report hydrodynamic fluctuations for the center-line velocity together with the corresponding relaxation times as a function of the size of the observing region and the Reynolds number. The effective diffusion coefficient for the fluid in the microchannel is also estimated (Deff = 1.43 x 10(-10) m2.s-1 for Re = 2), which is in accordance with measurements reported for a similar system (Stepisnik, J.; Callaghan, P. T. Physica B 2000, 292, 296-301; Stepisnik, J.; Callaghan, P. T. Magn. Reson. Imaging 2001, 19, 469-472).

In this paper a simple reaction-diffusion system, namely a binary fluid mixture with an association-dissociation reaction between the two components, is considered. Fluctuations at hydrodynamic spatiotemporal scales when a temperature gradient is present in this chemically reacting system are studied. First, fluctuating hydrodynamics when the system is in global equilibrium (isothermal) is reviewed. Comparing the two cases, an enhancement of the intensity of concentration fluctuations in the presence of a temperature gradient is predicted. The nonequilibrium concentration fluctuations are spatially long ranged, with an intensity depending on the wave number q. The intensity exhibits a crossover from a proportional, variantq(-4) to a proportional, variantq(-2) behavior depending on whether the corresponding wavelength is smaller or larger than the penetration depth of the reacting mixture. This opens a possibility to distinguish between diffusion- or activation-controlled regimes of the reaction by measuring these fluctuations. In addition, the possible observation of these fluctuations in nonequilibrium molecular dynamics simulations is considered.

Thermal fluctuations in fluids and fluid mixtures in stationary nonequilibrium states are spatially long ranged. This long-ranged nature of the non-equilibrium fluctuations is caused by a coupling between hydrodynamic modes through the non-equilibrium fluxes or gradients. The phenomena are discussed with some illustrative examples in fluids and fluid mixtures.

In this paper we evaluate the enhancement of nonequilibrium concentration fluctuations induced by the Soret effect when a binary fluid layer is subjected to a stationary temperature gradient. Starting from the fluctuating Boussinesq equations for a binary fluid in the large-Lewis-number approximation, we show how one can obtain an exact expression for the nonequilibrium structure factor in the long-wavelength limit for a fluid layer with realistic impermeable and no-slip boundary conditions. A numerical calculation of the wave-number dependence of the nonequilibrium enhancement and of the corresponding decay rate of the concentration fluctuations is also presented. Some physical consequences of our results are briefly discussed.

This chapter focuses on nonequilibrium fluctuations in the Rayleigh-Benard (RB) problem. The nonequilibrium fluctuations in isothermal one-component fluids under shear are presented. It is shown that the static structure factor of a one-component isothermal fluid subjected to uniform shear exhibits an anisotropic enhancement. It is found that while the equal-time fluctuations in the presence of shear are long-ranged compared to molecular scales, they do not become macroscopic and finite-size effects do not play a significant role unlike in the case of fluctuations in the presence of a temperature or concentration gradient. It is observed that in a two-phase system, like a liquid in equilibrium with its vapor, there are not only thermal fluctuations in the bulk phases, but also fluctuations of the location of the interface, especially capillary waves that broaden the width of the interface. The nonequilibrium fluctuations in nematic liquid crystals are analyzed. The concentration fluctuations in reaction–diffusion problems are also elaborated.

This chapter reviews thermal nonequilibrium fluctuations in one-component-fluid layers. The nonequilibrium structure factor is considered for a fluid layer between two free boundaries for which explicit analytic expressions can be obtained. For a fluid layer between two free boundaries, the nonequilibrium dynamic structure factor is obtained by substituting into the general expression. It is found that for the dimensionless nonequilibrium enhancement of the fluctuations in the case of rigid boundaries, a more compact result is obtained in the approximation of small-scattering angles. The exact analytical expressions for the limiting behavior of the nonequilibrium enhancement of the structure factor for rigid boundaries, are obtained. It is observed that the Galerkin method is equivalent to a finite-mode approach with approximate decay rates and corresponding modes. The advantage of this scheme is that, at the lowest order, it produces analytical expressions for the decay rates and the nonequilibrium structure factor. It is found that the Galerkin approximation reproduces for both limits the correct asymptotic behavior qualitatively. The contribution of nonequilibrium fluctuations to heat transfer is also elaborated.

This chapter presents the theory of fluctuations in fluids that are in thermodynamic equilibrium. The thermal fluctuations in fluids in thermodynamic equilibrium are spatially short ranged, except when the fluid is near a critical point. Thermal fluctuations are characterized by space-time functions that account for the correlation between the value of a quantity at defined position and time with its value at defined position and time. The intensity of the fluctuations is determined by equal-time correlation functions and the dynamical properties of the fluctuations by time-dependent correlation functions. The method of fluctuating hydrodynamics for dealing with fluctuations at hydrodynamic spatiotemporal scales in fluids in thermodynamic equilibrium is presented in the chapter. It is found that in a global-equilibrium state, the values of the fluctuating fluxes have a multivariate spatiotemporal Gaussian probability distribution. The fluctuation-dissipation theorem is specified for one-component fluids and for binary fluid mixtures that obey the hydrodynamic equations. It is found that applying a double inverse Fourier transform to the static structure factor can be related to the equal-time mass density autocorrelation function.

This chapter focuses on experiments on nonequilibrium fluctuations in the Rayleigh–Benard problem. The experimental techniques that have been employed to measure nonequilibrium fluctuations are presented. It is found that to obtain an expression for the intensity of scattered light, a plane-wave incident in a nonmagnetic nonconducting nonabsorbing medium is considered. An equation that relates the spectrum of light scattered by a fluid to the autocorrelation function of the refractive-index fluctuations is analyzed in the chapter. This autocorrelation function can, in turn, be related to the hydrodynamic fluctuations through the use of an equation of state. It is found that the spectrum of scattered light is proportional to the spectrum of the fluctuations. In particular, the total intensity of light scattered by the fluid will be obtained by integration over the frequency of the spectrum. Quantitative shadowgraph analysis is a promising alternative experimental technique for measuring the intensity of nonequilibrium fluctuations. The nonequilibrium light-scattering experiments in one-component fluids are also elaborated.

This chapter discusses finite-size effects in hydrodynamic fluctuations. The structure factor of a nonequilibrium fluid from the linearized random Boussinesq equations is calculated. It is shown that confinement effects become important in nonequilibrium fluctuations, even for fluid elements far away from the boundaries, when the wave number of the fluctuations is of the order of the inverse of the height of the layer. It is found that both sets of functions are bi-orthogonal not with respect to the usual scalar product, but with respect to a new scalar product, which can be defined for functions that obey the boundary conditions. There may be a problem in the orthogonality relationship, when there are different eigenfunctions corresponding to the same eigenvalue. It is assumed that the correlation functions between the different components of the random stress tensor and the random heat flux retain their equilibrium values given by the FDT for a divergence-free one-component fluid. The hydrodynamic modes and decay rates for two free-boundaries are also elaborated in the chapter.