Frame-invariant Fick diffusion matrices of multicomponent fluid mixtures.

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.

Definition of frame-invariant thermodiffusion and Soret coefficients for ternary mixtures.

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.

Thermodiffusion in multicomponent n-alkane mixtures.

Compositional grading within a mixture has a strong impact on the evaluation of the pre-exploitation distribution of hydrocarbons in underground layers and sediments. Thermodiffusion, which leads to a partial diffusive separation of species in a mixture due to the geothermal gradient, is thought to play an important role in determining the distribution of species in a reservoir. However, despite recent progress, thermodiffusion is still difficult to measure and model in multicomponent mixtures. In this work, we report on experimental investigations of the thermodiffusion of multicomponent n-alkane mixtures at pressure above 30 MPa. The experiments have been conducted in space onboard the Shi Jian 10 spacecraft so as to isolate the studied phenomena from convection. For the two exploitable cells, containing a ternary liquid mixture and a condensate gas, measurements have shown that the lightest and heaviest species had a tendency to migrate, relatively to the rest of the species, to the hot and cold region, respectively. These trends have been confirmed by molecular dynamics simulations. The measured condensate gas data have been used to quantify the influence of thermodiffusion on the initial fluid distribution of an idealised one dimension reservoir. The results obtained indicate that thermodiffusion tends to noticeably counteract the influence of gravitational segregation on the vertical distribution of species, which could result in an unstable fluid column. This confirms that, in oil and gas reservoirs, the availability of thermodiffusion data for multicomponent mixtures is crucial for a correct evaluation of the initial state fluid distribution.

Gravity effects on Soret-induced non-equilibrium fluctuations in ternary mixtures.

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.

Confinement effect on the dynamics of non-equilibrium concentration fluctuations far from the onset of convection.

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.

The NEUF-DIX space project - Non-EquilibriUm Fluctuations during DIffusion in compleX liquids.

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.

Non-equilibrium fluctuations induced by the Soret effect in a ternary mixture.

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.

Nonequilibrium velocity fluctuations and energy amplification in planar Couette flow.

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.

Concentration fluctuations in nonisothermal reaction-diffusion systems.

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.

Comment on "Shear-induced quench of long-range correlations in a liquid mixture".

Recently, Wada [Phys. Rev. E 69, 031202 (2004)] presented an analysis of the long-range nature of concentration fluctuations in a binary liquid mixture subjected to a concentration gradient in a uniform shear flow as a function of the wave number k of the fluctuations. Specifically, he argued that the presence of a uniform shear causes the intensity of the concentration fluctuations to crossover from the well-known k(-4) dependence at large wave numbers to a k(-4/3) dependence for small wave numbers. The purpose of this comment is to point out that the wave-number dependence of the concentration fluctuations to be expected in realistic experimental conditions will be affected by gravity and finite-size effects.

Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection.

We present experimental data and their theoretical interpretation for the decay rates of temperature fluctuations in a thin layer of a fluid heated from below and confined between parallel horizontal plates. The measurements were made with the mean temperature of the layer corresponding to the critical isochore of sulfur hexafluoride above but near the critical point where fluctuations are exceptionally strong. They cover a wide range of temperature gradients below the onset of Rayleigh-Bénard convection, and span wave numbers on both sides of the critical value for this onset. The decay rates were determined from experimental shadowgraph images of the fluctuations at several camera exposure times. We present a theoretical expression for an exposure-time-dependent structure factor which is needed for the data analysis. As the onset of convection is approached, the data reveal the critical slowing down associated with the bifurcation. Theoretical predictions for the decay rates as a function of the wave number and temperature gradient are presented and compared with the experimental data. Quantitative agreement is obtained if allowance is made for some uncertainty in the small spacing between the plates, and when an empirical estimate is employed for the influence of symmetric deviations from the Oberbeck-Boussinesq approximation which are to be expected in a fluid with its density at the mean temperature located on the critical isochore.

Boundary effects on the nonequilibrium structure factor of fluids below the Rayleigh-Bénard instability.

We consider a horizontal fluid layer between two rigid boundaries, maintained in a stationary thermal nonequilibrium state below the convective Rayleigh-Bénard instability. We derive an explicit expression for the nonequilibrium structure factor in a first-order Galerkin approximation valid for negative and positive Rayleigh numbers R up to the critical Rayleigh number R(c) associated with the appearance of convection. The results obtained for rigid boundaries by the Galerkin-approximation method are compared with exact results previously derived for the case of free boundaries. The nonequilibrium structure factor exhibits a maximum as a function of the wave number q of the fluctuations. This maximum is associated with a crossover from a q(-4) dependence for larger q to a q(2) dependence for small q. This maximum is present at both negative and positive R, becomes pronounced at positive R and diverges as R approaches the critical value R(c).