On the nonequilibrium thermodynamics of non-Fickian diffusion (original) (raw)
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Isothermal Nonstandard Diffusion in a Two-Component Fluid Mixture: A Hamiltonian Approach
Journal of Non-Equilibrium Thermodynamics, 1998
A nonlinear formalism describing non-Fickian diffusion in a two-component mixture at uniform temperature is presented. It is based on a Hamiltonian description and is achieved by introducing two potentials: the thermodynamic free energy potential and the so-called dissipative potential. The evolution equations are expressed in terms of Poisson's brackets whose generating potentials are the two above-mentioned potentials. In view of a better understanding, the paper is presented in a rather pedagogical way, starting with the simple Hamiltonian description of classical hydrodynamics before examining respectively classical Fickian diffusion and non-Fickian diffusion in ordinary fluids including non-local effects. The formalism rests on the introduction of a "flux" variable in the spirit of Extended Irreversible Thermodynamics (EIT). It is seen that this extra variable, which complements the classical variables (mass density, mass concentration and barycentric velocity), is related to the relative velocities of both constituants. The evolution equations for the whole set of variables are derived in a systematic manner and the correlation between mechanical properties and diffusion is emphasized. The present work generalizes earlier nonclassical models and opens the way to more complicated situations involving for example thermal effects, phase change, multicomponent mixtures and diffusion in polymeric systems.
Diffusion in stationary flow from mesoscopic nonequilibrium thermodynamics
Physical Review E, 2001
We analyze the diffusion of a Brownian particle in a fluid under stationary flow. By using the scheme of nonequilibrium thermodynamics in phase space, we obtain the Fokker-Planck equation that is compared with others derived from the kinetic theory and projector operator techniques. This equation exhibits violation of the fluctuation-dissipation theorem. By implementing the hydrodynamic regime described by the first moments of the nonequilibrium distribution, we find relaxation equations for the diffusion current and pressure tensor, allowing us to arrive at a complete description of the system in the inertial and diffusion regimes. The simplicity and generality of the method we propose makes it applicable to more complex situations, often encountered in problems of soft-condensed matter, in which not only one but more degrees of freedom are coupled to a nonequilibrium bath.
Journal of Non-Equilibrium Thermodynamics, 2000
All the time evolution equations describing approaches to thermodynamic equilibrium states on all levels of description share a common structure. This structure has been collected in an abstract equation called GENERIC. A time evolution equation describing a particular physical system is obtained as a particular realization of GENERIC. In this paper we work out a realization describing the time evolution of a mixture of two fluids in which both non-Fickean diffusion and heat conduction take place.
Nonlinear diffusion equations in fluid mixtures
Evolution Equations and Control Theory, 2016
The whole set of balance equations for chemically-reacting fluid mixtures is established. The diffusion flux relative to the barycentric reference is shown to satisfy a first-order, non-linear differential equation. This in turn means that the diffusion flux is given by a balance equation, not by a constitutive assumption at the outset. Next, by way of application, limiting properties of the differential equation are shown to provide Fick's law and the Nernst-Planck equation. Moreover, known generalized forces of the literature prove to be obtained by appropriate constitutive assumptions on the stresses and the interaction forces. The entropy inequality is exploited by letting the constitutive functions of any constituent depend on temperature, mass density and their gradients thus accounting for nonlocality effects. Among the results, the generalization of the classical law of mass action is provided. The balance equation for the diffusion flux makes the system of equations for diffusion hyperbolic, provided heat conduction and viscosity are disregarded. This is ascertained by the analysis of discontinuity waves of order 2 (acceleration waves). The wave speed is derived explicitly in the case of binary mixtures.
On the thermodynamics of volume/mass diffusion in fluids
arXiv (Cornell University), 2012
In reference [1], a kinetic equation for gas flows was proposed that leads to a set of four macroscopic conservation equations, rather than the traditional set of three equations. The additional equation arises due to local spatial random molecular behavior, which has been described as a volume or mass diffusion process. In this present paper, we describe a procedure to construct a Gibbs-type equation and a second-law associated with these kinetic and continuum models. We also point out the close link between the kinetic equation in [1] and that proposed previously by Klimontovich, and we discuss some of their compatibilities with classical mechanical principles. Finally, a dimensional analysis highlights the nature of volume/mass diffusion: it is a non-conventional diffusive process, with some similarities to the 'ghost effect', which cannot be obtained from a fluid mechanical derivation that neglects non-local-equilibrium structures, as the conventional Navier-Stokes-Fourier model does.
Diffusion processes in nonequilibrium thermodynamics
Rendiconti del Seminario Matematico e Fisico di Milano, 1988
This paper deals with two alternative formulations in irreversible thermodynamics: a microscopic, stochastic description is associated with every deterministic, macroscopic model. The association is done via semigroup theory, and is used in order to investigate basic issues such as equilibrium states and the stability thereof. In particular, the density of an invariant measure is characterized as an integrating factor for the infinitesimal generator of the process.
BGK models for diffusion in isothermal binary fluid systems
Physica A: Statistical Mechanics and its Applications, 2001
Two Bhatnagar-Gross-Krook (BGK) models for isothermal binary uid systems-the classical single relaxation time model and a split collision term model-are discussed in detail, with emphasis on the di usion process in perfectly miscible ideal gases. Fluid equations, as well as the constitutive equation for di usion, are derived from the Boltzmann equation using the method of moments and the values of the transport coe cients (viscosity and di usivity) are calculated. The Schmidt number is found to be equal to one for both models. The split collision term model allows the two uid components to have di erent values of the viscosity, while the single relaxation time model does not have this characteristic. The value of the viscosity does not depend on the density in the split collision term model, as expected from the classical kinetic theory developed by Maxwell. Possible extension of BGK models to non-ideal gases and ideal solutions (where the Schmidt number is larger than 1) is also investigated.
Mechanics and thermodynamics of diffusion
The relation between diffusive forces and fluxes is sometimes chosen on the basis of the entropy inequality. Since the form of the entropy inequality is influenced by the form of the thermal energy equation, a precise understanding of the latter is necessary when the matter of forces and fluxes is explored. Often the form of the thermal energy equation for multicomponent systems is developed on an intuitive basis, and this leads to uncertainty in the form of the entropy inequality. A detailed analysis of the thermal energy equation leads to an entropy transport equation which indicates that the use of the gradient of the chemical potential as a driving force for the diffusive flux is not justified.
Journal of Applied Mechanics, 2006
It is shown that extended irreversible thermodynamics (EIT) provides a unified description of a great variety of processes, including matter diffusion, thermo-diffusion, suspensions, and fluid flows in porous media. This is achieved by enlarging the set of classical variables, as mass, momentum and temperature by the corresponding fluxes of mass, momentum and heat. For simplicity, we consider only Newtonian fluids and restrict ourselves to a linear analysis: quadratic and higher order terms in the fluxes are neglected. In the case of diffusion in a binary mixture, the extra flux variable is the diffusion flux of one the constituents, say the solute. In thermo-diffusion, one adds the heat flux to the set of variables. The main result of the present approach is that the traditional equations of Fick, Fourier, Soret, and Dufour are replaced by time-evolution equations for the matter and heat fluxes, such generalizations are useful in high-frequency processes. It is also shown that the ...