Georgy Lebon

Extended irreversible thermodynamics revisited (1988-98)

We review the progress made in extended irreversible thermodynamics during the ten years that have elapsed since the publication of our first review on the same subject (Rep. Frog. Phys. 1988 51 1105-72). During this decade much effort has been devoted to achieving a better understanding of the fundamentals and a broadening of the domain of applications. The macroscopic formulation of extended irreversible thermodynamics is reviewed and compared with other non-equilibrium thermodynamic theories. The foundations of EIT are discussed on the bases of information theory, kinetic theory, stochastic phenomena and computer simulations. Several significant applications are presented, some of them of considerable practical interest (non-classical heat transport, polymer solutions, non-Fickian diffusion, microelectronic devices, dielectric relaxation), and some others of special theoretical appeal (superfluids, nuclear collisions, cosmology). We also outline some basic problems which are not yet completely solved, such as the definitions of entropy and temperature out of equilibrium, the selection of the relevant variables, and the status to be reserved to the H-theorem and its relation to the second law. In writing this review, we had four objectives in mind: to show (i) that extended irreversible thermodynamics stands at the frontiers of modern thermodynamics; (ii) that it opens the way to new and useful applications; (iii) that much progress has been achieved during the last decade, and (iv) that the subject is far from being exhausted.

An extension of the local equilibrium hypothesis

In order to extend the range of application of classical irreversible thermodynamics far from equilibrium, an extension of the Gibbs equation is presented. The new Gibbs equation is assumed to contain, besides its usual contributions, supplementary terms equal to the thermodynamic fluxes. The entropy flux and the entropy production also take more general forms than in classical non-equilibrium thermodynamics. As an illustration of the formalism, an isotropic viscous and non-isothermal two-fluid mixture is considered. The results are shown to be in agreement with the Boltzmann kinetic theory.

Long-wave instabilities of non-uniformly heated falling films

We consider the problem of a thin liquid layer falling down an inclined plate that is subjected to non-uniform heating. The plate temperature is assumed to be linearly distributed and both directions of the temperature gradient with respect to the flow are investigated. The film flow is not only influenced by gravity and mean surface tension, but in addition by the thermocapillary force acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. Applying the long-wave theory, a nonlinear evolution equation is derived. When the plate temperature is decreasing in the downstream direction, linear stability analysis exhibits a film stabilization, compared to a uniformly heated film. In contrast, for increasing temperature along the plate, the film becomes less stable. Numerical solution of the evolution equation indicates the existence of permanent finite-amplitude waves of different kinds. The shape of the waves depends mainly on the mean flow and the mean surface tension, but their amplitudes and phase speeds are influenced by thermocapillarity.

A nonlinear stability analysis of the Bénard–Marangoni problem

A nonlinear analysis of Benard–Marangoni convection in a horizontal fluid layer of infinite extent is proposed. The nonlinear equations describing the fields of temperature and velocity are solved by using the Gorkov–Malkus–Veronis technique, which consists of developing the steady solution in terms of a small parameter measuring the deviation from the marginal state. This work generalizes an earlier paper by Schluter, Lortz & Busse wherein only buoyancy-driven instabilities were handled. In the present work both buoyancy and temperature-dependent surface-tension effects are considered. The band of allowed steady states of convection near the onset of convection is determined as a function of the Marangoni number and the wavenumber. The influence of various dimensionless quantities like Rayleigh, Prandtl and Biot numbers is examined. Supercritical as well as subcritical zones of instability are displayed. It is found that hexagons are allowable flow patterns.

Buoyant-thermocapillary instabilities in medium-Prandtl-number fluid layers subject to a horizontal temperature gradient

Abstract Coupled buoyant and thermocapillary instabilities in a fluid layer of infinite horizontal extent bounded below by a rigid plane and above by a free flat surface and submitted to a temperature gradient are investigated. A general 3D mathematical formulation is used to determine the linearized perturbated equations of the steady state induced by the temperature gradient. Numerical results are obtained in the case of a horizontal temperature gradient, lower and upper surfaces are adiabatically isolated and the range of variation of the Prandtl number is selected as [10−2, 10]. The presence of travelling rolls is exhibited. The results display three kinds of behaviour according to the values taken by the Prandtl number: (a) 4 × 10−3 2.6.

Bénard–Marangoni instability in rigid rectangular containers

Thermocapillary convection in three-dimensional rectangular finite containers with rigid lateral walls is studied. The upper surface of the fluid layer is assumed to be flat and non-deformable but is submitted to a temperature-dependent surface tension. The realistic ‘no-slip’ condition at the sidewalls makes the method of separation of variables inapplicable for the linear problem. A spectral Tau method is used to determine the critical Marangoni number and the convective pattern at the threshold as functions of the aspect ratios of the container. The influence on the critical parameters of a non-vanishing gravity and a non-zero Biot number at the upper surface is also examined. The nonlinear regime for pure Marangoni convection ( Ra = 0 ) and for Pr = 10 4 , Bi = 0 is studied by reducing the dynamics of the system to the dynamics of the most unstable modes of convection. Owing to the presence of rigid walls, it is shown that the convective pattern above the threshold may be quite different from that predicted by the linear approach. The theoretical predictions of the present study are in very good agreement with the experiments of Koschmieder & Prahl (1990) and agree also with most of Dijkstras (1995 a, b ) numerical results. Important differences with the analysis of Rosenblat, Homsy & Davis (1982 b ) on slippery walls containers are emphasized.

A phenomenological scaling approach for heat transport in nano-systems

A phenomenological approach of heat transfer in nano-systems is proposed, on the basis of a continued-fraction expansion of the thermal conductivity, obtained within the framework of extended irreversible thermodynamics. Emphasis is put on the transition from the diffusive, collision-dominated heat transport to the ballistic heat transport, as a function of the mean free path and the length of the system.

Convective instability of a micropolar fluid layer by the method of energy

Abstract The convective instability of a micropolar incompressible fluid layer heated from below is treated within the framework of Serrin-Josephs energy method. In presence of coupling between temperature and micro-rotations, a region of subcritical instability is displayed. The influence of the various micropolar parameters on the onset of convection is also analyzed.

An extended thermodynamic model of transient heat conduction at sub-continuum scales

A thermodynamic description of transient heat conduction at small length and timescales is proposed. It is based on extended irreversible thermodynamics and the main feature of this formalism is to elevate the heat flux vector to the status of independent variable at the same level as the classical variable, the temperature. The present model assumes the coexistence of two kinds of heat carriers: diffusive and ballistic phonons. The behaviour of the diffusive phonons is governed by a Cattaneo-type equation to take into account the high-frequency phenomena generally present at nanoscales. To include non-local effects that are dominant in nanostructures, it is assumed that the ballistic carriers are obeying a Guyer–Krumhansl relation. The model is applied to the problem of transient heat conduction through a thin nanofilm. The numerical results are compared with those provided by Fourier, Cattaneo and other recent models.

Weakly Nonlocal And Nonlinear Heat Transport In Rigid Solids

A weakly nonlocal and nonlinear theory of heat conduction in rigid bodies is proposed. The constitutive equations generalize these of Fourier, Maxwell-Cattaneo and Guyer-Krumhansl. The proposed model uses the fundaments and the technique of extended irreversible thermodynamics. The main conclusion is that the presence of nonlocal terms in the transport equation for the heat flux implies a modification of the entropy flux; the latter is no longer given by its classical expression, i.e. the heat flux divided by the temperature, but contains extra contributions which are nonlinear in the heat flux and its gradient. These results arise as compatibility conditions with the second law of thermodynamics. A nonequilibrium temperature depending on the heat flux and generalizing the local equilibrium temperature is also emerging naturally from the formalism.