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The analysis of coupled processes is one of the most outstanding aspects of the classical theory of non-equilibrium thermodynamics. To emphasize this feature, this chapter is dedicated to the study of coupled transport processes, involving two fluxes and two thermodynamic forces. Emphasis is put on some practical applications in thermoelectricity (coupling of heat and electricity), thermodiffusion (coupling of heat and mass motion) and transport of matter through membranes. In the precedent chapter, we have already outlined, in general terms, the importance of the Onsager's reciprocal relations ruling the coefficients that describe such couplings. Here, we will discuss direct applications. [ABSTRACT FROM AUTHOR]

The previous chapters were essentially concerned with the average behaviour of thermodynamic variables without reference to their fluctuations. The latter are not only a subject of intense study in statistical mechanics, but also play a significant role in non-equilibrium thermodynamics,like near critical points,non-equilibrium instabilities and,more generally,in processes taking part at small time and space scales. They played an important role in the original derivation of Onsager—Casimir's reciprocal relations,and in modern formulations of the fluctuation—dissipation relations expressing the transport coefficients in terms of the correlation functions of the fluctuations of the fluxes.Furthermore,when the fluctuations become important,they must be included amongst the set of independent variables.The status of fluctuations being intermediate between the macroscopic and the microscopic,it is justified to refer to them as mesoscopic variables. [ABSTRACT FROM AUTHOR]

In the theories discussed so far, the limitations on the form of the constitutive equations arise essentially from the application of the second law of thermodynamics. However, no restrictions were placed on the reversible terms of the time evolution equations, as they do not contribute to the entropy production. Such terms may be either gyroscopic forces, as Coriolis force, or convected time derivatives, as Maxwell or corotational derivative, which are of frequent use in rheology. It is shown here that requirement of a Hamiltonian structure leads to restrictions on the reversible part of the evolution equations. [ABSTRACT FROM AUTHOR]

In Chaps. 7 and 8, it was assumed that the instantaneous local state of the system out of equilibrium was characterized by the union of classical variables and a number of additional variables (fluxes in EIT, internal variables in IVT). Only their instantaneous value at the present time was taken into account and their evolution was described by a set of ordinary differential equations. An alternative attitude, followed in the early developments of rational thermodynamics (RT), is to select a smaller number of variables than necessary for an exhaustive description. The price to be paid is that the state of the material body will be characterized not only by the instantaneous value of the variables, but also by their values taken in the past, namely by their history. [ABSTRACT FROM AUTHOR]

An alternative approach of non-equilibrium thermodynamics may be carried out through the internal variables theory (IVT). This phenomenological description has been very successful in the study of a wide variety of processes in rheology, deformable bodies, physico-chemistry and electromagnetism. IVT may be considered as a generalization of the classical theory of irreversible processes, just like extended irreversible thermodynamics. In the classical theory, the state of the system is described by the local equilibrium variables, say energy, volume or deformation, but it is well known that many non-equilibrium effects cannot be adequately described by this set of variables. To avoid, among others, the inconvenience of a heavy formalism in terms of functionals, one has, in extended irreversible thermodynamics, supplemented the local equilibrium variables by the corresponding fluxes. In the IVT, one adopts a similar attitude by adding a certain number of internal variables to the local equilibrium variables. Such variables are called internal because they are connected either to internal motions or to local microstructures. They should not be confused with order parameters as introduced in Chap. 6, which are describing some broken symmetry related to long-range interactions as in phase transitions. We should also add that unlike the fluxes in extended irreversible thermodynamics, the internal variables are arbitrary extensive quantities, not identified from the outset. It is the freedom in the choice of the internal variables that explains the wide domain of applicability of the formalism. [ABSTRACT FROM AUTHOR]

With this chapter, we begin a panoramic overview of non-equilibrium thermodynamic theories that go beyond the local equilibrium hypothesis, which is the cornerstone of classical irreversible thermodynamics (CIT). We hope that this presentation, covering Chaps. 7-11, will convince the reader that non—equilibrium thermodynamics is a fully alive and modern field of research, combining practical motivations and conceptual questions. Indeed, such basic topics as the definition and meaning of temperature and entropy, the formulation of the second law, and its consequences on the admissible transport equations are still open questions nowadays. [ABSTRACT FROM AUTHOR]

The notion of irreversibility is frequently associated with a tendency of the systems towards spatial homogeneity and rest, i.e. towards equilibrium. Although this observation is true in isolated systems, it is no longer verified in closed and open systems, which may exhibit organized structures in space and time. This organization arises as a consequence of occurrence of instabilities in systems driven far from equilibrium by external forces. Instabilities may be considered as phase transition phenomena, because they manifest under the form of a discontinuous change of the physical behaviour at a critical value of an external control parameter. [ABSTRACT FROM AUTHOR]

One of the motivations of the early developments of thermodynamics was the optimization of heat engines, as steam engines, which transform partially heat into work, and which led to the first industrial revolution between the end of the eighteenth century and the beginning of the nineteenth century. This provides an interesting illustration of how an engineering problem coupled with economical needs stimulated the occurrence of a new fundamental science. Nowadays, we live in a similar situation: we are experiencing an industrial revolution based on information processing and miniaturized engines and, on the other side, we are conscious of the need to incorporate ecological restrictions into the economical progress. To convey some flavour of these problems, we present in this chapter applications of non-equilibrium thermodynamics to heat engines working with a finite non-vanishing rate. This approach is referred to as finite-time thermodynamics and received a strong impetus during the last quarter of the twentieth century. [ABSTRACT FROM AUTHOR]

Chemical reactions are among the most widespread processes influencing life and economy. They are extremely important in biology, geology, environmental sciences, and industrial developments (energy management, production of millions of different chemical species, search for new materials). Chemical kinetics is a very rich but complex topic, which cannot be fully grasped by classical irreversible thermodynamics, whose kinetic description is restricted to the linear regime, not far from equilibrium. [ABSTRACT FROM AUTHOR]

Equilibrium thermodynamics is concerned with ideal processes taking place at infinitely slow rate, considered as a sequence of equilibrium states. for arbitrary processes, it may only compare the initial and final equilibrium states but the processes themselves cannot be described. To handle more realistic situations involving finite velocities and inhomogeneous effects, an extension of equilibrium thermodynamics is needed. [ABSTRACT FROM AUTHOR]

Equilibrium or classical thermodynamics deals essentially with the study of macroscopic properties of matter at equilibrium. A comprehensive definition of equilibrium will be given later; here it is sufficient to characterize it as a time-independent state, like a column of air at rest in absence of any flux of matter, energy, charge, or momentum. By extension, equilibrium thermodynamics has also been applied to the description of reversible processes: they represent a special class of idealized processes considered as a continuum sequence of equilibrium states. [ABSTRACT FROM AUTHOR]

This paper is a review of some recent experimental and theoretical works about the deflection of the upper surface in thin layers heated from below and subject to Bénard-Marangoni instability. [ABSTRACT FROM AUTHOR]

This work deals with Bénard-Marangoni convection in a thin horizontal fluid layer heated from below. The lower surface of the layer is in contact with a uniformly heated rigid plate and the upper face is deformed and submitted to a temperature-dependent surface tension. Two situations are analyzed: in the first one, the container is at rest; in the second one, it is uniformly rotating about a vertical axis. The role of the combined effects of surface deflection and rotation is examined. Two problems receive a particular attention: overstability and the observation that in pure surface-tension driven instability, the system is unconditionally unstable with an infinite critical wavelength. [ABSTRACT FROM AUTHOR]

In the first part of this note, one briefly, recalls the motivations and the basic hypotheses underlying Extended Irreversible Thermodynamics. To illustrate the application of the formalism, one considers the problem of a viscous fluid subject to thermal heating: the relevant constitutive equations are established and their relation with the classical formulation is discussed. Extended Thermodynamics appears to be useful in treating materials involving large relaxation times (like dielectrics at low temperature, polymers,...) and high-frequency and short-wavelength processes (ultrasounds, light and neutron scattering,...) However to deal with short-wavelength phenomena, it is imperious to introduce non-locality in space. This is achieved by including supplementary variables taking the form of fluxes of the fluxes. The corresponding changes in the expression of the constitutive equations and the entropy flux are discussed. [ABSTRACT FROM AUTHOR]