A mesoscopic approach to diffusion phenomena in mixtures

The mesosocpic concept is applied to the theory of mixtures. The aim is to investigate the diffusion phenomenon from a mesoscopic point of view. The domain of the field quantities is extended by the set of mesoscopic variables, here the velocities of the components. Balance equations on this enlarged space are the equations of motion for the mesoscopic fields. Moreover, local distribution functions of the velocities are introduced as a statistical element, and an equation of motion for this distribution function is derived. From this equation of motion differential equations for the diffusion fluxes, and also for higher order fluxes are obtained. These equations are of balance type, as it is postulated in Extended Thermodynamics. The resulting evolution equation for the diffusion flux generalizes the Fick's law

WAVE SOLUTIONS IN RHEOLOGICAL MEDIA

The propagation of linear acoustic waves in isotropic media in which mechanical relaxation phenomena occur are investigated. The irreversible mechanical processes in the medium due to viscosity and to changes in a tensorial internal variable are analysed with the aid of non-equilibrium thermodynamics. In particular, in this paper, a wave solution applying the method of Laplace transform is proposed. Moreover, the corresponding solutions in Poynting-Thomson, Maxwell, Kelvin-Voigt, Hooke and Newton media are calculated

Generalized heat-transport equations: Parabolic and hyperbolic models

We derive two different generalized heat-transport equations: The most general one, of the first order in time and second order in space, encompasses some well known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman-Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed

A thermodynamical description of third grade fluids mixtures

A complete thermodynamical analysis for a non-reacting binary mixtures exhibiting the features of a third grade fluid is analyzed. The constitutive functions are allowed to depend on the mass density of the mixture and the concentration of one of the constituents, together with their first and second order gradients, on the specific internal energy of the mixture with its first order gradient, as well as on the symmetric part of the gradient of barycentric velocity. Compatibility with second law of thermodynamics is investigated by applying the extended Liu procedure. An explicit solution of the set of thermodynamic restrictions is obtained by postulating a suitable form of the constitutive relations for the diffusional mass flux, the heat flux and the Cauchy stress tensor. Taking a first order expansion in the gradients of the specific entropy, the expression of the entropy flux is determined. It includes an additional contribution due to non non-local effects.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:2109.1070

The Role of the Second Law of Thermodynamics in Continuum Physics: A Muschik and Ehrentraut Theorem Revisited

In continuum physics, constitutive equations model the material properties of physical systems. In those equations, material symmetry is taken into account by applying suitable representation theorems for symmetric and/or isotropic functions. Such mathematical representations must be in accordance with the second law of thermodynamics, which imposes that, in any thermodynamic process, the entropy production must be nonnegative. This requirement is fulfilled by assigning the constitutive equations in a form that guaranties that second law of thermodynamics is satisfied along arbitrary processes. Such an approach, in practice regards the second law of thermodynamics as a restriction on the constitutive equations, which must guarantee that any solution of the balance laws also satisfy the entropy inequality. This is a useful operative assumption, but not a consequence of general physical laws. Indeed, a different point of view, which regards the second law of thermodynamics as a restriction on the thermodynamic processes, i.e., on the solutions of the system of balance laws, is possible. This is tantamount to assuming that there are solutions of the balance laws that satisfy the entropy inequality, and solutions that do not satisfy it. In order to decide what is the correct approach, Muschik and Ehrentraut in 1996, postulated an amendment to the second law, which makes explicit the evident (but rather hidden) assumption that, in any point of the body, the entropy production is zero if, and only if, this point is a thermodynamic equilibrium. Then they proved that, given the amendment, the second law of thermodynamics is necessarily a restriction on the constitutive equations and not on the thermodynamic processes. In the present paper, we revisit their proof, lighting up some geometric aspects that were hidden in therein. Moreover, we propose an alternative formulation of the second law of thermodynamics, which incorporates the amendment. In this way we make this important result more intuitive and easily accessible to a wider audience

Thermoelectric efficiency of silicon–germanium alloys in finite-time thermodynamics

We analyze the efficiency in terms of a thermoelectric system of a one-dimensional Silicon–Germanium alloy. The dependency of thermal conductivity on the stoichiometry is pointed out, and the best fit of the experimental data is determined by a nonlinear regression method (NLRM). The thermoelectric efficiency of that system as function of the composition and of the effective temperature gradient is calculated as well. For three different temperatures (T = 300K, T = 400K, T = 500K), we determine the values of composition and thermal conductivity corresponding to the optimal thermoelectric energy conversion. The relationship of our approach with Finite-Time Thermodynamics is pointed out

Analysis of a Hyperbolic Heat Transfer Model in Blood-perfused Biological Tissues with Laser Heating

This paper proposes a hyperbolic heat transport model for a homogeneously perfused biological tissue irradiated by a laser beam. In particular, involving two local energy equations, one for the blood vessel and the other for the tissue, a non-Fourier-like heat equation is introduced and solved analytically using the Laplace transform method. The generalized hyperbolic model obtained is reduced to Pennes' heat transport equation in case the thermal delay time is zero and the solution obtained is in accordance with the numerical and experimental data existing in the literature. In addition, the achieved results also show that the effects of thermal relaxation and blood perfusion on temperature distribution are similar; indeed the highest temperature is expected when the delay time IR increases during tissue cooling. Finally, the consequences of the change in the values of the physical parameters characterizing the model are described and the effect of thermal relaxation on the temperature profile in the tissue during and after laser application is investigated

Differential consequences of balance laws in extended irreversible thermodynamics of rigid heat conductors

We consider a system of balance laws arising in extended irreversible thermodynamics of rigid heat conductors, together with its differential conse- quences, namely the higher-order system obtained by taking into account the time and space derivatives of the original system. We point out some mathematical properties of the differential consequences, with particular attention to the problem of the propagation of thermal perturbations with finite speed. We prove that, under an opportune choice of the initial conditions, a solution of the Cauchy problem for the system of differential consequences is also a solution of the Cauchy problem for the original system. We investigate the thermodynamic compatibility of the system at hand by applying a generalized Coleman–Noll procedure. On the example of a generalized Guyer–Krumhansl heat-transport model, we show that it is possible to get a hyperbolic system of evolution equations even when the state space is non-local