Weakly nonlocal fluid mechanics - the Schrodinger equation

A weakly nonlocal extension of ideal fluid dynamics is derived from the Second Law of thermodynamics. It is proved that in the reversible limit the additional pressure term can be derived from a potential. The requirement of the additivity of the specific entropy function determines the quantum potential uniquely. The relation to other known derivations of Schr\"odinger equation (stochastic, Fisher information, exact uncertainty) is clarified.Comment: major extension and revisio

Thermodynamics of non-local materials: extra fluxes and internal powers

The most usual formulation of the Laws of Thermodynamics turns out to be suitable for local or simple materials, while for non-local systems there are two different ways: either modify this usual formulation by introducing suitable extra fluxes or express the Laws of Thermodynamics in terms of internal powers directly, as we propose in this paper. The first choice is subject to the criticism that the vector fluxes must be introduced a posteriori in order to obtain the compatibility with the Laws of Thermodynamics. On the contrary, the formulation in terms of internal powers is more general, because it is a priori defined on the basis of the constitutive equations. Besides it allows to highlight, without ambiguity, the contribution of the internal powers in the variation of the thermodynamic potentials. Finally, in this paper, we consider some examples of non-local materials and derive the proper expressions of their internal powers from the power balance laws.Comment: 16 pages, in press on Continuum Mechanics and Thermodynamic

Computational analysis of heat rectification in composition-graded systems: From macro-to-nanoscale

The heat rectification coefficient of a composition-graded system of the type AxB1−x, with Aand B being theoretical materials,and composition x changing along the length of the system,is considered.By starting from a mathematical model for the thermal conductivity of the material λ in terms of tem- perature T and composition x, the influence of compositionspatial distribution, heat flux, length of the system,and minimum of λ(T, x) on the rectification coefficient is explored.In some circumstances, a reversal in the direction of the rectification is observed for increasing heat flux

Phase-field evolution in Cahn–Hilliard–Korteweg fluids

In this paper, the diffusion of different phases in a third-grade Korteweg fluid is modeled by introducing a phase-field as a new independent thermodynamic variable. The constitutive equations are supposed to depend on the mass density and its spatial derivatives up to the second order, as well as on specific internal energy, barycentric velocity and phase-field, together with their first-order spatial derivatives. The compatibility of the model with the second law of thermodynamics is exploited by applying a generalized Liu procedure. For isothermal and isochoric phases, a general evolution equation for the phase-field, which generalizes the classical Cahn–Hilliard equation, is derived. Specific entropy and free energy are proved to depend on the basic unknown fields as well as on their gradients. A general constitutive equation for the Cauchy stress, which encompasses the classical one postulated by Korteweg in 1901, is obtained