Irreversible Thermodynamics and Modelling of Random Media

Abstract

Engineering and Nature very often are concerned with media that do contain some randomness. One particularly well known example is given by turbulent flows, where random fluctuations are the results of the growth of unstable motions from small perturbations in the initial or boundary conditions, when the velocity gradients or/and temperature gradients are too large in some place within the flow. These fluctuating motions result in effective additional friction, effective additional diffusion of species or heat, and additional energy dissipation (i.e. transfer from organized kinetic energy to turbulent kinetic energy, and finally to internal energy). Another example of random flows is seen in two-phase flows, either with particles dispersed into a liquid or gaseous continuous phase, or built with a bubbly liquid. Here, a first reason of randomness is given by the fact that the locations and velocities of the particles (or bubbles) at initial time and in entrance sections cannot be known or controlled, and the induced fluctuations are not rapidly damped with time. Generally, this randomness becomes rapidly uncorrelated with these initial conditions due to the complex interactions between the flow of the continuous phase and the moving inclusions, interactions that modify also the number and sizes of these inclusions. Consequently, the flow conditions in both phases are or become turbulent, with interconnected fluctuations of velocities. Again, the effective result of the fluctuations is additional friction, additional diffusion or dispersion, and additional energy dissipation. Two-phase or multiphase flows with a large amount of solid particles are used in the industrial devices called as “Fluidized Beds”, and are encountered naturally in “Granular Flows”, and in this case there are many lasting contacts between the particles. The so called “Granular Media”, even without actual flow but just with some deformations or slow motions, are also subjected to randomness, due to the preparation of the medium but also to the differences between the grains shapes, and even the global properties of these media are very difficult to predict. The properties of these kinds of media and flows have been studied since more than hundred years, and very useful prediction methods have been proposed so far, with methodologies becoming more and more clearly similar. Although all these media are showing clearly irreversible and dissipative processes, the typical reasoning of irreversible thermodynamics has never been used, at least explicitly (it might be present, but not consciously, in the brain of the authors). Our purpose here is to show that the Irreversible Thermodynamics and the “Second Principle” can justify many features of the models that