Non-equilibrium thermodynamics in multiphase flows
The book is devoted to the description and to the study of non-equilibrium thermodynamics problems occurring in multiphase flows. It is divided into 11 chapters completed with 7 appendices. Each chapter ends with some problems whose solutions are gathered in the last appendix. The first chapter studies the notion of local equilibrium. The author starts with the description of the fluctuations of a thermodynamic quantity around its constant equilibrium value for a system of \(N\) particles at equilibrium and then for an isolated system macroscopically at equilibrium. He then computes the entropy \(S_{tot}\) of the system. In few examples he computes the entropy changes \(\Delta S_{tot}\). Chapter 2 presents the notion of microscopic reversibility. The author introduces the notions of probability distribution and of joint probability and their properties. Then he presents the notion of microscopic reversibility for an \(N\)-body system. Onsager's reciprocity relations are discussed and applied in order to derive a fluctuation-dissipation theorem through the use of Fourier's transform. Chapter 3 starts with the description of Brownian motions in the case of suspended particles, considered as random walks. Langevin's equation is established first in the 1D case and then in a generalized situation. Chapter 4 studies the Fokker-Planck equation. The author starts with a short description of the Markov processes from which he derives the Fokker-Planck equation. The general solution of this equation is given and the chapter ends with some considerations on Wiener processes. Chapter 6 presents some tools of the stochastic differential calculus. As mentioned in the introduction of this chapter, the author here intends to explain this topic ``in a way that is readily understandable also to a non-mathematician''. He presents Ito's and Stratinovich's stochastic calculus applied to the context of equilibrium and fluctuations. Considering a Brownian particle, he describes some notions of stochastic mechanics. In Chapter 6, the author briefly reviews some notions of path integrals. He considers free Brownian motions and Brownian motions in a field of force. He introduces the notion of minimal path and the chapter ends with the application of the path integral approach in three special cases. Chapter 7 moves to the derivation of the balance equations for a multicomponent system: conservation of mass, of chemical species, of momentum, of energy and of angular momentum. In Chapter 8, the author presents the constitutive relations for pure fluids, multicomponent mixtures, metals and non-isotropic media. He here uses the Onsager reciprocity relations. The chapter ends with the description of the fluctuations of the thermodynamic fluxes. Chapter 9 presents the equations of motion for multiphase flows. The author starts with the description of the equilibrium conditions for such materials. In order to illustrate its computations, the author plots phase diagrams considering different couples of variables. He then introduces the notion of diffuse interfaces assuming that the molar density of the material is not constant and introducing series expansion of the density. He writes the equations of motion first in the case of a dissipation-free fluid, then in a more general case. He concludes the chapter studying a multicomponent system. In Chapter 10, the author considers two phase materials for which one phase is randomly dispersed in the other, leading to different types of composite materials. He first studies the heat conductivity of composite materials and he computes the effective Fourier law. Then the effective viscosity of suspensions and the permeability of porous media are computed. The case of a suspension of neutrally buoyant Brownian particles is considered and the author here studies the diffusion in colloidal suspensions. The chapter ends with the dynamic definitions of transport coefficients, using Fourier transforms. Chapter 11 follows the study of multiphase materials, the author here introducing the multiple scale tools. The volume averaging method is presented in order to derive the effective equations which describe the flow through a porous medium. Using Onsager's reciprocity relations, the author derives Brinkman's equation. He then shows how the Fokker-Planck equation, written in the phase space, reduces to Smoluchowsky's equation. He illustrates his computations for different materials. The chapter ends with a short description of the homogenization techniques applied to transport in homogeneous random velocity fields or to dispersion in nonhomogeneous random fields. The first six appendices gather some further tools which are useful for the present study, while the last one presents the solutions of the problems which end each chapter. The book presents an up-to-date and complete description of the multiphase flows from a thermodynamic point of view.
scientific article; zbMATH DE number 6107092