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Overview of the Entropy Production of Incompressible and Compressible fluid Dynamics View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PORTO Publications Open Repository TOrino Politecnico di Torino Porto Institutional Repository [Article] Overview of the entropy production of incompressible and compressible fluid dynamics Original Citation: Asinari, Pietro; Chiavazzo, Eliodoro (2015). Overview of the entropy production of incompressible and compressible fluid dynamics. In: MECCANICA. - ISSN 0025-6455 Availability: This version is available at : http://porto.polito.it/2624292/ since: November 2015 Publisher: Kluwer Academic Publishers Published version: DOI:10.1007/s11012-015-0284-z Terms of use: This article is made available under terms and conditions applicable to Open Access Policy Article ("Public - All rights reserved") , as described at http://porto.polito.it/terms_and_conditions. html Porto, the institutional repository of the Politecnico di Torino, is provided by the University Library and the IT-Services. The aim is to enable open access to all the world. Please share with us how this access benefits you. Your story matters. (Article begins on next page) Meccanica manuscript No. (will be inserted by the editor) Overview of the entropy production of incompressible and compressible fluid dynamics Pietro Asinari · Eliodoro Chiavazzo Received: date / Accepted: date Abstract In this paper, we present an overview of the Keywords Thermodynamics of Irreversible Processes entropy production in fluid dynamics in a systematic (TIP) · Compressible fluid dynamics · Incompress- way. First of all, we clarify a rigorous derivation of ible limit · Entropy production · Second law of the incompressible limit for the Navier-Stokes-Fourier thermodynamics system of equations based on the asymptotic analy- sis, which is a very well known mathematical technique used to derive macroscopic limits of kinetic equations 1 Introduction and motivation (Chapman-Enskog expansion and Hilbert expansion are popular methodologies). This allows to overcome the Entropy is a fundamental concept in both physics and theoretical limits of assuming that the material deriva- engineering. It plays a prominent role in thermodynam- tive of the density simply vanishes. Moreover, we show ics, statistical mechanics, continuum physics, informa- that the fundamental Gibbs relation in classical ther- tion theory and, more recently, also in biology, sociology modynamics can be applied to non-equilibrium flows and economics. More details about the broad spectrum for generalizing the entropy and for expressing the sec- of involved disciplines can be found in Ref. [1] (and ref- ond law of thermodynamics in case of both incompress- erences therein) and some historical notes in Ref. [2]. ible and compressible flows. This is consistent with the One of the reasons of the fascination that scientists have Thermodynamics of Irreversible Processes (TIP) and it always felt with regards to entropy is due to the fact is an essential condition for the design and optimiza- this concept is one of the few (maybe the only one) tion of fluid flow devices. Summarizing a theoretical dealing with irreversibility, the arrow of time and, ul- framework valid at different regimes (both incompress- timately, the evolution of life. As an example, to this ible and compressible) sheds light on entropy produc- respect we notice that recently the evolutionary strat- tion in fluid mechanics, with broad implications in ap- egy of cave spiders has been interpreted by means of plied mechanics. an entropic argument [3]. In fact, most of the quantum equations of motion are time-reversible (with few excep- P. Asinari tions, see [4]), as pointed out by the Loschmidt’s para- multi-Scale ModeLing Laboratory (SMaLL), dox which is still source of discussions nowadays (par- Energy Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy tially solved by the fluctuation-dissipation theorems, Tel.: +39-011-0904434 see next). The term entropy was introduced in 1865 Fax: +39-011-0904499 by the German physicist Rudolf Clausius. The idea E-mail: [email protected] was inspired by an earlier formulation by Sadi Carnot E. Chiavazzo of what is now known as the second law of thermo- multi-Scale ModeLing Laboratory (SMaLL), dynamics. The Austrian physicist Ludwig Boltzmann Energy Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy and the American scientist Willard Gibbs put entropy Tel.: +39-011-0904530 into the probabilistic framework of statistical mechan- Fax: +39-011-0904499 ics (around 1875). This idea was later developed by E-mail: [email protected] Max Planck. Entropy was extended to quantum me- 2 chanics in 1932 by John von Neumann. Later this led scopic interpretation of molecular results is far from to the invention of entropy as a term in probability the- obvious [9]. ory by Claude Shannon (1948), popularized in a joint Enhancing our understanding about entropy pro- book with Warren Weaver, that provided foundations duction may have an impact beyond molecular sim- for information theory. The concept of entropy in dy- ulations. An example is given by some trends in the namical systems was introduced by Andrei Kolmogorov mesoscopic numerical methods. In particular, the lat- and more deeply formulated by Yakov Sinai in what is tice Boltzmann method [12–16] is a powerful numerical now known as the Kolmogorov-Sinai entropy. The for- method applied much beyond rarefied flows, including mulation of Maxwell’s paradox (by James C. Maxwell, thermal radiation [17], thermal conduction [18], com- around 1871) started a search for the physical meaning bustion [19–22], porous media [23,24], multi-component of information, which resulted in the finding by Rolf flows [25,26] and turbulence [27], to mention a few. Per- Landauer (1961) of the heat equivalent for the erasure tinently to this paper, the entropic lattice Boltzmann of one bit of information, which brought the notions method was invented by Ilya Karlin and co-workers in of entropy in thermodynamics and information theory 1998, by applying the maximum entropy principle to together. References for the above historical overview lattice kinetic equations [28]. Hence, the concept of en- can be found in [1]. Remarkably, entropy is experienc- tropy production boosted a tremendous development ing a new revival in computational science, driven by of further refinements [29–31]. A more complete re- the recent developments in quantum computing (e.g. view about the original development of entropic lat- see [5]). tice Boltzmann method for hydrodynamic simulations can be found in Ref. [32]. Also in the case of numerical Many attempts have been made in order to ratio- schemes for fluid dynamics, the entropy can be used as nalize the concept of entropy and its application to the a design tool for generalizing the local equilibrium to second law of thermodynamics (e.g. see the efforts by include hydrodynamic moments beyond the conserved Jakob Yngvason [6]). Beyond the axiomatic approach, ones [33]. This allows one to design novel collisional the interpretation of entropy based on statistical me- kernels with improved stability [34]. Moreover, entropy chanics still remains the most popular one [7]. Here, in may give a systematic guideline to derive advanced lat- particular, we focus on some fundamental results, with tices by factorization symmetry [35], and it also reveals broad practical applications, obtained in the attempt a key notion in dissipative dynamical systems for dis- to rationalize the physical meaning of irreversibility and cerning between fast and slow processes [36–41]. entropy production. We present two examples: The fluc- In spite of the broad implications of entropy produc- tuation theorems and the mesoscopic numerical meth- tion (both in terms of fundamental theorems and prac- ods. Concerning the first implication (fluctuation the- tical numerical tools), its rigorous derivation in fluid orems) studies in this context have been carried out dynamics for generic hydrodynamic regimes deserves a over the past 15 years, and have led to fundamental more detailed analysis. The need to clarify such elemen- breakthroughs in our understanding of how irreversibil- tary issues in the fluid dynamics community should not ity emerges from reversible dynamics [8]. In 1993 Evans, be a surprise [9]. The mathematical theory of fluids is Cohen and Morriss [10] considered the fluctuations of in a very primitive state and the fluid dynamic equa- the entropy production rate in a shearing fluid, and pro- tions do not have a fundamental nature [42]. Based on posed the so called Fluctuation Relation. This pioneer- the former theoretical framework proposed by Lars On- ing work has experienced an extensive development by sager, the pioneering idea by Ilya Prigogine to extend different authors (see Ref. [11] and references therein). the fundamental Gibbs relation also to non-equilibrium The original result has been extended to many different states paved the way to the Thermodynamics of Irre- cases and it is now a whole new theoretical framework versible Processes (TIP) [43,44]. The previous straight- which encompasses the previous linear response theory forward extension is based on the assumption of local and goes beyond that, to include far from equilibrium equilibrium, i.e. all subparts of the system are close phenomena, such as turbulence and the dynamics of enough to equilibrium condition to be still described granular materials [11]. These results might have impor- by classical thermodynamic
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