sign in sign up
<<
Home , 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 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 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 and theoretical limits of assuming that the material deriva- . It plays a prominent role in thermodynam- tive of the simply vanishes. Moreover, we show ics, , 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- Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy tially solved by the fluctuation- 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 . 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 [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 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 [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 variables and relationships. tant implications also from the computational point of On the other hand, the Extended Irreversible Thermo- view. Indeed, these theorems aim at quantifying micro- dynamics (EIT) [45,46] is an active field of research, scopic and understand how a system responds to where scientists make an effort to overcome the cur- external perturbations, using techniques such as molec- rent limitations of the local equilibrium assumption. ular dynamics simulations [8]. Molecular dynamics sim- The basic idea is that thermodynamic quantities (e.g. ulations are powerful tools, even though the macro- entropy) depend also on dissipative fluxes (e.g. fluid 3 velocity) which are regarded as independent variables. briefly summarized. In Section 3 the main results for The differential form of the generalized entropy leads the incompressible limit are reported, substantially sup- to the generalized Gibbs relation for non-equilibrium porting the basic assumption of the Thermodynamics states. Any non-equilibrium Gibbs relation should be of Irreversible Processes which assumes the Gibbs re- considered as an assumption which satisfies the follow- lation valid for non-equilibrium conditions as well. In ing requirements: i) it is Galilean invariant; 2) it re- Section 4 the main results for the compressible limit duces to the equilibrium Gibbs relation in the limit of are reported. In Section 5 some consequences are de- quasi-static processes; 3) the conclusions derived from it rived from the previous results. Finally, in Section 6, are not contradictory and agree with experiments [47]. the conclusions are reported. An example of the above procedure for studying non- equilibrium interfaces can be found in Refs. [48–50]. 2 Conservation equations of computational Compressible fluid dynamics has paid little atten- fluid dynamics tion to investigate entropy production (with very few exceptions, see Ref. [51]). Compressible fluid dynam- Let us consider the fundamental equations stipulating ics deals with flows having significant changes in fluid the balance of mass, and total energy of the density. Significant occurs if the Mach fluid [52,53], namely number, which is the ratio between the characteristic ∂ρ speed and the local speed, is greater than about + ∇ (ρu) = 0, (1) 0.2 [52,53]. The range of applications of compressible ∂t fluid dynamics includes aircrafts, spacecrafts (for re- ∂(ρu) + ∇ (ρ u ⊗ u)+ ∇p = ∇ Πν + ρ a, (2) entry), turbo-machinery and hypersonic plasmas. Com- ∂t pressible flows are usually characterized by shocks, i.e. ∂(ρet) + ∇ (ρet u + p u)= ∇ (−q + Πν u), (3) an abrupt discontinuity in the flow field. Entropy condi- ∂t tion (e.g. by Lax) is the typical admissibility condition where ρ is the fluid density, u is the fluid velocity, p used to discriminate the unique single-valued physical is the , Πν is the viscous part of the solution. Hence entropy is part of the state-of-the-art tensor, a is some spatially varying, external acceleration numerics for solving compressible flows [54]. However, (e.g. the gravity), et is the total fluid energy per unit in spite of this, little attention has been paid to rig- of mass and q is the thermal flux vector. Clearly the orous entropy production in compressible flows, which previous system of equations is not closed, and some could be helpful in guiding design and optimizations, phenomenological expressions are needed for Πν and as already happens in similar fields [55,56]. q. Some popular expressions for Newtonian fluids are This paper aims to present an overview of the en- given by tropy production in fluid dynamics in a more system- T 2 Πν = ρν ∇u + ∇u − (∇ u) I + ρζ (∇ u) I, (4) atic way. First of all, we present a rigorous derivation of  3  the incompressible limit for the Navier-Stokes-Fourier − ∇ system of equations based on the asymptotic analy- q = λ T, (5) sis, which is a very well known mathematical technique where ν is the kinematic (shear) of the fluid, used to derive macroscopic limits of kinetic equations ζ is the bulk viscosity of the fluid and λ is the ther- (Chapman-Enskog expansion and Hilbert expansion are mal conductivity. Eqs. (1, 2, 3), together with the phe- popular methodologies). This allows to overcome the nomenological correlations (4) for Πν and (5) for q theoretical limits of assuming that the material deriva- represent the so-called Navier-Stokes-Fourier system of tive of the density simply vanishes. Moreover, we show equations. The previous system of equations can be that the fundamental Gibbs relation in classical ther- equivalently written in terms of Lagrangian total deriva- modynamics can be applied to non-equilibrium flows tives, namely D/Dt = ∂/∂t + u ∇, as reported in A. for generalizing the entropy and for expressing the sec- The Lagrangian derivative is the fundamental opera- ond law of thermodynamics in case of both incompress- tor ensuring Galilean invariance for the fluid dynamics ible and compressible flows. This is consistent with the (see B for details). Equations (1, 2, 3) are written in Thermodynamics of Irreversible Processes (TIP) and it standard form (see A for details) and this allows one is an essential condition for the design and optimization to say that mass and total energy are conserved, while of fluid flow devices. momentum is conserved if and only if there is no ex- The paper is organized as follows. In Section 2, con- ternal field, i.e. a = 0. The physical meaning is that servation equations of computational fluid dynamics, a conserved quantity can change its amount inside a which represent the starting point of our analysis, are fixed closed only by either accumulation inside 4 the same volume and/or fluxes at the border. Accu- 3 Results for the incompressible limit mulation is described by the Eulerian time derivative (∂ϕ/∂t) and border effects by the divergence of fluxes The conservation equations of computational fluid dy- (∇ f(ϕ)), according to the Gauss theorem. namics can describe many different fluid flow regimes The total energy is the sum of the mechanical energy at the same time. On the other hand, different ap- em and the internal energy ei, namely et = em +ei. Let plications are interested usually in only few of them. us derive first the for the mechanical energy For example, fluid dynamics of ground vehicles can be 2 em, defined as the sum of the ek = u /2 studied by the incompressible limit of Navier-Stokes- and the potential energy ep such that a = −∇ep. Mul- Fourier equations, while supersonic vehicles should be tiplying the Lagrangian form of (2) (see A for details) addressed with the compressible inviscid limit. These by u and recalling that u ∇u u = u ∇ek, it follows asymptotic limits can be rigorously defined by means ∂e of characteristic quantities. There are different ways to ρ k + ρu ∇e − ρ u a = −(∇ Π) u, (6) ∂t k define characteristic quantities depending on what they where Π = p I−Πν is the total stress tensor. Assuming refer to, i.e. fields or operators. Typically the magnitude that potential energy does not depend on time yields of the velocity field is given by the characteristic quan- De tity uc, where uc = max(u) on the considered domain. ρ m = −(∇ Π) u, (7) Dt The same definition applies to the characteristic viscos- and consequently ity νc, namely νc = max(ν). Concerning operators, the definitions are slightly more difficult. For example, the Dem ρ = −∇ (Π u)+ Π : ∇u. (8) characteristic time t and the characteristic length l Dt c c Coming back to standard form (see A for details) yields are defined by

∂(ρem) O(ϕ) + ∇ (ρe u + Π u)= Π : ∇u, (9) tc = , (15) ∂t m O(∂ϕ/∂t) where it is shown that mechanical energy is not con- served with the source (sink) term being at the right- O(ϕ) O(ϕ) lc = = max , (16) hand side of (9). O(∇ϕ) i O(∂ϕ/∂xi) ¿From (9), it is straightforward to derive the equa- respectively. tion for internal energy e = e − e , by subtraction i t m It is possible to use the previous characteristic quan- with Eq. (3), namely tities to define new operators. For example, in case of ∂(ρei) + ∇ (ρei u + q)= −Π : ∇u, (10) the Eulerian time derivative, it is possible to define ∂t ∂/∂tˆ= tc ∂/∂t. From (15), it follows or equivalently, in Lagrangian form, ˆ Dei tc O(∂ϕ/∂t)= O(∂ϕ/∂t)= O(ϕ). (17) ρ = −p∇ u −∇ q + Πν : ∇u. (11) Dt The previous expression means that scaled operators We notice that internal energy is not conserved either, (with “hat” notation), obtained by multiplying the orig- and the corresponding source/sink is equal in modulus inal operator by the corresponding characteristic quan- (but with opposite sign) to the one for mechanical en- tity, preserve the order of magnitude of the function ergy. The latter observation ensures that total energy, they are applied to. Let us introduce the characteristic which is the sum of macroscopic (e ) and microscopic m quantities and the scaled operators in Eq. (1), namely (ei) mechanical , is conserved instead. Finally, as typically done in the engineering community when lc ∂ρ + ∇ˆ (ρuˆ) = 0. (18) dealing with open systems, we derive the expression for tc uc ∂tˆ h, defined as h = ei + pv, where v = 1/ρ. Introducing the St = lc/(tc cs) and Taking into account that the Ma = uc/cs, where cs is the sound Dh De Dp p Dρ De Dp ρ = ρ i + − = ρ i + +p∇u,(12) speed, yields Dt Dt Dt ρ Dt Dt Dt St ∂ρ the equation for enthalpy can be derived by (11), namely + ∇ˆ (ρuˆ) = 0. (19) Ma ∂tˆ Dh Dp ρ = −∇ q + + Π : ∇u, (13) Dt Dt ν Similarly, we can proceed for the momentum equation or equivalently, in standard form, (2), namely 2 ∂(ρh) Dp uc ∂(ρuˆ) u 1 uc νc uc + ∇ (ρh u + q)= + Πν : ∇u. (14) c ∇ˆ ⊗ ∇ˆ ∇ˆ Πˆ ∂t Dt + (ρ uˆ uˆ)+ p = 2 ν + ρ aˆ,(20) tc ∂tˆ lc lc lc tc 5 and consequently ¿From (27), it is possible to derive all other equa- tions in the incompressible limit. Introducing the pre- StMa ∂(ρuˆ) Ma2 ˆ 1 ˆ + ∇ (ρ uˆ ⊗ uˆ)+ 2 ∇p = vious expressions into Eq. (21) yields ∂tˆ cs 2 Ma ∂(ρuˆ) ˆ ˆ 1 ˆ Πˆ ∇ˆ Πˆ StMa aˆ + ∇ (ρ uˆ ⊗ uˆ)+ ∇pˆ = ∇ ν + ρ aˆ. (28) = Re ν + ρ , (21) ∂tˆ Re where Re = uc lc/νc is the . For the sake of simplicity, let us consider constant vis- We notice two issues in the latter expression: (a) cosity in the considered domain. In this case, the in- There are multiple dimensionless numbers, i.e. St, Ma compressible momentum equation becomes Re and defining the asymptotic regime; (b) One term ∂uˆ 1 1 2 2 ∇ˆ 2 + uˆ ∇ˆ uˆ + ∇ˆ pˆ = ∇ˆ uˆ + aˆ + O(Ma ). (29) is not properly scaled by them, i.e. p/cs. The first ∂tˆ ρ0 Re problem can be solved by selecting the Mach number Ma Similarly, we can proceed with the enthalpy h = h(ρ, T ). as the asymptotic parameter, i.e. ≪ 1, and by spec- 2 ifying how all other dimensionless numbers scale with Expanding the enthalpy around γh cs, where γh is an- other dimensionless constant depending on the initial it. In particular, we set tc = lc/uc and hence St = Ma. Moreover, we assume that the Reynolds number is in- conditions, yields dependent on Mach, i.e. Re = O(1). These choices allow 2 ∂h ∂h h − γ c = (ρ − ρ0)+ (T − T0). (30) one to solve the second problem too: All terms in Eq. h s ∂ρ ∂T 2 0 0 (21) become O(Ma ) and hence also the remaining term 2 2 Taking into account Eqs. (25, 26), the scaling for en- must have the same magnitude, i.e. ∇ˆ p/c = O(Ma ). s thalpy follows, namely Consequently the pressure field can be split into two parts h 2 = γ + Ma h.ˆ (31) c2 h p Ma2 s 2 = γp + p,ˆ (22) ρ0 cs Introducing the previous characteristic quantities and where ρ0 is a proper constant depending on the initial scaled operators in Eq. (13) yields conditions, γp is a dimensionless constant (which is not 2 2 1 Dhˆ ρ0 γ0 Ma 1 Dpˆ Ma the heat capacity ratio) andp ˆ is the part dictating the ρ = − ∇ˆ qˆ + + Πˆ : ∇ˆ uˆ,(32) c2 ˆ Pr Re α c2 ˆ Re ν fluid flow. The previous relation can be generalized to s Dt s Dt all thermodynamic quantities. Let us assume density ρ where Pr = νc/αc is the , αc = λc/(ρ0 c0) and T as the two thermodynamic coordi- is the thermal diffusivity, c0 is the specific heat capacity 2 nates used to identify the thermodynamic state, namely and γ0 = c0 T0/cs is a constant. Introducing the scal- p = p(ρ, T ). The previous equation highlights that the ing relations given by Eqs. (31, 22) into the previous pressure field is characterized by small deviations from equation yields 2 the constant γp ρ0 cs, which can be expressed by the Dˆhˆ ρ0 γ0 Dˆpˆ 1 thermodynamic coordinates, namely ρ = − ∇ˆ qˆα + + Πˆ ν : ∇ˆ uˆ, (33) Dtˆ Pr Re Dtˆ Re 2 ∂p ∂p p = γ ρ0 c + (ρ − ρ0)+ (T − T0), (23) and consequently, taking into account all previous asymp- p s ∂ρ ∂T 0 0 totic results (in particular Eq. (27)), the enthalpy equa- or equivalently tion for incompressible flows reads

∂p ∂p 2 Dˆhˆ γ0 1 Dˆpˆ (ρ − ρ0)+ (T − T0)= O(Ma ). (24) − ∇ˆ q ∂ρ ∂T = Pr Re ˆα + + ... 0 0 Dtˆ ρ0 Dtˆ 1 Hence the largest possible deviations are given by ∇ˆ u ∇ˆ uT ∇ˆ u O Ma2 ... + Re( ˆ + ˆ ): ˆ + ( ). (34) ρ 2 =1+ Ma ρ,ˆ (25) ρ0 It is possible to simplify the notations in (34) by focus- T 2 ing on the notion of strain rate, as shown below. Let us =1+ Ma T.ˆ (26) S T T0 introduce the strain rate tensor ∇ u =(∇u + ∇u )/2 and the tensor ∇W u = (∇u − ∇uT )/2. It is Substituting these new scalings into Eq. (19) yields easy to prove that ∇u = ∇Su+∇W u and, more impor- 2 ∇ˆ uˆ = O(Ma ), (27) tantly, ∇Su : ∇W u = 0. Making use of these definitions which is the fundamental relation for defining the in- in the previous expression yields ˆ ˆ ˆ compressible limit of Navier-Stokes-Fourier system of Dh γ0 1 Dpˆ 2 S 2 2 = − ∇ˆ qˆα+ + (∇ˆ uˆ) +O(Ma ),(35) equations. Dtˆ Pr Re ρ0 Dtˆ Re 6 where (∇ˆ Suˆ)2 = ∇ˆ Suˆ : ∇ˆ Suˆ. For the sake of simplic- where µ = ρν is the dynamic viscosity. Elaborating ity, let us consider constant in the further on the first term at the right hand side of the considered domain, namely previous expression yields ˆ ˆ ˆ Dh γ0 ˆ 2 ˆ 1 Dpˆ 2 ˆ S 2 2 Ds 1 λ 2 2 µ S 2 = ∇ T + + (∇ uˆ) + O(Ma ).(36) ρ + ∇ q = 2 (∇T ) + (∇ u) ≥ 0.(44) Dtˆ Pr Re ρ0 Dtˆ Re Dt T  T T In the asymptotic limit Ma → 0, coming back to physi- Taking into account the general property given by Eq. cal units assuming constant thermo-physical properties (59), the previous equation can be rewritten in standard for the sake of simplicity, the incompressible limit of form, namely the Navier-Stokes-Fourier system of equations can be ∂(ρs) written as + ∇ f(ρs)= σ + σ ≥ 0, (45) ∂t α ν ∇ u = 0, (37) where ∂u 1 2 1 + u ∇u = − ∇p + ν∇ u + a, (38) f(ρs)= ρs u + q = ρs u + k β q, (46) ∂t ρ0 T B ∂h λ 2 2 1 Dp + u ∇h = ∇ T + 2ν(∇Su) + . (39) kB is the Boltzmann constant, β = 1/(kB T ) (as com- ∂t ρ0 ρ0 Dt mon in statistical mechanics) and the entropy produc- Towards an effort of exploring the (slightly) non- tion terms are defined by equilibrium thermodynamics of the incompressible limit, c0 ∇ 2 ≥ it is essential to derive an equation for entropy and, in σα = ρα 2 ( T ) 0, (47) T  particular, for entropy production. Let us follow the 2 S 2 guidelines of the Thermodynamics of Irreversible Pro- σν = ρν (∇ u) ≥ 0. (48) cesses (TIP) [43,44]. The key idea is to assume that T  local equilibrium holds, i.e. all subparts of the system Each entropy production term describes the entropy are close enough to equilibrium condition to be still produced by a specific transport phenomenon: σα takes described by classical thermodynamic variables and re- into account the entropy produced by , lationships. In a state of thermodynamic equilibrium, which is ruled by temperature , and σν the en- there is a fundamental relation (Gibbs relation) which tropy produced by fluid flow, which is ruled by strain provides the connection between entropy and other ther- rate. Clearly each entropy production mechanism satis- modynamic potentials, namely Tds = dei + pdv. The fies the second law of thermodynamics, namely σα ≥ 0 first simple (but powerful) idea is to assume that the and σν ≥ 0, and so it does the global entropy produc- same relation holds in non-equilibrium conditions as tion, namely σα + σν ≥ 0. This confirms the validity of well. In non-equilibrium states, the previous expression the assumption given by Eq. (40) for non-equilibrium becomes the definition of entropy. Of course, this defini- states in the incompressible limit. tion must be Galilean invariant and hence Lagrangian time derivatives must be considered for this goal (see B for details), namely 4 Results for the compressible limit

Ds Dei Dv Dh Dp T = + p = − v , (40) In the generic compressible case, the situation is slightly Dt Dt Dt Dt Dt more complex. Let us rewrite Eq. (42) as follows or equivalently Ds Dh Dp Ds S 2 1 2 ρT = ρ − . (41) ρT = −∇ q + 2 ρν (∇ u) − (∇ u) + ... Dt Dt Dt Dt  3  2 Substituting Eq. (13) into Eq. (41) yields ... + ρζ (∇ u) . (49)

Ds S 2 2 2 Following Ref. [44], let us consider the following equiv- ρT = −∇q+2 ρν (∇ u) +ρ ζ − ν (∇u) .(42) Dt  3  alence In the incompressible limit, namely at very low Mach 2 1 2 2 1 2 (∇Su) − (∇ u) =(∇Su) + (∇ u) + ... number flows, according to Eq. (37), the last term of the 3 3 2 1 previous expression can be neglected. Taking into ac- ... − (∇ u)2 = ∇Su : ∇Su + (∇ u)2 I : I + ... count the definition of thermal flux according to Fourier’s 3 9 1 1 law given by Eq. (5) and assuming constant thermo- ... − (∇ u) I : ∇Su − (∇ u) ∇Su : I = physical properties for the sake of simplicity yield 3 3 2 Ds λ 2 µ S 1 ρ = ∇2T + (∇Su)2, (43) = ∇ u − (∇ u) I , (50) Dt T T  3  7 where the property I : I = 3 was used. Using the equiv- been selected equal to the dimensionality of the cor- alence given by Eq. (50) in Eq. (49) and proceeding responding thermodynamic (scalar k = 0, vec- 2 as before, the final entropy balance equation, which is tor k = 1, tensor k = 2). Consequently Xk = valid for any fluid dynamic regimes, becomes Xk ∗Xk where the generalized product ∗ means sim- ple product for k = 0, scalar product for k = 1 and ∂(ρs) ′ + ∇ f(ρs)= σα + σν + σζ = σ ≥ 0, (51) saturation product : for k = 2. Equation (55) can ∂t be further rewritten as where c0 σ = Jk ∗ Xk, (56) σ = ρα (∇T )2 ≥ 0, (52) α 2 Xk T  2 where Jk = ηk Xk is the generalized thermodynamic ′ 2 1 σ = ρν ∇Su − (∇ u) I ≥ 0, (53) flux. For the sake of simplicity, in this work, we con- ν T   3  sidered Jk = ηk Xk, which only takes into account 1 2 the diagonal terms of the more general expression σζ = ρζ (∇ u) ≥ 0. (54) T  Jk = l LklXl with Lkl being the Onsager’s matrix of phenomenologicalP coefficients. Hence, our deriva- The previous derivation allows one to identify the gen- tion is also consistent with the general formalism of eralized thermodynamic forces, as discussed in the fol- linear irreversible processes in non-equilibrium ther- lowing section. modynamics. However, it is worth the effort to point out that, in the case of compressible flows, the same intensive quantity u is generating two generalized 5 Discussion S forces, namely X0 = ∇u and X2 = ∇ u−(∇ u/3) I. Hence the corresponding entropy production must Equation (45), as a consequence of the assumption (40), be described by two phenomenological coefficients, confirms the applicability of the Thermodynamics of Ir- namely η0 = ρζ/T and η2 = ρν 2/T . This is defini- reversible Processes to incompressible flows. Moreover, tively not surprising, taking into account the dimen- assumption (40) is also valid for compressible flows, as ′ sionality of the intensive quantity u in comparison far as the entropy production σ given by Eq. (53) is ν with other scalar quantity 1/T . used instead of σν given by Eq. (48).

1. Equation (45) extends the original expressions for entropy balance to fluid flow (see Refs. [43,44] for 6 Conclusions mass diffusion) by the assumption (40), which is the simplest possible extension of the fundamental In the present paper, we aim to present an overview of the entropy production in fluid dynamics in a more sys- thermodynamics relation Tds = dei + pdv to non- equilibrium fluid flows. tematic way. First of all, we clarify the rigorous deriva- 2. In case of highly compressible flows, there are three tion of the incompressible limit of Navier-Stokes-Fourier terms which are responsible of the entropy produc- system of equations by means of the asymptotic analy- sis, which is a very well known mathematical technique tion, as reported in Eq. (51): in addition to σα in ′ used to derive macroscopic limits of kinetic equations. (47) and σν in (53), which is the generalization of This allows to overcome the theoretical limits of assum- the former σν , a novel term σζ is derived in (54), which is proportional to the bulk viscosity of the ing that the of the density simply fluid. All these terms (and consequently their sum) vanishes. are consistent with the second law of thermodynam- Secondly, we show how the fundamental Gibbs re- ics (i.e. positively defined). lation in classical thermodynamics can be applied to 3. The entropy production in Eq. (51) can be rewritten non-equilibrium flows for generalizing the concept of as entropy and hence the second law of thermodynamics for both incompressible and compressible flows. This 2 σ = ηk Xk , (55) is consistent with the Thermodynamics of Irreversible Xk Processes (TIP). The second law of thermodynamics where ηk are the phenomenological coefficients of given by Eq. (51) is an essential condition for the de- the irreversible phenomena (in particular, η0 = ρζ/T , sign and optimization of fluid flow devices (e.g. by the 2 η1 = ρα c0/T and η2 = ρν 2/T ) and Xk the gener- entropy generation minimization approach [55,56]). alized thermodynamic forces (X0 = ∇u, X1 = ∇T Providing an overview of the theoretical framework S and X2 = ∇ u−(∇ u/3) I). The subscript k has for irreversibilities, consistently valid at different regimes 8

(both incompressible and compressible), sheds light on where t, x, u are time, position vector and velocity vector ∗ ∗ ∗ entropy production in fluid mechanics, with broad im- in the reference frame assumed at rest, t , x , u are time, plications in applied mechanics. position vector and velocity vector in the moving frame and c is the velocity of the moving frame (with respect to the rest one). Once the quantities in the frame at rest are known, the previous transformation allows one to compute the cor- A Lagrangian formulation responding quantities in the moving frame. An operator is said to be Galilean invariant if and only if it is invariant with Conservation equations in computational fluid dynamics can regards to the application of the previous transformation. be conveniently written in standard form, namely Let us consider the Lagrangian time derivative (see Eq. (58) in A). The generic quantity ϕ is measured first in the ∂ϕ + ∇ f(ϕ)= s(ϕ), (57) frame at rest and hence it can be considered a function of ∂t t and x, namely ϕ = ϕ(t, x). However, by means of the where ϕ is a generic quantity (density, momentum, total en- transformation given by Eqs. (61), t and x themselves can ergy, mechanical energy, kinetic energy, internal energy, en- be considered as functions of the quantities in the moving ∗ ∗ ∗ tropy, etc.), f(ϕ) is the corresponding flux and s(ϕ) the cor- frame by parameter c, namely t = t(t , x , u ; c) and x = ∗ ∗ ∗ ∗ ∗ responding source/sink. x(t , x , u ; c). Consequently also ϕ = ϕ(t , x ; c) holds. ∗ ∗ If a quantity has a balance equation written in standard When computing the derivative of ϕ = ϕ(t , x ; c) with re- form such that s(ϕ) = 0, then this quantity is conserved gards to D/Dt expressed in the frame at rest, some mixed and the corresponding equation is a conservation equation. derivatives appear which can be solved by the chain rule, The physical meaning is that a conserved quantity changes namely inside a fixed closed volume only by accumulation inside the Dϕ ∂ϕ Dϕ(t∗, x∗) same volume and/or fluxes at the border. Accumulation is = + u ∇ϕ = = described by the Eulerian time derivative (∂ϕ/∂t) and border Dt ∂t Dt ∗ x∗ effects by the divergence of fluxes (∇ f(ϕ)), according to the ∂ϕ ∂t ∇∗ ∂ ∗ ∂ϕ ∇ ∗ ∇∗ ∇ ∗ = ∗ + ϕ +(u + c) ∗ t + ϕ x = Gauss theorem. ∂t ∂t ∂t ∂t ∗ There is also another useful form for conservation equa- ∂ϕ ∗ ∗ ∗ D ϕ = + ∇ ϕ (−c)+(u + c) ∇ ϕ = . (62) tions, which involves the Lagrangian derivative Dϕ/Dt, namely ∂t∗ Dt∗ Dϕ ∂ϕ The latter expression proves that the value of the Lagrangian ˙= + u ∇ϕ. (58) Dt ∂t time derivative is the same if computed in any inertial frame and hence it is Galilean invariant. The Lagrangian time derivative is Galilean invariant (see B for details) and this ensures that it has the same values in all inertial frames. Taking into account the mass conservation equation (1), the following property holds Acknowledgments ∂(ρϕ) ∂ϕ + ∇ (ρϕu) = ρ + ρu ∇ϕ... Authors would like to acknowledge the THERMALSKIN project: ∂t ∂t ∂ρ Dϕ Revolutionary coatings by carbon nanotubes for high ... + ϕ + ϕ∇ (ρu)= ρ . (59) heat transfer efficiency (FIRB 2010 - “Futuro in Ricerca”, ∂t Dt Ministero dell’Istruzione, dell’Universit`ae della Ricerca). Consequently the Lagrangian formulation of the Navier-Stokes- Fourier system of equations is given by Dρ References = −ρ∇ u, (60a) Dt Du 1. T. Downarowicz, Entropy, Scholarpedia 2:11, 3901 ρ = −∇ Π + ρ a, (60b) Dt (2007). Det 2. I. M¨uller, A History of Thermodynamics: The Doctrine ρ = −∇ (q + Π u). (60c) of Energy and Entropy, Springer-Verlag, 2007. Dt 3. E. Chiavazzo, M. Isaia, S. Mammola, E. Lepore, L. Ven- From the previous expressions, it is easy to prove that the tola, P. Asinari, N.M. Pugno, Cave spiders choose opti- Navier-Stokes-Fourier system of equations is Galilean invari- mal environmental factors with respect to the generated ant. entropy when laying their cocoon, Scientific Reports 5, 7611 (2015). 4. G.P. Beretta, E.P. Gyftopoulos, J.L. Park, G.N. Hat- B Galilean invariance sopoulos, Quantum thermodynamics. A new equation of motion for a single constituent of matter, Il Nuovo Ci- mento B Series 11, 82:2, 169-191 (1984). Galilean invariance states that the laws of motion are the 5. L. Rio, J. Aberg,˚ R. Renner, O. Dahlsten, V. Vedral, same in all inertial frames. Galileo Galilei first described this The thermodynamic meaning of negative entropy, Nature principle in 1632. In non-, it is possible 474, 61-63 (2011). to introduce a Galilean transformation in order to verify if an 6. E.H. Lieb, J. Yngvason, The physics and of operator is Galilean invariant, namely the second law of thermodynamics, Physics Report 310:1, t∗ = t, 1-96 (1999). ∗  x = x − c t, (61) 7. G. Gallavotti, Statistical Mechanics: A Short Treatise,  u∗ = u − c, Springer-Verlag, 1999.  9

8. E.M. Sevick, R. Prabhakar, S.R. Williams, D.J. Sear- 27. P. Asinari, L.-S. Luo, A Consistent Lattice Boltzmann les, Fluctuation Theorems, Annual Review of Physical Equation with Baroclinic Coupling for Mixtures, Journal Chemistry 59, 603-633 (2008). of 227, 3878-3895 (2008). 9. P. Asinari, E. Chiavazzo, The notion of energy through 28. I.V. Karlin, A.N. Gorban, S. Succi, V. Boffi, Maximum multiple scales: From a molecular level to fluid flows and entropy principle for lattice kinetic equations, Phys. Rev. beyond, Energy 68, 870-876 (2014). Lett. 81, 6 (1998). 10. D.J. Evans, E.G.D. Cohen, G.P. Morriss, Probability of 29. I.V. Karlin, A. Ferrante, H.C. Ottinger, Perfect entropy second law violations in shearing steady states, Physical functions of the Lattice Boltzmann method, Europhys. Review Letters 71:15, 2401404 (1993). Lett. 47, 18288 (1999). 11. U. Marini Bettolo Marconi, A. Puglisi, L. Rondoni, A. 30. S. Ansumali, I.V. Karlin, H.C. Ottinger, Minimal en- Vulpiani, Fluctuation-Dissipation: Response Theory in tropic kinetic models for hydrodynamics, Europhys. Lett. , Physics Reports 461, 111-195 (2008). 63, 79804 (2003). 12. G.R. McNamara, G. Zanetti, Use of the Boltzmann equa- 31. S.S. Chikatamarla, S. Ansumali and I.V. Karlin, Entropic tion to simulate lattice- automata, Phys. Rev. Lett. lattice Boltzmann models for hydrodynamics in three di- 61, 2332 (1988). mensions, Phys. Rev. Lett. 97, 010201 (2006). 13. Y. Qian, D’d’Humi`eres, P. Lallemand, Lattice BGK mod- 32. S. Succi, I.V. Karlin, H.D. Chen, Colloquium: Role of els for Navier-Stokes equation, Europhys. Lett. 17, 479 the H theorem in lattice Boltzmann hydrodynamic simu- (1992). lations, Reviews of 74, 1203-1220 (2002). 14. X. He, L.-S. Luo, A priori derivation of the lattice Boltz- 33. P. Asinari, I.V. Karlin, Generalized Maxwell state and H mann equation, Phys. Rev. E 55, R6333 (1997). theorem for computing fluid flows using the lattice Boltz- 15. S. Chen, G.D. Doolen, Lattice Boltzmann method for mann method, E, Statistical, Nonlinear, fluid flows, Annu. Rev. Mech. 30, 329 (1998). and Soft Matter Physics 73, 036703 (2009). 16. S. Succi, The Lattice Boltzmann Equation for Fluid Dy- 34. P. Asinari, I.V. Karlin, Quasiequilibrium lattice Boltz- namics and Beyond, Oxford University Press, 2001. mann models with tunable bulk viscosity for enhancing 17. P. Asinari, S.C. Mishra, R. Borchiellini, A lattice Boltz- stability, , Statistical, Nonlinear, and mann formulation to the analysis of radiative heat trans- Soft Matter Physics 81, 01670 (2010). fer problems in a participating medium, Numerical Heat 35. I.V. Karlin, P. Asinari, Factorization symmetry in the lat- Transfer Part B-Fundamentals 57, 1-21 (2010). tice Boltzmann method, Physica A: Statistical Mechanics 18. E. Chiavazzo E., P. Asinari P., Reconstruction and mod- and its Applications 389, 389-1548 (2010). eling of 3D percolation networks of carbon fillers in a 36. A.N. Gorban, I.V. Karlin, Invariant manifolds for phys- matrix, International Journal of Thermal Sci- ical and , Springer-Verlag, 2005. 37. E. Chiavazzo, I.V. Karlin, A.N. Gorban, The role of ther- ences 49, 2272-2281 (2010). 19. A.F. Di Rienzo, P. Asinari, E. Chiavazzo, N.I. modynamics in model reduction when using invariant Prasianakis, J. Mantzaras, Lattice Boltzmann model for grids, Communications in Computational Physics 8, 701- reactive flow simulations, Europhysics Letters 98, 34001 734 (2010). 38. E. Chiavazzo, I.V. Karlin, C.E. Frouzakis, K. Boulouchos, (2012). Method of invariant grid for model reduction of hydrogen 20. E. Chiavazzo, I.V. Karlin, A. N. Gorban, K. Boulou- , Proceedings of the Combustion Institute 32, chos, Combustion via lattice Boltzmann and 519-526 (2009). reduced chemical kinetics, JSTAT P06013, (2009). 39. E. Chiavazzo, I.V. Karlin, Adaptive simplification of com- 21. E. Chiavazzo, I.V. Karlin, A. N. Gorban, K. Boulou- plex multiscale systems, Physical Review E 83, 036706 chos, Coupling of the model reduction technique with the (2011). lattice Boltzmann method for combustion simulations, 40. E. Chiavazzo, Approximation of slow and fast dynamics Combust. Flame 157, 1833-1849 (2010). in multiscale dynamical systems by the linearized Relax- 22. E. Chiavazzo, I.V. Karlin, A. N. Gorban, K. Boulouchos, ation Redistribution Method, Journal of Computational Efficient simulations of detailed combustion fields via the Physics 231, 1751-1765 (2012). lattice Boltzmann method, International Journal of Nu- 41. E. Chiavazzo, M. Fasano, P. Asinari, Inference of analyti- merical Methods for Heat and Fluid Flow 21, 494-517 cal thermodynamic models for biological networks, Phys- (2011). ica A: Statistical Mechanics and its Applications 392, 23. P. Asinari, M.R. von Spakovsky, M. Cal´ıQuaglia, V. Ka- 1122-1132 (2013). sula, Direct numerical calculation of the kinematic tortu- 42. G. Gallavotti, Foundations of , Springer- osity of reactive mixture flow in the anode layer of Verlag, 2002. oxide fuel cells by the lattice Boltzmann method, Journal 43. I. Prigogine, Introduction to Thermodynamics of Irre- of Power Sources 170:2, 359-375 (2007). versible Processes, Interscience, 1961. 24. U.R. Salomov, E. Chiavazzo, P. Asinari, Pore-scale mod- 44. S.R. de Groot, P. Mazur, Non-Equilibrium Thermody- eling of fluid flow through gas diffusion and catalyst layers namics, North-Holland, 1962. for high temperature proton exchange membrane (HT- 45. D. Jou, L. Restuccia, Mesoscopic transport equations and PEM) fuel cells, Computers & Mathematics with Appli- contemporary thermodynamics: An introduction, Con- cations 67, 393-411 (2014). temporary Physics 52:5, 465-474 (2011). 25. P. Asinari, Lattice Boltzmann scheme for mixture mod- 46. D. Jou, J. Casas-V´azquez, G. Lebon, Extended Irre- eling: analysis of the continuum diffusion regimes recov- versible Thermodynamics, Springer Verlag, 2012. ering Maxwell-Stefan model and incompressible Navier- 47. K. Glavatskiy, Multicomponent interfacial transport as Stokes equations, Physical Review E, Statistical, Nonlin- described by the square gradient model: evaporation and ear, and Soft Matter Physics 80, 056701 (2009). condensation, PhD Dissertation, Trondheim, 2009. 26. P. Asinari, Multiple-relaxation-time lattice Boltzmann 48. D. Bedeaux, E. Johannessen, A. Røsjorde, The nonequi- scheme for homogeneous mixture flows with external librium van der Waals square gradient model. (I). The force, Physical Review E, Statistical, Nonlinear, and Soft model and its numerical solution, Physica A: Statistical Matter Physics 77, 056706 (2008). Mechanics and its Applications 330:3-4, 329-353 (2003). 10

49. E. Johannessen, D. Bedeaux, The nonequilibrium van der Waals square gradient model. (II). Local equilibrium of the Gibbs surface, Physica A: Statistical Mechanics and its Applications 330:3-4, 354-372 (2003). 50. E. Johannessen, D. Bedeaux, The nonequilibrium van der Waals square gradient model. (III). Heat and mass trans- fer coefficients, Physica A: Statistical Mechanics and its Applications 336:3-4, 252-270 (2004). 51. A.V. Kolesnichenko, On the thermodynamic derivation of differential equations for turbulent flow transfer in a com- pressible heat-conducting fluid, Research 44:4, 334-347 (2010). 52. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000. 53. H.W. Liepmann, A. Roshko, Elements of Gasdynamics, John Wiley & Sons Inc, 2002. 54. E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009. 55. A. Bejan, Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-size Systems and Finite-time Processes, CRC Press, 1995 56. A. Bejan, M. J. Moran, Thermal Design and Optimiza- tion, John Wiley & Sons Inc, 1996.