A variational formulation for uid dynamics with irreversible processes

07/11/2017
Publication GSI2017
OAI : oai:www.see.asso.fr:17410:22346
contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit
 

Résumé

In this paper, we present a variational formulation for heat conducting viscous uids, which extends the Hamilton principle of continuum mechanics to include irreversible processes. This formulation follows from the general variational description of nonequilibrium thermodynamics introduced in [3, 4] for discrete and continuum systems. It relies on the concept of thermodynamic displacement. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of the entropy production associated to the irreversible processes involved.

A variational formulation for  uid dynamics with irreversible processes

Collection

application/pdf A variational formulation for uid dynamics with irreversible processes François Gay-Balmaz, Hiroaki Yoshimura
Détails de l'article
contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit

In this paper, we present a variational formulation for heat conducting viscous uids, which extends the Hamilton principle of continuum mechanics to include irreversible processes. This formulation follows from the general variational description of nonequilibrium thermodynamics introduced in [3, 4] for discrete and continuum systems. It relies on the concept of thermodynamic displacement. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of the entropy production associated to the irreversible processes involved.
A variational formulation for  uid dynamics with irreversible processes

Média

Voir la vidéo

Métriques

0
0
349.75 Ko
 application/pdf
bitcache://5d77e4fae8b6d6f1a6ebf161d2b639a3b32bef44

Licence

Creative Commons Aucune (Tous droits réservés)

Sponsors

Sponsors Platine

alanturinginstitutelogo.png
logothales.jpg

Sponsors Bronze

logo_enac-bleuok.jpg
imag150x185_couleur_rvb.jpg

Sponsors scientifique

logo_smf_cmjn.gif

Sponsors

smai.png
gdrmia_logo.png
gdr_geosto_logo.png
gdr-isis.png
logo-minesparistech.jpg
logo_x.jpeg
springer-logo.png
logo-psl.png

Organisateurs

logo_see.gif
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/17410/22346</identifier><creators><creator><creatorName>François Gay-Balmaz</creatorName></creator><creator><creatorName>Hiroaki Yoshimura</creatorName></creator></creators><titles>
            <title>A variational formulation for  uid dynamics with irreversible processes</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2018</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><subjects><subject>Nonequilibrium thermodynamics</subject><subject>variational formalism</subject><subject>viscosity</subject><subject>heat conduction</subject></subjects><dates>
	    <date dateType="Created">Sun 18 Feb 2018</date>
	    <date dateType="Updated">Sun 18 Feb 2018</date>
            <date dateType="Submitted">Tue 13 Nov 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">5d77e4fae8b6d6f1a6ebf161d2b639a3b32bef44</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>37001</version>
        <descriptions>
            <description descriptionType="Abstract">In this paper, we present a variational formulation for heat conducting viscous uids, which extends the Hamilton principle of continuum mechanics to include irreversible processes. This formulation follows from the general variational description of nonequilibrium thermodynamics introduced in [3, 4] for discrete and continuum systems. It relies on the concept of thermodynamic displacement. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of the entropy production associated to the irreversible processes involved.
</description>
        </descriptions>
    </resource>
.

A variational formulation for fluid dynamics with irreversible processes François Gay-Balmaz1 and Hiroaki Yoshimura2 1 CNRS & Ecole Normale Supérieure, LMD, IPSL, 24 Rue Lhomond 75005 Paris gaybalma@lmd.ens.fr, 2 Waseda University, School of Science and Engineering, Okubo, Shinjuku, Tokyo 169-8555, Japan yoshimura@waseda.jp, Abstract. In this paper, we present a variational formulation for heat conducting viscous fluids, which extends the Hamilton principle of con- tinuum mechanics to include irreversible processes. This formulation fol- lows from the general variational description of nonequilibrium thermo- dynamics introduced in [3, 4] for discrete and continuum systems. It re- lies on the concept of thermodynamic displacement. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expres- sion of the entropy production associated to the irreversible processes involved. Keywords: Nonequilibrium thermodynamics, variational formalism, vis- cosity, heat conduction 1 Variational principle for discrete systems In this section we review the variational formulation for nonequilibrium thermo- dynamics of discrete (i.e., finite dimensional) systems developed in [3]. 1.1 Variational formulation of nonequilibrium thermodynamics of simple systems We shall present the variational formulation by first considering simple thermo- dynamic systems before going into the general setting of the discrete systems. We follow the systematic treatment of thermodynamic systems presented in [12], to which we also refer for the precise statement of the two laws of thermodynamics. Simple discrete systems. A discrete thermodynamic system Σ is a collection Σ = ∪N A=1ΣA of a finite number of interacting simple thermodynamic systems ΣA. By definition, a simple thermodynamic system is a macroscopic system for which one (scalar) thermal variable and a finite set of mechanical variables are sufficient to describe entirely the state of the system. Variational formulation. Let Q be the configuration manifold associated to the mechanical variables of the simple system and denote by TQ and T∗ Q its tangent and cotangent bundles. The Lagrangian of a simple thermodynamic system is a function L : TQ × R → R, (q, v, S) 7→ L(q, v, S), where S ∈ R is the entropy. We assume that the system is subject to exterior and friction forces given by fiber preserving maps Fext , Ffr : TQ × R → T∗ Q, and to an external heat power supply Pext H (t). We say that a curve (q(t), S(t)) ∈ Q × R, t ∈ [t1, t2] ⊂ R is a solution of the variational formulation of nonequilibrium thermodynamics if it satisfies the variational condition δ Z t2 t1 L(q, q̇, S)dt+ Z t2 t1 Fext (q, q̇, S), δq dt = 0, Variational Condition (1) for all variations δq(t) and δS(t) subject to the constraint ∂L ∂S (q, q̇, S)δS = Ffr (q, q̇, S), δq , Variational Constraint (2) with δq(t1) = δ(t2) = 0, and also if it satisfies the phenomenological constraint ∂L ∂S (q, q̇, S)Ṡ = Ffr (q, q̇, S), q̇ − Pext H , Phenomenological Constraint (3) where q̇ = dq dt and Ṡ = dS dt . From this variational formulation, we deduce the system of evolution equa- tions for the simple thermodynamic system as        d dt ∂L ∂q̇ − ∂L ∂q = Ffr + Fext , ∂L ∂S Ṡ = Ffr , q̇ − Pext H . (4) We note that the energy function, defined by E = D ∂L ∂q̇ , q̇ E − L verifies d dt E = hFext , q̇i + Pext H , i.e., the first law of thermodynamics. Remark 1 (Phenomenological and variational constraints). The explicit expres- sion of the constraint (3) involves phenomenological laws for the friction force Ffr , this is why we refer to it as a phenomenological constraint. The associated constraint (2) is called a variational constraint since it is a condition on the variations to be used in (1). Note that the constraint (3) is nonlinear and also that one passes from the variational constraint to the phenomenological con- straint by formally replacing the variations δq, δS by the time derivatives q̇, Ṡ. Such a systematic correspondence between the phenomenological and variational constraints still holds for the general discrete systems, as we shall recall below. For the case of adiabatically closed systems (i.e., Pext H = 0), the evolution equations (4) can be geometrically formulated in terms of Dirac structures in- duced from the phenomenological constraint and from the canonical symplectic form on T∗ Q or on T∗ (Q × R), see [6]. 1.2 Variational formulation of nonequilibrium thermodynamics of discrete systems Discrete systems. We now consider the case of a discrete system Σ = ∪N A=1ΣA, composed of interconnecting simple systems ΣA, A = 1, ..., N that can exchange heat and mechanical power, and interact with external heat sources. We follow the description of discrete systems given in [12] and [7]. The state of the discrete system Σ is described by geometric variables q ∈ QΣ and entropy variables SA, A = 1, ..., N. The Lagrangian is a function L : TQΣ × RN → R, (q, q̇, S1, ..., SN ) 7→ L(q, q̇, S1, ..., SN ). (5) We assume that the system is subject to external forces Fext = PN A=1 Fext→A : TQΣ × RN → T∗ QΣ and external heat power supply Pext H = PN A=1 Pext→A H . The friction force associated to system ΣA is Ffr(A) : TQΣ × RN → T∗ QΣ and we define the total friction force Ffr := PN A=1 Ffr(A) . The internal heat power exchange between ΣA and ΣB can be described by PB→A H = κAB(q, SA , SB )(TB − TA ), where κAB = κBA ≥ 0 are the heat transfer phenomenological coefficients. For simplicity, we ignore internal and external matter exchanges in this sec- tion. Hence, in particular, the system is closed. A typical, and historically relevant, example of a discrete (non-simple) system is the adiabatic piston. We refer to [8] for a systematic treatment of the adiabatic piston from Stueckelberg’s approach. Variational formulation. Our variational formulation is based on the intro- duction of new variables, called thermodynamic displacements, that allow a sys- tematic inclusion of all the irreversible processes involved in the system. In our case, since we only consider the irreversible processes of mechanical friction and heat conduction, we just need to introduce (in addition to the mechanical dis- placement q) the thermal displacements3 , ΓA , A = 1, ..., N such that Γ̇A = TA , where ΓA are monotonically increasing real functions of time t and hence the temperatures TA of ΣA take positive real values, i.e., (T1 , ..., TN ) ∈ RN + . Each of these variables is accompanied with its dual variable ΣA whose time rate of change is associated to the entropy production of the simple system ΣA. We say that a curve q(t), SA(t), ΓA (t), ΣA(t)  ∈ QΣ × R3N , t ∈ [t1, t2] ⊂ R is the solution of the variational formulation of nonequilibrium thermodynamics 3 The notion of thermal displacement was first used by [13] and in the continuum setting by [9]. We refer to the Appendix of [11] for an historical account. if it satisfies the variational condition δ Z t2 t1 h L(q, q̇, S1, ...SN ) + N X A=1 (SA − ΣA)Γ̇A i dt + Z t2 t1 Fext , δq dt = 0, (6) for all variations δq(t), δΓA (t), δΣA(t) subject to the variational constraint ∂L ∂SA δΣA = D Ffr(A) , δq E − N X B=1 κAB(δΓB − δΓA ), (no sum on A) (7) with δq(ti) = 0 and δΓ(ti) = 0, for i = 1, 2, and also if it satisfies the nonlinear phenomenological constraint ∂L ∂SA Σ̇A = D Ffr(A) , q̇ E − N X B=1 κAB(Γ̇B − Γ̇A ) − Pext→A H . (no sum on A) (8) From this variational formulation, we deduce the system of evolution equations for the discrete thermodynamic system as              d dt ∂L ∂q̇ − ∂L ∂q = N X A=1 Ffr(A) + Fext , ∂L ∂SA ṠA = D Ffr(A) , q̇ E + N X B=1 κAB  ∂L ∂SB − ∂L ∂SA  − Pext→A H , A = 1, ..., N. We refer to [3] for the details regarding the treatment of discrete systems. In a similar way with the situation of simple thermodynamic systems, one passes from the variational constraint (7) to the phenomenological constraint (8) by formally replacing the δ-variations δq, δΣA, δΓA by the time derivatives q̇, Σ̇A, Γ̇A (see Remark 1). This is possible thanks to the introduction of the thermodynamic displacements ΓA. 2 The heat conducting viscous fluid We shall now systematically extend to the continuum setting the previous vari- ational formulation by focalising on the case of a heat conducting viscous fluid. We refer to [4] and [2] for the extension of this approach to the case of diffu- sion, chemical reaction, and phase changes in fluid dynamics, and to [5] for the variational formulation in terms of the free energy. Configuration space and geometric setting. We assume that the domain occupied by the fluid is a smooth compact manifold D with smooth boundary ∂D. The configuration space is Q = Diff0(D), the group of all diffeomorphisms4 of D 4 In this paper we do not describe the functional analytic setting needed to rigorously work in the framework of infinite dimensional manifolds. For example, one can as- sume that the diffeomorphisms are of some given Sobolev class, regular enough (at least of class C1 ), so that Diff0(D) is a smooth infinite dimensional manifold and a topological group with smooth right translation, [1]. that keep the boundary ∂D pointwise fixed. This corresponds to no-slip boundary conditions. We assume that the manifold D is endowed with a Riemannian metric g. The Levi-Civita covariant derivative, the sharp and flat operator associated to g are denoted as ∇g , [g : TD → T∗ D, and ]g : T∗ D → TD. Given a curve ϕt of diffeomorphisms, starting at the identity at t = 0, we denote by x = ϕt(X) = ϕ(t, X) ∈ D the current position of a fluid particle which at time t = 0 is at X ∈ D. The mass density %(t, X) and the entropy density S(t, X) in the Lagrangian (or material) description are related to the corresponding quantities ρ(t, x) and s(t, x) in the Eulerian (or spatial) descrip- tion as %(t, X) = ρ(t, ϕt(X))Jϕt (X) and S(t, X) = s(t, ϕt(X))Jϕt (X), (9) were Jϕt denotes the Jacobian of ϕt relative to the Riemannian metric g, i.e., ϕ∗ t µg = Jϕt µg, with µg the Riemannian volume form. From the conservation of the total mass, we have %(t, X) = %ref (X), i.e., the mass density in the material description is time independent. It therefore appears as a parameter in the Lagrangian function and in the variational formulation. Lagrangian. In a similar way to the case of discrete systems in (5), the La- grangian in the material description is a map L%ref : T Diff0(D) × F(D) → R, (ϕ, ϕ̇, S) 7→ L%ref (ϕ, ϕ̇, S), where T Diff0(D) is the tangent bundle to Diff0(D) and F(D) is a space of real valued functions on D with a given high enough regularity, so that all the formulas used below are valid. The index notation in L%ref is used to recall that L depends parametrically on %ref . Consider a fluid with a given state equation ε = ε(ρ, s) where ε is the internal energy density. The Lagrangian is given by L%ref (ϕ, ϕ̇, S)= Z D 1 2 %ref (X)|ϕ̇(X)|2 gµg(X)− Z D ε  %ref (X) Jϕ(X) , S(X) Jϕ(X)  Jϕ(X)µg(X) = Z D L(ϕ(X), ϕ̇(X), TX ϕ, %ref (X), S(X))µg(X), (10) where TXϕ : TXD → Tϕ(X)D is the tangent map to ϕ. The first term of L%ref represents the total kinetic energy of the fluid, computed with the help of the Riemannian metric g, and the second term represents the total internal energy. The second term is deduced from ε(ρ, s) by using the relations (9). In the second line we defined the Lagrangian density L(ϕ, ϕ̇, Tϕ, %ref , S)µg as the integrand of the Lagrangian L. The material temperature is given by T = − ∂L ∂S = ∂ε ∂s (ρ, s) ◦ ϕ = T ◦ ϕ, where T is the Eulerian temperature. The derivative of L with respect to TXϕ is the conservative Piola-Kirchhoff stress tensor Pcons := −  ∂L ∂TXϕ ]g . Variational formulation in material description. The continuum version of the variational formulation (6)–(8) reads δ Z t2 t1 h L%ref (ϕ, ϕ̇, S) − Z D (Ṡ − Σ̇)Γµg i dt = 0, Variational Condition (11) with variational and phenomenological and constraint Γ̇δΣ = (Pfr )[g : ∇g δϕ − JS · dδΓ Variational constraint (12) Γ̇Σ̇ = (Pfr )[g : ∇g ϕ̇ − JS · dΓ̇ + %ref R, Phenomenological constraint (13) where Pfr (t, X) is the friction Piola-Kirchhoff tensor, JS(t, X) is the entropy flux density, and %ref (X)R(t, X) is the heat power supply density. In (12) and (13), the double point “ : ” indicates the contraction with respect to both indices. In the same way as the case of discrete systems, the introduction of the vari- ables Γ and Σ allows us to propose a variational formulation with a very simple and physically meaningful structure: - The criticality condition (11) is an extension of the Hamilton critical action principle for fluid dynamics in material representation. - The nonholonomic constraint (13) is the expression of the power density as- sociated to all the irreversible processes involved (heat transport and viscosity in our case) in the entropy production. This constraint is of phenomenological nature, each of the “thermodynamic forces” being related to the fluxes charac- terizing an irreversible process via phenomenological laws, see Remark 2 below. The introduction of the variable Γ allows to write this constraint as a sum of force densities acting on velocities, namely Pfr “acting” on d dt ϕ and JS “acting” on d dt Γ, resulting in a power or rate of work density. - Concerning the virtual constraint (12), the occurrence of the time derivative in (13), also allows us to systematically replace all velocities by “δ-derivatives”, i.e., virtual displacements and to formulate the variational constraint as a sum of virtual thermodynamic work densities associated to each of the irreversible processes. It is important to note that this interpretation is possible thanks to the introduction of the variable Γ(t, X) whose time derivative is identified with the temperature T(t, X): d dt Γ = − ∂L ∂S =: T, from the stationarity condition associated with the variation δS in the variational principle. By computing the variations in (11), using the condition δϕ|∂D = 0 associated to no-slip boundary conditions, and using the variational and phenomenological constraints (12) and (13), we get the following result, see [3] for details. Proposition 1. In material representation, the evolution equations for a heat conducting viscous fluid given by    %ref DV Dt = DIV(Pcons + Pfr ), T(Ṡ + DIV JS) = (Pfr )[g : ∇g ϕ̇ − JS · dT + %ref R, with no-slip boundary conditions, follow from the variational formulation for non-equilibrium thermodynamics (11) with variational and phenomenological con- straints (12), (13), and with δΓ|∂D = 0. If the constraint δΓ|∂D = 0 is removed, then it implies JS · n[g = 0, where n is the outward pointing unit vector field on ∂D. If, in addition, %ref R = 0, then the fluid is adiabatically closed. Variational formulation in spatial description. In terms of the Eulerian velocity v = ϕ̇ ◦ ϕ−1 and Eulerian variables ρ and s, the Lagrangian (10) reads `(v, ρ, s) = Z S 1 2 ρ|v|2 gµg − Z S ε(ρ, s)µg. The material variational formulation (11)–(13) induces the spatial variational formulation δ Z t2 t1 h `(v, ρ, s) − Z S [Dt(s − σ)] γ µg i dt = 0 (14) with respect to variations δv = ∂tζ + [v, ζ], δρ = − div(ρζ), δγ, and δσ, (15) and subject to the variational and phenomenological constraints DtγD̄δσ = (σfr )[g : ∇ζ − jS · dDδγ, (16) DtγD̄tσ = (σfr )[g : ∇v − jS · dDtγ + ρr, (17) where γ, σ, jS, and σfr are the Eulerian quantities associated to Γ, Σ, JS, and Pfr , and we used the notations Dtf = ∂tf + v · df, Dδf = δf + ζ · df, D̄tf = ∂tf +div(fv) and D̄δf = δf +div(fζ) for the Lagrangian time derivatives and variations of scalar fields and density fields. The first two expressions in (15) are obtained by taking the variations with respect to ϕ, v, and ρ, of the relations v = ϕ̇ ◦ ϕ−1 and ρ = (%ref ◦ ϕ−1 )Jϕ and by defining the vector field ζ := δϕ ◦ ϕ−1 . These formulas can be directly justified by employing the Euler- Poincaré reduction theory on Lie groups, [10]. By computing the variations in (14), using the condition ζ|∂D = 0 associated to no-slip boundary conditions, and using the variational and phenomenological constraints (16) and (17), we get the following result, see [3] for details. Proposition 2. In spatial representation, the evolution equations for a viscous heat conducting fluid given by        ρ(∂tv + ∇vv) = − grad p + div σfr , p = ∂ε ∂ρ ρ + ∂ε ∂s s − ε ∂tρ + div(ρv) = 0 T(∂ts + div(sv) + div jS) = (σfr )[ : ∇v − jS · dT + ρr, T = ∂ε ∂s , (18) with no-slip boundary conditions, follow from the variational condition for non- equilibrium thermodynamics given in (14), with variational and phenomenologi- cal constraints (15), (16), (17), and with δγ|∂D = 0. If the constraint δγ|∂D = 0 is removed, then it implies jS · n[g = 0. If, in addition, %ref R = 0, then the fluid is adiabatically closed. Remark 2 (Thermodynamic phenomenology). In order to close the system (18), it is necessary to provide phenomenological expressions of the thermodynamic fluxes in terms of the thermodynamic affinities, compatible with the second law of thermodynamics. In our case, the thermodynamic fluxes are σfr and js and we have the well-known relations: σfr = 2µ(Def v)] +  ζ − 2 3 µ  (div v)g] and Tj[ s = −κdT (Fourier law), where Def v = 1 2 (∇v + ∇vT ), µ ≥ 0 is the first coefficient of viscosity (shear viscosity), ζ ≥ 0 is the second coefficient of viscosity (bulk viscosity), and κ ≥ 0 is the thermal conductivity. Generally, these coefficients depend on ρ and T. Acknowledgements. F.G.B. is partially supported by the ANR project GE- OMFLUID, ANR-14-CE23-0002-01; H.Y. is partially supported by JSPS Grant- in-Aid for Scientific Research (26400408, 16KT0024, 24224004), Waseda Univer- sity (SR2017K-167), and the MEXT “Top Global University Project”. References 1. Ebin, D.G. and Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. 92 (1970), 102–163. 2. Gay-Balmaz, F.: A variational derivation of the thermodynamics of a moist atmo- sphere with irreversible processes (2017), https://arxiv.org/pdf/1701.03921.pdf 3. Gay-Balmaz, F. and Yoshimura, H.: A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: discrete systems, J. Geom. Phys., 111 (2016), 169–193. 4. Gay-Balmaz, F. and Yoshimura, H.: A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: continuum systems, J. Geom. Phys., 111 (2016), 194–212. 5. Gay-Balmaz, F. and Yoshimura, H.: A free energy Lagrangian variational formula- tion of the Navier-Stokes-Fourier system (2017), https://arxiv.org/pdf/1706.09010.pdf. 6. Gay-Balmaz, F. and Yoshimura, H.: Dirac structures in nonequilibrium thermody- namics, preprint (2017), https://arxiv.org/pdf/1704.03935.pdf 7. Gruber, C.: Thermodynamique et Mécanique Statistique, Institut de physique théorique, (1997), EPFL. 8. Gruber, C.: Thermodynamics of systems with internal adiabatic constraints: time evolution of the adiabatic piston, Eur. J. Phys. 20 (1999), 259–266. 9. Green, A. E. and Naghdi, P. M.: A re-examination of the basic postulates of thermo- mechanics, Proc. R. Soc. London., Series A: Mathematical, Physical and Engineering Sciences, 432, No. 1885 (1991), 171–194. 10. Holm, D. D., Marsden, J. E., and Ratiu, T. S.: The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math. 137 (1998), 1–81. 11. Podio-Guidugli, P.: A virtual power format for thermomechanics, Continuum Me- chanics and Thermodynamics, 20(8) (2009), 479–487. 12. Stueckelberg, E. C. G. and Scheurer, P. B.: Thermocinétique phénoménologique galiléenne, Birkhäuser, (1974). 13. von Helmholtz, H.: Studien zur Statik monocyklischer Systeme. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1884), 159– 177.