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An essential part in modeling out-of-equilibrium dynamics is the formulation of irreversible dynamics. In the latter, the major task consists in specifying the relations between thermodynamic forces and fluxes. In the literature, mainly two distinct approaches are used for the specification of force-flux relations. On the one hand, quasi-linear relations are employed, which are based on the physics of transport processes and fluctuation-dissipation theorems (de Groot and Mazur in Non-equilibrium thermodynamics, North Holland, Amsterdam, 1962 , Lifshitz and Pitaevskii in Physical kinetics. Volume 10, Landau and Lifshitz series on theoretical physics, Pergamon Press, Oxford, 1981 ). On the other hand, force-flux relations are also often represented in potential form with the help of a dissipation potential (Silhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997 ). We address the question of how these two approaches are related. The main result of this presentation states that the class of models formulated by quasi-linear relations is larger than what can be described in a potential-based formulation. While the relation between the two methods is shown in general terms, it is demonstrated also with the help of three examples. The finding that quasi-linear force-flux relations are more general than dissipation-based ones also has ramifications for the general equation for non-equilibrium reversible-irreversible coupling (GENERIC: e.g., Grmela and Öttinger in Phys Rev E 56:6620-6632, 6633-6655, 1997 , Öttinger in Beyond equilibrium thermodynamics, Wiley Interscience Publishers, Hoboken, 2005 ). This framework has been formulated and used in two different forms, namely a quasi-linear (Öttinger and Grmela in Phys Rev E 56:6633-6655, 1997 , Öttinger in Beyond equilibrium thermodynamics, Wiley Interscience Publishers, Hoboken, 2005 ) and a dissipation potential-based (Grmela in Adv Chem Eng 39:75-129, 2010 , Grmela in J Non-Newton Fluid Mech 165:980-986, 2010 , Mielke in Continuum Mech Therm 23:233-256, 2011 ) form, respectively, relating the irreversible evolution to the entropy gradient. It is found that also in the case of GENERIC, the quasi-linear representation encompasses a wider class of phenomena as compared to the dissipation-based formulation. Furthermore, it is found that a potential exists for the irreversible part of the GENERIC if and only if one does for the underlying force-flux relations.[PUBLICATION ABSTRACT]
Continuum Mech. Thermodyn. (2013) 25:803816
DOI 10.1007/s00161-012-0289-y
ORIGINAL ARTICLE
Received: 28 September 2012 / Accepted: 21 December 2012 / Published online: 25 January 2013 Springer-Verlag Berlin Heidelberg 2013
Abstract An essential part in modeling out-of-equilibrium dynamics is the formulation of irreversible dynamics. In the latter, the major task consists in specifying the relations between thermodynamic forces and uxes. In the literature, mainly two distinct approaches are used for the specication of forceux relations. On the one hand, quasi-linear relations are employed, which are based on the physics of transport processes and uctuationdissipation theorems (de Groot and Mazur in Non-equilibrium thermodynamics, North Holland, Amsterdam, 1962, Lifshitz and Pitaevskii in Physical kinetics. Volume 10, Landau and Lifshitz series on theoretical physics, Pergamon Press, Oxford, 1981). On the other hand, forceux relations are also often represented in potential form with the help of a dissipation potential (ilhav in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997). We address the question of how these two approaches are related. The main result of this presentation states that the class of models formulated by quasi-linear relations is larger than what can be described in a potential-based formulation. While the relation between the two methods is shown in general terms, it is demonstrated also with the help of three examples. The nding that quasi-linear forceux relations are more general than dissipation-based ones also has ramications for the general equation for non-equilibrium reversibleirreversible coupling (GENERIC: e.g., Grmela and ttinger in Phys Rev E 56:66206632, 66336655, 1997, ttinger in Beyond equilibrium thermodynamics, Wiley Interscience Publishers, Hoboken, 2005). This framework has been formulated and used in two different forms, namely a quasi-linear (ttinger and Grmela in Phys Rev E 56:66336655, 1997, ttinger in Beyond equilibrium thermodynamics, Wiley Interscience Publishers, Hoboken, 2005) and a dissipation potentialbased (Grmela in Adv Chem Eng 39:75129, 2010, Grmela in J Non-Newton Fluid Mech 165:980986, 2010, Mielke in Continuum Mech Therm 23:233256, 2011) form, respectively, relating the irreversible evolution to the entropy gradient. It is found that also in the case of GENERIC, the quasi-linear representation encompasses a wider class of phenomena as compared to the dissipation-based formulation. Furthermore, it is found that a potential exists for the irreversible part of the GENERIC if and only if one does for the underlying forceux relations.
Communicated by Andreas chsner.
M. Htter (B)
Department of Mechanical Engineering, Materials Technology (MaTe), Eindhoven University of Technology,P. O. Box 513, 5600 MB Eindhoven, The Netherlands
E-mail: [email protected]
B. Svendsen
Material Mechanics, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany
B. Svendsen
Microstructure Physics and Alloy Design, Max-Planck Institute for Steel Research, Max-Planck Str. 1, 40237 Dsseldorf, Germany
Markus Htter Bob Svendsen
Quasi-linear versus potential-based formulations of forceux relations and the GENERIC for irreversible processes: comparisons and examples
804 M. Htter, B. Svendsen
Keywords Non-equilibrium thermodynamics GENERIC Irreversible processes Forceux relation
Dissipation potential
1 Introduction
Over the years, a number of approaches to the thermodynamic formulation of models for material behavior have been developed. In the phenomenological realm, one of the most common of these is continuum thermodynamics (e.g., [1]) as based on the ClausiusDuhem inequality and ColemanNoll dissipation principle (e.g., [2,3]). Another is based on the entropy inequality of MllerLiu (e.g., [4,5]). Classically, such approaches have been based on the assumption of local thermodynamic equilibrium to formulate models for thermoelastic materials with heat conduction and viscosity. More general approaches such as extended thermodynamics (e.g., [6]) or extended linear irreversible thermodynamics (e.g., [7]) relax this assumption and model the approach of the system to thermodynamic equilibrium. Alternatively, for the case of history-dependent, and in particular inelastic, materials, such models have generally been based on the concepts of strong fading memory and internal variables (e.g., [8]). This concept also lies at the heart of so-called generalized standard or standard dissipative materials (e.g., [9,10]). From the point of view of irreversible thermodynamics, the goal of such formulations is to model the approach of non-equilibrium systems to thermodynamic equilibrium (if it exists) (e.g., [11]). Perhaps the most prominent example of such models is offered by the Ginzburg-Landau equation as based on the free energy.
More recently, an alternative approach to the thermodynamics of solids has been developed as an application of the so-called general equation for non-equilibrium reversibleirreversible coupling (GENERIC: e.g., [1216]). Originally developed for (complex) uids, this formalism has been applied to derive models for anisotropic elastic and elasto-(visco)plastic solids in an Eulerian setting [17,18]. An alternative approach to the formulation of GENERIC-based models for inelastic solids was pursued in [19,20], who considered thermoelastic solids with heat conduction and viscosity, as well as the case of viscoplasticity, in a Lagrangian setting. Yet another GENERIC-based approach to formulate models for inelastic materials (e.g., viscoelastic, elastoplastic) has been discussed in [21].
A cornerstone of both continuum thermodynamics and the GENERIC is the modeling of irreversible processes and entropy production via thermodynamic uxforce relations. Via the physics of transport processes (e.g., [11,22]), such relations are derived from uctuationdissipation and coarse-graining considerations, resulting in their dependence on transport properties and a mathematical form quasi-linear in the forces. In some cases, such uxforce relations may also be representable in potential form with the help of a dissipation potential (e.g., Chapter 12 in [1]). Formally speaking, such a representation is analogous to that of evolution-constitutive relations for internal variables based on such a potential [2328] when the internal variable rates involved are interpreted as thermodynamic uxes. A formulation of the GENERIC based on a dissipation potential has also been advocated in [15,16] and more recently in [21]. On the other hand, it is not clear whether all uxforce relations derived from transport theory are representable in potential form.
One purpose of the current work is to formulate conditions which uxforce relations must satisfy in order for a potential representation of these to exist. A second is the investigation of the connections between the potential representation of uxforce relations and that of the GENERIC for irreversible processes. Among other things, we show that the GENERIC-based model for irreversible processes can be represented in potential form if and only if such a potential form exists for the underlying thermodynamic uxforce relations. To explore the implications of these basic results in more detail, we consider three examples: (i) heat conduction in anisotropic solids, (ii) slippage in complex uids (GordonSchowalter derivative), and (iii) homogeneous chemical reactions. As shown by the application of the general results, the models of (i) and (iii) can be formulated in terms of a dissipation potential under certain conditions. On the other hand, since the irreversible process of slippage is dissipation-free, the dissipation potential for the model of (ii) is identically zero, and hence, the concept of the dissipation potential is inappropriate for its description.
For the purposes of the current work, it is sufcient to work with the purely local form of the GENERIC (Chapters 2 and 3 in [14]) in terms of differential operators rather than generalized functions and integration. As well, this involves working with the energy and entropy densities. Before we begin, a word on notation. For clarity and ease of understanding for continuum mechanicians and physicists alike, a number of results in this work will be expressed in both direct (i.e., symbolic) and (Cartesian) component notation. To this latter end, let upper latin indices K, L, M, . . . = 1, 2, 3, represent Cartesian components of referential or Lagrangian
tensors. The summation convention on repeated such indices will be used throughout. Likewise, we use the
Quasi-linear versus potential-based formulations 805
notation L = /rL for the components of the gradient of any eld dened with respect to the reference
conguration of the material in three-dimensional Euclidean point space E3 with translation vector space V 3. These are functions of referential position r = o + rK iK E3 with respect to the Cartesian basis vectors
i1, i2, i3 V 3 and origin o E3. As is common in solid mechanics for large deformation, all model relations
to follow are represented in referential or Lagrangian form relative to some reference conguration. Finally, for a tensor A of arbitrary rank and two vectors a and b, the notation aA b implies a differentiation of A and subsequent contraction between the two vectors a and b.
The paper is organized as follows. In Sect. 2, forceux relations are discussed, and their quasi-linear and dissipation potentialbased formulations are compared. The analogous comparison is made in Sect. 3 for the GENERIC framework, where also both types of formulations exist. The nding that the cases of forceux relations and the GENERIC are closely related is examined in Sect. 4 in more detail. To illustrate the abstract results, examples are studied in Sect. 5. We close with a discussion in Sect. 6.
2 Fluxforce relations
2.1 Basic considerations
For simplicity, attention is restricted in this work to continuous thermodynamic systems containing no discontinuities (e.g., singular surface). In this context, consider the local or strong form
= div (1) of referential entropy balance for supply-free processes (e.g., Chapters 3 and 9 in [1]). Here, represents the referential entropy density, is the referential entropy ux density, and is the referential entropy production rate density. In particular, represents the (net) ux of entropy per unit area from the environment into the system, and is the entropy production rate in the system per unit volume. In the current context, the second law is expressed in the local form
[greaterorequalslant] 0 (2) relative to . Given Gibbs relation and the assumption of local equilibrium (e.g., [1,11,14]), the well-known constitutive form
= j f (3) for follows in terms of thermodynamic uxes j (e.g., heat ux) and forces f (e.g., gradient of reciprocal temperature). Constitutive relations between j and f are generally formulated in the form
j = j(. . . , f ), (4) where . . . indicate a possible dependence on additional quantities besides f . For simplicity, these will be suppressed in the notation and we will just write j(f ) for this relation for the time being; this applies as well to all related forms of this relation to be considered in what follows.
2.2 Quasi-linear transport relation
In the common context of transport theory (e.g., [11,14,22]), for example, the constitutive form
j(f ) = L(f ) f (5) quasi-linear in f is obtained. In particular, in the classical special case of linear irreversible thermodynamics for systems near equilibrium, the transport operator L(f ) is independent of f . In what follows, it will be useful to work with the split
L = Lsym + Lskw
= sym(L) + skw(L)
=
1
2 (L + LT) +
1
2 (L LT) (6)
of L into symmetric Lsym and skew-symmetric Lskw parts. This induces the corresponding split
806 M. Htter, B. Svendsen
j = jsym + jskw = Lsym f + Lskw f (7)
of j into dissipative jsym and non-dissipative jskw parts. Indeed, since
jskw f = Lskw f f = 0 (8)
follows via the skew symmetry of Lskw, substitution of (7) into (3) yields
= j f = jsym f = Lsym f f . (9)
In the context of (5) and (6), then, clearly only the symmetric part Lsym of L contributes to . On this basis, non-negative entropy production (2) is satised sufciently by requiring Lsym(f ) to be non-negative definite;
if Lsym is in fact independent of f (i.e., linear irreversible special case), non-negative-definiteness of Lsym is also necessary for a potential representation to exist.
2.3 Potential-based transport relation
Assume now that a particular form of the uxforce constitutive relation j(f ), for example, the transport form (5), has been derived via physical considerations and is known. As discussed elsewhere [1,29], any variational formulation of the corresponding initial-boundary-value problem is then contingent on whether or not a potential representation for j(f ) can be found. Specifically, this means that the forceux relation can be written in the form
j(f) = f p(f) (10)
in terms of the so-called dissipation potential p. In order to ensure a non-negative rate of entropy production, (2), it is sufcient to require that the dissipation potential is non-negative and convex. In this case, one obtains
= f f p [greaterorequalslant] p [greaterorequalslant] 0 (11)
as required.
Whether or not a potential representation for j(f ) can indeed be found is basically a mathematical problem (e.g., integrability) for which there may be in general no solution. If the mathematical form of j(f ) satises certain conditions, however, then such a potential representation can be found. One possibility in this regard can be formulated with the help of a generalization of the Helmholtz theorem1 due to [26] (see also [1,31]). To this end, assume that the domain of the uxforce relation j(f ) is convex or star-shaped. In particular, this implies that there exists a path cf(s) = sf (0 [lessorequalslant] s [lessorequalslant] 1) in the space of all forces connecting equilibrium
cf(0) = 0 with any f , that is, cf(1) = f . On this basis, one can introduce for j(f ) the unique (via linearity and
orthogonality) additive split into symmetric and skew-symmetric parts
j(f ) = f pj(f ) + sj(f ), (12)
in terms of the scalar-valued non-negative function
pj(f ) =
1
2 skw(cf(s) j(cf(s))) cf(s) ds (14)
which is non-dissipative, that is, sj(f ) f = 0. One thus obtains
1 Related to the de Rham decomposition in differential geometry as based on the Poincar theorem (e.g., [30]).
[integraldisplay]
0
1
j(cf(s)) f ds (13)
(assuming pj(0) = 0 for simplicity without loss of physical generality) and the vector-valued function
sj(f ) =
[integraldisplay]
0
Quasi-linear versus potential-based formulations 807
1
f s sym(cf(s) j(cf(s))) f ds, (15)
where (13) has been employed to derive the second equality. In this form, it is evident that if sym(f j(f )) is non-negative definite, the integral in this last relation is non-negative, and
(f ) = f pj(f ) f [greaterorequalslant] pj(f ) [greaterorequalslant] 0, (16)
is obtained, that is, pj(f ) as given by (13) is convex. If f j(f ) is in fact symmetric, sj(f ) vanishes identically, and pj(f ) is a potential for j(f ). Conversely, if a potential for j(f ) exists, skw(f j(f )) vanishes identically (i.e., via Eulers theorem: [30]).
In the context of (12), then, the deviation sj(f ) of j(f ) from being potential results in no entropy production. Indeed, (12) splits j(f ) into dissipative f pj(f ) and non-dissipative sj(f ) parts. As we saw in the previous section, the split (6) of the transport operator induces an analogous split (7) of j(f ) into dissipative jsym(f ) = Lsym(f ) f and non-dissipative jskw(f ) = Lskw(f ) f parts. To look into this more closely, consider
now the representation of (5) via (12). One obtains
pL(f ) =
and
s2 f (cf(s)Lsym(cf(s)) a) f ds (18)
for all vectors a via the skew symmetry of Lskw. In particular, this latter result for sL(f ) implies that, even if Lskw is identically zero, assuming non-negative definiteness of Lsym(f ) is necessary, but generally not sufcient, for the transport form (5) of j(f ) to be represented solely by pL(f ). Indeed, the additional condition
a (f Lsym(f ) f ) f = f (f Lsym(f ) a) f (19)
on the functional form of Lsym(f ) must hold, in which case f jsym(f ) is symmetric. Clearly, this represents an additional constitutive restriction on the form of Lsym(f ) going beyond those of OnsagerCasimir symmetry and non-negative definiteness. It also has implications for approaches to the formulation of models for non-equilibrium systems which are based on thermodynamic uxforce relations j(f ). This includes, for example, the case of the GENERIC, to which we turn in Sect. 3.
2.4 Potential-based representations in quasi-linear form
In the previous section, we have shown that not every quasi-linear forceux relation can be cast into potential form, but that conditions on L apply, namely Lskw = 0 and the conditions (19). In this section, we prove that
in turn any potential-based forceux relation can be written in quasi-linear form. This implies that the class of constitutive relations from potential-based formulations is a subset of the models captured by quasi-linear relations.
Close to equilibrium, the dissipation potential can be approximated to be quadratic in the force [32,33],
p =
1
2f [parenleftbig]
(f ) = f pj(f ) f = pj(f ) +
[integraldisplay]
0
[integraldisplay]
0
1
f s Lsym(sf ) f ds (17)
a sL(f ) = a Lskw(f ) f
+
[integraldisplay]
0
1
1
s2 a (cf(s)Lsym(cf(s)) f ) f ds
[integraldisplay]
0
ff p|0[parenrightbig]
f, (20)
808 M. Htter, B. Svendsen
because at f = 0 the general properties of p require p(0) = 0 and f p = 0. Since p is convex, ff p|0 is not
only symmetric but also non-negative definite. A straightforward calculation leads to the forceux relation (5) with the constant matrix
L = Lsym = ff p|0 . (21)
Therefore, any constitutive relation close to equilibrium can both be derived from the dissipation potential and be written in the (quasi-)linear form.
For the general case beyond the proximity to equilibrium, however, the situation requires a more careful discussion. Let us assume that a constitutive relation is written in the form with a dissipation potential p. Then, that same constitutive rule can also be written in the quasi-linear form (5) with the special choice
L = Lsym =
1 j j =
1 f p f p, (22)
with the entropy production rate density dened above. By construction, L is not only symmetric, but also non-negative by virtue of the non-negative rate of entropy production (2). The only possible caveat is that (22) may be ill dened mathematically for = 0. While the (close to) equilibrium case (with 0) has already
been covered in (21), we need to discuss the possibility of a vanishing dissipation rate out of equilibrium, that is, = j f = 0 for a certain value of the driving force vector f = 0. If such a non-vanishing driving force
really exists, then the dissipation potential assumes its minimum value p(f ) = pmin = 0, since p is bracketed
between and 0 according to (11). However, because p is minimal, one obtains j j(f ) = f p|f=f
= 0.
In other words, = 0 for a nite f requires j = 0, rather than j f for nite j . Analyzing (22) upon
j 0, one nds the well-dened limit L = 0 at f . In summary, any constitutive relation written in terms of
the dissipation potential can also be written in the quasi-linear form (5). This is also true in the context of the GENERIC (see below), that is, any potential form for the irreversible part of the GENERIC can be expressed in quasi-linear form as well.
For completeness, we point out that the rank of L in tensor product form (22) is unity. Often, the rank of L is related on physical grounds to the number of independent irreversible processes that are at play simultaneously. However, such an argument is foreign to a dissipation potential formulation from the start. Since the main goal of this section was to demonstrate the existence of a quasi-linear formulation for any potential form, we do not go into further details about the number of independent processes. We just mention that, using physical arguments, the relation (22) can be generalized to a form with rank larger than unity, that is, with multiple processes.
3 GENERIC-based formulation for irreversible processes
3.1 Basics
In the context of the GENERIC, the total energy E and total entropy S are modeled as functionals of a set of variables (i.e., elds) x (e.g., temperature) characterizing the system under consideration. Let A represent either E or S, and let a be the density of A. In the current work, attention will be focused on the class of models given by the form
A(x) = [integraldisplay] a(x, x) dv (23)
of this functional in which a depends on both x and (one or more of) their spatial gradients x. In all what
follows, boundary terms are neglected for simplicity which is appropriate if either the boundary conditions are chosen appropriately or the elds vanish at the (innitely remote) boundary. Since A is a functional of time-dependent elds x,
A = [integraldisplay] [parenleftBig]xa x + x a x[parenrightBig]dv
= [integraldisplay] ax x dv (24)
Quasi-linear versus potential-based formulations 809
follows for its time rate-of-change via the divergence theorem and neglect of boundary effects as just discussed above, where
ax xa div x a (25) represents the rst-order variational derivative of a.
As they embody the physics of transport processes, uxforce relations like (4) lie at the heart of many approaches to the modeling of non-equilibrium systems such as the GENERIC [1214]. In the spirit of the Ginzburg-Landau equation, this is a model for the evolution of system variables x (e.g., temperature) in non-equilibrium toward equilibrium. In the context of the GENERIC, the evolution relation
x = xrev + xirr (26)
for x splits into reversible xrev and irreversible xirr parts. In turn, (26) induces the split
A = Arev + Airr
= [integraldisplay] ax xrev dv + [integraldisplay] ax xirr dv (27)
of A from (24). In the current work, attention is focused in particular on xirr and the corresponding part Airr
of A for the cases of total energy (A = E , a = e) and entropy (A = S, a = ). Analogous to the uxforce
relation j(. . . , f ) from (4), a formulation based on the GENERIC works with the constitutive form
xirr = xirr(. . . , x) (28)
for xirr; as above in the case of the uxforce relation (4), the dots . . . indicate a possible dependence on
additional quantities (e.g., x) besides the entropy gradient x. For simplicity, these will be suppressed in the notation and we will just write xirr(x) for this relation for the time being; this applies as well to all related
forms of this relation to be considered in what follows.
3.2 Quasi-linear relation
In one version of the GENERIC, advocated by ttinger [13,14], the irreversible evolution (28) for xirr is
modeled via the transport-theoretic form
xirr(x) = M(x) x (29)
quasi-linear in the GENERIC-based derivative x of the entropy density with respect to x. Here, M represents the so-called friction operator. Analogous to the case of the uxforce relation (5) above, besides on x, xirr
and related quantities like M depend in general on additional elds like x. Because they play no direct role in the following, however, we dispense with them in the notation for simplicity. It should be noted that in the phase-eld-like case, is a constitutive function of x and its spatial gradient x, for example, and thus
x = x div x cannot be simply recast in terms of the local elds x alone. Therefore, x and x are in
general distinct quantities, of which x is of prime importance for the relation to the forceux relations.
The correspondence of the form of (29) with that (5) for transport-based thermodynamic uxforce relations is no coincidence, as will be discussed in more detail below. In the context of the dissipation bracket
[A, B] := [integraldisplay] ax M bx dv (30)
on functionals A, B of the form (23) induced by (29), consider the split
M = Msym + Mskw
= sym(M) + skw(M)
=
1
2 (M + MT) +
1
2 (M MT) (31)
810 M. Htter, B. Svendsen
of M into symmetric and skew-symmetric parts, formally analogous to that (6) of the transport operator L above. In this context, the GENERIC-based form2
Sirr = [integraldisplay] dv = [integraldisplay] x xirr dv = [integraldisplay] x Msym(x) x dv (32)
of the entropy production rate is obtained which depends only on Msym. This is analogous to the reduced form (9) of following from split (6) of L analogous to that (31) of M just discussed. Note that a non-negative rate of entropy production (32) can be achieved by requiring that Msym be positive semi-definite.
3.3 Potential-based formulation
Another form of the GENERIC, alternative to (29), is advocated by Grmela [12,15,16]. It is based on assuming that there exists a potential representation for xirr(x) a priori, that is, from the start. For later convenience,
we write this form of the GENERIC as
xirr(x) = x pirr(x) + sirr(x), (33)
where Grmela considers sirr(x) = 0. Such a potential-based formulation of the irreversible part of the
GENERIC has been studied from a mathematical perspective in [21].
To compare (29) and (33) in more detail, we could proceed analogously to the discussion about the forceux relations, Sects. 2.3 and 2.4. One would then nd that (i) sirr(x) is skew-symmetric and thus dissipation-less, analogous to (14), and that (ii) a potential representation exists only for a certain class of relations
xirr(x). An alternative route, demonstrated in the following (Sect. 4), consists in relating the quasi-linear and dissipation potential formulations of the GENERIC to the corresponding formulations of the forceux relations.
4 Connection between uxforce relations and the GENERIC
The parallels between the formulation of thermodynamic uxforce relations j(f ) in Sect. 2 and the GENERIC-based relation xirr(x) in Sect. 3 alluded to, and clearly evident, in the development up to this point are no
coincidence. Indeed, from a physical point of view, the form of the uxforce relation j(f ) determines that of xirr(x). In particular, this is the case for the transport-theoretic forms (5) and (29) of these relations. The
purpose of the current section is to delve into this in more detail.To begin, note that any connection between j(f ) and xirr(x) clearly involves in particular relations between
x, f , j, and xirr. Consider, for example, the case of heat conduction in which the internal energy density
is chosen as an element of x. In this case, the GENERIC-based entropy gradient = 1 in (29) is the reciprocal temperature, and its spatial gradient f 1 is the thermodynamic force driving heat
conduction. Consequently, is mapped into f via the gradient operator . More generally, assume that
there exists an operator C (e.g., Chapters 2 and 3 in [14,34]) independent of x such that3
f (x) = CTx (34) holds. Since C is independent of x, f (x) is linear in x. In terms of operator transposition
[integraldisplay] Cj ax dv := [integraldisplay] j CT ax dv, (35)
the common relation
Sirr = [integraldisplay] dv = [integraldisplay] j f dv = [integraldisplay] xirr x dv (36)
2 By orthogonality of reversible and irreversible processes, x annihilates the reversible part xrev of x in the context of the
GENERIC [1214].3 Using the opposite sign convention to ttinger (Section 3.1.1 in [14]).
Quasi-linear versus potential-based formulations 811
for the entropy production rate density from (3) and (32) induces the basic connection
xirr(x) = C j(f (x)) = C j(CTx) (37)
For quasi-linear forceux relations (5), the basic connections (34) and (37) imply via the quasi-linear
GENERIC (29) the form
M(x) = C L(f (x)) CT (38) for the friction operator M in terms of the transport operator L [14,3437]. As such, the functional form of M(x) is clearly induced by that of L(f ).
For forceux relations of the form (12), (34) and (37) imply the connections
x pirr(x) = C f(x) pj(f (x)),
sirr(x) = C sj(f (x)),
(39)
via the GENERIC (33). On the basis of this last result, one can conclude that, for general C, sirr(x) vanishes iff sj(f ) does. As such, a potential form for xirr(x) exists iff one for j(f ) does. In particular, this then applies
to the transport-based forms (5) and (29), respectively, of these relations.
Specic expressions for CT can be determined in view of the relation (34) between the entropy gradient x and the thermodynamic force f . For completeness, we mention that an additional condition on CT emerges from the GENERIC requirement of the conservation of energy. Specifically, since Erev = 0 is determined by
Hamiltonian dynamics and vanishes identically in the context of the GENERIC,
E = Eirr = [integraldisplay] xirr ex dv = [integraldisplay] j CTex dv (40)
follows from (37) as well. Consequently, conservation of energy E = 0 is ensured identically if
CTex = 0 (41) holds identically [14,34]. Note that this induces a dependence of C on the components of the GENERIC-based energy gradient ex.
5 Examples
In the following, examples are given to study possible formulations in quasi-linear or potential-based forms. It has been demonstrated above that the GENERIC approach is closely related to the forceux relations, specifically also with respect to the (non-)existence of a potential representation. Therefore, we concentrate in the following only on the underlying forceux relations and possible potential representations thereof.
5.1 Example: anisotropic rigid heat conductor
Although not terribly realistic, an anisotropic rigid heat conductor represents perhaps the simplest example of non-isothermal solid behavior. As such, it is ideal for the purpose of illustrating the derivation of specic forms of the general results obtained in the last two sections. As is well known, such a solid deforms via translation and rotation alone; stretch, strain, thermal expansion, and so on are excluded. In this case, the material behavior depends solely on heat conduction, that is, on the temperature and its gradient .
As a model for heat conduction, consider the transport-based generalized Fourier relation
q = K ( , ) = 2 K ( , 1) 1 (42) for the heat ux q quasi-linear in the gradient of the (reciprocal) temperature. Since this is the only uxforce relation for the current material class, we have
j = (j) := (q), L = [L] := [2 K ], f = (f) := (1). (43)
between xirr and j via C.
812 M. Htter, B. Svendsen
To discuss the issue of possible potential representations, we proceed as follows. Since L = 2 K is sym
metric and non-negative definite, Lskw vanishes identically, and L = Lsym is symmetric and non-negative
definite. On this basis, the general requirement (19) for sL(f ) sK (f) to vanish reduces toa (f L(f) f) b = f (f L(f) a) b (44)
for all a, b. As discussed above, this is clearly a condition going beyond symmetry and non-negative definiteness. For example, viewing J = f L as a third-order tensor-valued quantity J = JK L M iK iL iM, the symmetry of L = 2 K implies JK L M = JL K M , whereas sK (f) will vanish if JK L M = JM L K holds. This
is trivially satised of course by the standard Fourier form K ( ) = k( ) I for K independent of 1 in
terms of the coefcient of thermal conductivity k( ). In the Fourier case, then,
pL(f ) = f
follows from (17) with pL(0) 0.
5.2 Example: slippage in complex uids
Complex uids consist of discrete constituents (e.g., macromolecules) which can move relative to each other. If one imagines, for example, ellipsoids immersed in a uid, then this means that their motion is not determined in an afne (i.e., homogeneous) fashion by the (Eulerian) velocity eld v of the surrounding uid. Rather, they may slip relative to one another (e.g., [3840]). Such slippage is characterized constitutively by a slip parameter which is related to the aspect ratio of ellipsoids. Slippage is also relevant to the modeling of polymeric uids (e.g., [35,41]). As it turns out, slippage is an irreversible process resulting in no dissipation. In other words, the skew-symmetric part Lskw of L does not vanish in this case.
The Eulerian or spatial formulation of complex uids with slippage (e.g., Section 4.2.1 in [14]) involves the velocity eld v or the spatial momentum density m = v, respectively, with the spatial mass density.
In addition, the spatial conformation tensor4 C is a measure of the (internal) deformation state of the uid (microstructure) relative to the continuum as a whole. Recall that the time-dependent deformation or ow x =
(r, t ) of the material determines the deformation gradient F(r, t ) = (r, t ), the material velocity eld (r, t ) = t (r, t ) = v( (r, t ), t ) = v(x, t ), and the material velocity gradient F(r, t ) = L(r, t ) F(r, t )
via the chain rule, with L(x, t ) = v(x, t ). In what follows, the usual split L = D +W of L into its symmet
ric D = 12 (L + LT) and skew-symmetric W = 12 (L LT) parts is utilized. In particular, this latter quantity
is known in general continuum mechanics and in solid mechanics as the spin tensor, and as the vorticity tensor in uid mechanics.
Let A be a time-dependent spatial second-order tensor eld associated with the material. If the evolution of any such tensor is determined solely by the motion or ow of the material as a whole, then its pullback F1 AFT to the reference conguration is constant. In the language of rheology due to Oldroyd, such behavior is referred to as upper convected. In this case, A = L A + ALT holds, with A the material-time derivative.
For example, the left Cauchy-Green deformation tensor B = F FT from solid mechanics is such a tensor,
that is, B = L B + B LT. Due to slippage, however, the conformation tensor C deviates from being upper
convected in this sense. Indeed, the more general evolution relation
C = LC + C LT (DC + C D) (46) holds in terms of the so-called slippage parameter [14,35,41]. The time derivative in (46) depending in particular on is known as the GordonSchowalter derivative [42,43]. As discussed by Beris and Edwards [44] and later by ttinger [14], only in the case of = 0 (upper-convected behavior) or = 2 (i.e., lower-
convected behavior: FTC F constant) is the slippage process reversible or controllable. Specifically, they have shown that any other value of is inconsistent with the Poisson structure of reversible dynamics because the so-called Jacobi identity is violated. So in general, slippage involves irreversible dynamics. On the other hand,
4 Not to be confused with the right Cauchy-Green deformation C = FT F.
[integraldisplay]
0
1
s L f ds =
1
2 1 2 K 1 =
1
2 K (45)
Quasi-linear versus potential-based formulations 813
since it does not result in an increase of entropy, slippage is not dissipative; for the full details, the reader is referred to [35]. As will be seen in what follows, in the context of the transport-based form (5) of the uxforce relation and (6), this implies that the skew-symmetric part Lskw of the transport operator L is non-zero.
The explicit form of Lskw is discussed in the following. The full GordonSchowalter derivative can be split into two contributions,
Crev = LC + C LT,
Cirr = (DC + C D) =: X v,
(47)
and so C = Crev + Cirr, where the last equality in (47) denes the tensor X. For simplicity, all viscous effects,
thermal conduction, and diffusion effects will be ignored here and attention will be focused solely on slippage. Since the slippage involves both the conformation tensor C and the momentum density m (or velocity eld), there are also force and ux contributions corresponding to these variables. Specifically, one can make the following identications for the forces and uxes (see also 4.2.1 in [14]),
f = [parenleftbigg]
[parenrightbigg] = [parenleftbigg]
[parenrightbigg] = [parenleftbigg]
with the entropy density . From this, one obtains
L = [bracketleftbigg]
which is indeed skew-symmetric.
Given these results, we can now address the issue of a possible potential representation for slippage in complex uids. Since L = Lskw is skew-symmetric, Lsym is identically zero, according to (17). In this case,
pL(f ) = pL(0) 0 follows from (17), and sL(f ) = L f from (18). In conclusion, then, no dissipation potential
exists for the forceux relation representative of the GordonSchowalter derivative of a complex uid with slippage.
5.3 Example: chemical reactions
Our nal example represents the prototype of models for non-linear dynamics, namely homogeneous chemical reactions. In particular, we consider a system consisting of n species; for our purposes, it sufces in this context to restrict attention to a single reaction in this system including both forward and reverse reaction. Generalization to multiple reactions is straightforward but lends no new insight into the possible potential representation of the model.
From chemical kinetics (e.g., Chapter 4 in [45]), one has the basic evolution relation
Ni = i i (50)
for Ni in terms of the corresponding extent of reaction i and stoichiometric coefcient i = i Ni. In par
ticular, i = i i is the difference between the stoichiometric coefcients i and i of the product and
reactant species, respectively, during forward reaction. Again, for simplicity, assume that i is the same
for all i = 1, . . . , n. Neglecting further changes in the total volume, this system implies the interpretations ofJ =
N
[summationdisplay]
i=1
fm fC
1 v
C
,
(48)
j = [parenleftbigg]
jm jC
X CX v [parenrightbigg] =
Lf .
0 X
X 0
[bracketrightbigg] , (49)
,
F = kB A,
(51)
for the thermodynamic ux and force, respectively. The symbol A denotes the system chemical afnity in
terms of the species chemical potential i,
A = A + A,
A
=
1 kB
ii,
(52)
A
=
1 kB
N
[summationdisplay]
i=1
ii,
814 M. Htter, B. Svendsen
where A and A denote the afnities of the reactant and product species, respectively. The absolute tem
perature is denoted by , and kB is Boltzmanns constant. Lastly, the quasi-linear uxforce relation (5) then reduces to the scalar relation
J (F ) = L(F ) F . (53)
On this basis, the potential representation (17) reduces to
PL(F ) = F
1
[integraldisplay]
0
s L(s F ) F ds, (54)
assuming PL(0) = 0, and (18) to SL(F ) = 0 identically. Any form for L(F ) satisfying the conditions of the
representation induces PL(F ) in this fashion. One possibility in this regard is that
L(x, F ) = kB R(x) eA F1(eF/k
B
1) = R(x) A1 (eA
eA ) (55)
in terms of any non-negative function R(x) of the state variables x. Further,
PL(F ) = kB R eA [braceleftBig]
kB(eF/kB 1) F [bracerightBig] (56)
then follows from (54). Assuming that the chemical potentials i depend on Ni as an ideal gas, the mass action law is recovered.
Conversely, Grmela [15] works from the start with a potential of the form
(x, F ) = W (x) [braceleftBig]
eF/2kB + eF/2k
B
2[bracerightBig]
. (57)
The ux derived from this dissipation potential is
J = F =
W 2 kB
eF/2kB eF/2kB[parenrightBig] =
W2 kBeA/2 eA/2 [parenleftBig]eF/kB 1[parenrightBig]. (58)
This will be compatible with (55) when
W 2 kB
eA/2 eA/2 = kB R eA (59)
holds.
In summary, we observe that highly non-linear chemical reactions can be described equally well both by the quasi-linear uxforce relation (5) and also in terms of the dissipation potential. In the quasi-linear approach, the reaction rate L, (55), has been adjusted by an appropriate function in terms of the state variables x, in order to arrive at the usual form of the reaction equations. Analogously, in the case of the dissipation potential, the function W needed to be chosen appropriately as a function of the state variables x, (59). We close by noting that in the example of chemical reactions, the distinction between a dependence on the state variables x and the driving force F is blurred, because the latter is only a non-linear function of the former, and hence both potentials, PL(F) in (56) and (F) in (57), are equally admissible.
6 Discussion
The relation (37) documents clearly the dependence of the irreversible part (29) of a GENERIC on the transport relation (5). Related to this is the basic dependence (38) of the friction operator M on the transport coefcient L (e.g., [14]). As also seen in the examples discussed in this work, in particular, this dependence implies that M(x) depends on the entropy gradient x if and only if L(f ) depends on the thermodynamic force f . In addition, this state of affairs carries over to the case of a potential representation; indeed, as demonstrated by (39), such a representation exists for the GENERIC if and only if one does for the underlying uxforce relations.
Quasi-linear versus potential-based formulations 815
The extension of the basic representation (12) for general uxforce relations j(f ) on star-shaped regions in force space to deal with the possibility of non-symmetric transport coefcients L(f ) facilitated the treatment of a broader class of physical models. In particular, these include models for dissipation-free irreversible processes such as the case of slippage in complex uids, for which the dissipation potential is identically zero (i.e., no dissipation). In the more general case of irreversible and dissipative processes, the existence of a potential representation pL(f ) is contingent on the satisfaction of the higher-order symmetry restrictions (19) on the functional form of Lsym(f ). Again, in the context of (39), (19) are then also necessary for the existence of the analogous potential representation for the irreversible part of the GENERIC. It is worth emphasizing that this represents a mathematical problem which one has to solve if one is interested in deriving such a potential representation for the purpose of a variational formulation of the corresponding initial-boundary-value problem. Indeed, no new physics is involved.
Perhaps not surprisingly, it has been clearly demonstrated above that the class of constitutive relations based on a potential is a subset of the models captured by quasi-linear relations. In other words, by using a potential-based formulation, one uses a more restricted setup as compared to a quasi-linear setting. As stated by Grmela [46], one may deliberately choose to employ the more restrictive procedure, in a similar spirit as one also employs the Hamiltonian structure with the restrictive Jacobi identity [12,13,47] for formulating the reversible dynamics. On the one hand, it should be mentioned that the dissipation potential plays an important role in the formulation of out-of-equilibrium dynamics in the context of differential geometry and so-called Legendre time evolution [15]. Furthermore, it has recently been suggested [48] that the potential form of GENERIC emerges from an optimization principle. On the other hand, we point out that the quasi-linear form is a result of projection-operator techniques and of systematically accounting for the separation of timescales [14,49,50], and thus has a rm statistical basis. In addition, the quasi-linear form is closer to experimental procedures than is the potential-based form, as mentioned above. For example, the dynamic behavior of materials is often described in terms of their transport coefcients, that is, in terms of L(f). In the non-linear case, L(f) plays no direct role in the dissipation potential, and hence, it is difcult to formalize L(f) in a dissipation potential. Particularly, the potential p = (1/2)f L(f)f eventually results in a matrix of transport coefcients
different from L.
Acknowledgments We gratefully acknowledge stimulating discussions with Miroslav Grmela on the issue of dissipation potentials.
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