Bracket formulation as a source for the development of dynamic equations in continuum mechanics

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Abstract

The bracket formulation of the dynamic equations in nonequilibrium thermodynamics is examined here. After a short review of the early historical development of the subject, we present an introduction to the theoretical foundations of the one-generator bracket formalism, where the one generator is the Hamiltonian. The Hamiltonian represents the system’s total (extended) internal energy but, through a Legendre transform similar to that used in equilibrium thermodynamics, its derivatives can also be calculated from the available expression for the system’s extended free energy. First, the conservative component of the bracket is recognized as the Poisson bracket of Hamiltonian mechanics. Its properties are briefly reviewed and justified based on its connection to Hamiltonian mechanics. The Poisson structure of the conservative bracket is also manifested in a variety of other formalisms of the dynamic equations of nonequilibrium thermodynamics, thus providing a unilying connection. Next, the nonconservative, dissipative component is presented in the one-generator formalism, a more extensive treatment of which can be found in the research monograph by Beris and Edwards [A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, New York, 1994]. We further address here some of the more recent developments that have taken place since the publication of the above-mentioned work. Hence, the two-generator form of the bracket equations that corresponds to the GENERIC framework of description is considered next. The two generators are the Hamiltonian (or total energy), which drives the conservative part of the dynamics, and the entropy, which drives the dissipative part. Differences and similarities between the two- and the one-generator formalisms are pointed out. Finally, the advantage of the use of the bracket formulation is illustrated, as a number of recent applications are reviewed.

Introduction

The quest to formalize to basic governing equations in dynamics has been a long one dating at least from Newton’s Principia [1]. Since that time, the basic governing equations for classical dynamics have been reformulated several times giving rise to, among others, the Euler–Lagrange, Hamilton’s equations of motion and Poisson bracket formulations of classical mechanics [2], [3], [4], [5], [6] — see also Table 1 given in Appendix A for an example of their application to discrete particle dynamics. However, these reformulations have not just been an elegant mathematical exercise, expressing simply the same physics in a slightly different way. Instead, they have offered powerful new physics through the generalization and systematic extension and refinement of old concepts. As an example, it suffices to mention here (classical) quantum mechanics, which emerged from the Poisson bracket formulation of classical mechanics about three quarters of a century ago. In fact, for the principal theoretical constructs of quantum mechanics, the commutators statements like ‘Commutators in quantum mechanics are not only analogous to Poisson brackets, they are Poisson brackets,’ have been made [5].

However, although applications of the Poisson brackets in mechanics date as far back as 1812 when they were discovered by Poisson [7] as quoted in [4], their extension to continuum mechanics took almost a century to take place, till the work of Lie by the end of the 19th century and Poincaré in the beginning of the 20th century [5]. Nevertheless, it was not until even much later, in the middle 1960s, through the work of Arnold on ideal fluids [8], and especially in the beginning of the 1980s, primarily through the work of Marsden and his coworkers [9], [10], that noncanonical Poisson formalism for conservative continnum systems became widely known. Already by mid 1980s several Poisson bracket formulations of equations in continnum mechanics were known, like the compressible and incompressible Euler equations [8], [10], Maxwell–Vlasov [9], [11], KdV equations and others [12] — see also [5] for a brief account of the history of the development of the Poisson brackets in continuum mechanics.

As already alluded to, the main characteristics of all the systems for which Poisson bracket formalisms developed was that they were conservative, thus severely limiting their applicability to engineering practice. This limitation is inherent to a basic property of the Poisson bracket (antisymmetry). Indeed, if we look at the fundamental equation for a Poisson bracket dynamics, this is very simply describing the dynamics of any arbitrary functional F, in terms of a Poisson bracket operating on F and the system’s total (extended) internal energy E (also known as Hamiltonian, H):dFdt={F,E};conservativesystems.

Since the Poisson bracket is a bilinear antisymmetric operator, if it operates on E twice it gives zero. Thus, the total system’s energy is by necessity conserved, as we get trivially by applying Eq. (1) for F=E. Because of this inherent limitation, the Poisson bracket formalism was primarily used in order to construct additional flow invariants for Hamiltonian Systems and through them develop novel stability analysis methods [13], [14], [15]. The critical moment for the development of bracket formulations for general continuum Systems came in 1984 when almost simultaneously three papers appeared by Kaufman [16], Morrison [17] and Grmela [18] with an extended bracket description. In their bracket equation, the Poisson bracket discussed before still played a dominant role, but it was not any longer the only player. The dynamics was controlled by both the Poisson bracket, {.,.} and by another new bracket, [.,.], called ‘dissipative bracket’, which, however, was assumed to operate on F and S, where S is the system’s entropy:dFdt={F,E}+[F,S];non-conservationsystems.

The Poisson bracket appearing in Eq. (2) has exactly the same properties as the Poisson bracket mentioned above and in fact it is obtained exactly in the same way, i.e. by considering only conservative (usually convective) phenomena. However, by adding the dissipative bracket, Kaufman, Morrison and Grmela have allowed the description of nonconservative, dissipative dynamics and thus have significantly expanded the utility of the Poisson bracket itself, by making it an inherent component to nonequilibrium thermodynamics. Some of the most spectacular successes of these new formulations of nonequilibrium thermodynamics that were developed during the last decade as a consequence and/or in continuation to the Kaufman, Morrison, and Grmela work have indeed resulted by solely exploiting the properties of the Poisson bracket entering in these formulations [19], [20] — for further discussion see Section 3. Thus emerges the justification for devoting Appendix A for a review and analysis of the key properties of the Poisson bracket.

The original development of Kaufman et al. was noteworthy for yet one more reason: it opened the possibility for the merging of thermodynamics (manifested here in the presence of two thermodynamic potentials, E and S, which have now to be defined (extended) even outside thermodynamic equilibrium) with Hamiltonian mechanics. Moreover, it is of interest to point out here that all the key properties of the dissipative bracket (that were the subject later on of a much more detailed analysis [21], [22], [23], [24], [25]) were nevertheless brought-up in these three seminal early publications. Those properties are bilinearity, symmetry and positive definiteness [16], [17], [18] but also the introduction of ‘dissipative invariants’ [16], [17], [18], i.e. functionals D for which the dissipative bracket [D, B] for any functional B is identically zero. The energy and momentum functionals were proposed as dissipative invariants in plasma physics applications [16], [17], whereas the energy functional was proposed as a dissipative invariant for the Edelen dissipative equations in fluid mechanics that represent a generalization of the viscous (Newtonian) fluids [18]. In addition, although not recognized as such at the time, this particular bracket formulation expressed by Eq. (2) is an example of what we call now [24], [25] a ‘two-generator’ formalism, since two thermodynamic potentials are used to ‘drive’ the nonequilibrium dynamics.

The approach taken in [21] was more macroscopic (with the meaning that the applications have focused on variable fields defined on regular three-dimensional space as opposed to an extended space which also includes structural coordinates). The corrective ‘dissipation bracket’ was simply taken as the most general description that could represent dissipative dynamics subject to the conservation of energy constraint. In addition, the Onsager–Casimir reciprocal relations were systematically imposed and the nonnegative character of the entropy production required from the final equations. That led to a consistent description of the dynamics in terms of the Hamiltonian and its derivatives [21], what was later called a ‘one-generator’ formalism [24], [25].

This ‘two-generator’ bracket formalism found its most formal treatment and analysis in the GENERIC formalism [22], [23]. In these publications, a more systematic (and more general) way to construct the dissipative bracket was proposed, by using the degeneracy relations [22], [23] that arose in consequence of the energy dissipative invariants. However, the one- and two-generator formalisms usually provide the same results. First, in the presence of a local thermal equilibrium, in which case energy and entropy are interrelated and a temperature can be defined by their partial derivative, it is, in general, possible to express the entropy in terms of the energy (i.e. the Hamiltonian) and thus we can deduce from the ‘two-generator’ an equivalent ‘one-generator’ bracket formalism. As shown in a recent publication [24], for macroscopic systems, this ‘one-generator’ formalism leads to identical results as the one-generator formalism postulated in [21]. Nevertheless, outside a local thermal equilibrium the two potentials are not, in general, related (under which conditions even a temperature cannot be defined, as, for example, is the case of the Boltzmann equation [26]), and in this case only a two-generator formalism is possible which cannot be further reduced to a one-generator formalism. Moreover, only the two-generator formalism was found to generate results consistent with polymer kinetic theory results. As a recent detailed study of the microscopic equations corresponding to the kinetic theory of polymer dynamics showed [25], it is still possible to extend the development of the one-generator bracket formalism following the procedure outlined in [21] even for those extended phase space systems. However, there were small but noticeable differences with the results obtained from the two-generator formalism that were the only ones in agreement with kinetic theory [25]. Nevertheless, the differences were limited to a term weighting spatial gradients of the chain distribution function, so one does not expect them to be important but only when phenomena occurring within a length scale smaller than a typical chain length are present [25]. An overview of these two bracket formalisms is offered in Section 2.

Grmela was the first to offer the bracket reformulation of the dynamics for elastic and viscoelastic continua using the Poisson bracket that he developed empirically for a nonlinear elastic system and the dissipative bracket for the Upper-Convected Maxwell (UCM) fluid [17]. The bracket formulation of the dynamics of several other viscoelastic fluids quickly followed, first by Grmela himself [27], [28], [29], [30] and then by others [31], as well as for other macroscopic continuum systems like polymeric liquid crystals [32], [33], [34], [35]. This activity on bracket formulations of continuum systems culminated to the publication of the research monograph by Beris and Edwards that focused particularly on a special one-generator generalized brackets [21]. In there, a first attempt was made for a systematic development of the bracket formulation based on Hamiltonian mechanics and irreversible thermodynamics (the basics of which were reviewed in the first part of the above mentioned research monograph). Moreover, in the second part, a rather detailed analysis of the application of the bracket approach for the modeling of the dynamics of complex continuum systems (continua endowed with an internal microstructure) was offered. Particular emphasis was placed in the application of the generalized bracket to polymer fluid mechanics and liquid crystals, further expanding the works already mentioned above. In addition, more complex cases were also examined, such as the influence of surface effects on the flow and conformation of polymer solutions, thermal and mass relaxation, two-fluid description of plasmas, and others [21].

Although not easily recognized (in part because they were provided amidst results that have also been obtained before by other theoretical means of investigation) several contributions were made in that work related to the modeling of the dynamic behavior of complex continua. First, a variety of well-known (and experimentally successful) constitutive models for viscoelastic and liquid crystalline systems were shown to exhibit a Hamiltonian structure and to be thermodynamically consistent simply by being amenable to a bracket formulation description. Second, models for which in the past it was shown that under certain conditions an aphysical behavior could arise (such as, for example, the White–Metzner model of viscoelasticity [36]) were also shown to be inconsistent with the bracket formalism. Third and most interesting, more complex models, which have never before been tested in complex flow simulations, were shown to exhibit inconsistencies. Those however could be removed through suitable modifications of the original models, guided by the physics and the bracket formalism (as, for example, in the case of the Marrucci–Acierno model [37], [38], discussed in Chapter 8 of [21]). Work is currently in progress in collaboration with R. Keunings and X. Gallez, trying to assess the value of those formulation-driven model modifications.

A special case of these formulation-driven model corrections that needs to be emphasized more is the one that occurred when a serious flaw in a previous approximation of a finer model by a coarser one was detected and corrected. This is to be attributed not only to the constraints imposed by the formalism but also in this case to the systematic presentation of the modeling of the same physical system at several levels of description, which is facilitated by the bracket formalism with further consistency equations and conditions specified to assure their mutual compatibility. The case at hand is the second order tensor representation for the dynamics of rigid rods (a popular model for liquid crystalline solutions [39]) where the bracket formalism has shown that it is important to allow for a mixed convected time derivative in the evolution equation for the second order structure tensor, as discussed in Chapter 11 of [5]. Previous approximations using the upper convected time derivative [39] were already seen to be defective in that the resulting equations could not generate some of the experimentally measured dynamic behavior (in this particular case, tumbling of the rigid rods [40]). Previous investigators did not know how to correct that flaw of the second order tensor approximation and they had to resort to the solution of the full distribution function of the rigid rod diffusion equation [40], [41]. However, as later calculations with the bracket-modified equations have shown [42], [43], that is not necessary when the proper mixed time derivative is used, thus saving considerable computational work.

Since the publication of [21], the major new development that affected the bracket description of continua has no doubt been the development of the very systematic GENERIC formalism [22], [23] that formalized the two-generator bracket formalism and also expressed it in terms of operator matrices. As it turns out from the analysis offered in two recent papers [24], [25], the two formulations, one-generator bracket and GENERIC are very much related. Indeed, as far as the Poisson bracket is concerned, it is fair to say that there are identical. However, substantial new information was brought up as far as the second operator, the one governing dissipation, is concerned. By using the entropy as the generator in that case, and by reformulating the old ‘dissipative invariants’ as new degeneracy conditions, GENERIC offered a distinctly different approach for constructing models for dissipation. Furthermore, by performing a thorough study of microscopic models from polymer kinetic theory it was convincingly shown that the rules postulated made physical sense. Thus, when a separate study [25] showed that under certain circumstances it is possible to have differences between the one and two-generator approaches, even when a one-generator formalism can be constructed (i.e. in the presence of local thermal equilibrium), the need to use in the most general case a two-generator bracket formalism was evident. The two-generator bracket formalism is therefore also described at the end of the next section.

Since the appearance of [21], several new applications using properties of the brackets have emerged, primarily taking advantage of the structure of the Poisson bracket. Those are discussed briefly in Section 3 of this report. This brings us to the discussion of the objectives of the present work. These are multiple in number. First, we want to offer a comprehensive review of the history and theoretical foundation of the subject. This is primarily accomplished in the introduction and the two Appendices. Second, we want to present the essential elements of the bracket formalism in nonequilibrium thermodynamics as we understand them now and in order to make the distinction between the one and two-generator bracket formalisms as well as between the modeling approach followed in [21] and that proposed through the operator matrices in [22], [23]. Thus, in Section 2, the bracket is also presented in a more general, ‘two-generator,’ form than the ‘one-generator’ to be found in [21]. However, in most cases the two formalisms give identical results [24], which therefore still leaves a considerable domain of applicability and usefulness to the one-generator formalism, in particular given its higher simplicity and higher order structure that it possesses as compared to the two-generator approach. Third, we want to offer a brief review for the applications of the bracket formalism, especially the new ones established since 1994. We have reviewed a selection of pre-1994 applications in this section; more can be found in [21]. For the new ones since 1994, we offer a brief discussion in Section 3. Finally, in Section 4 we offer our concluding remarks.

Section snippets

Elements of the bracket formulation

The bracket formulation of the dynamic equations in continuum mechanics concerns the dynamics of a continuum system between ‘quasi-equilibrium’ states, in the sense that the states can be described in terms of a small, finite number, of variables, x. At a given time, t, those variables are assumed to be continuous fields in an extended (in general) space Ω. This extended space can possibly involve, in addition to the usual 3-dimensional space, R3, spaces representing internal degrees of freedom

New applications since 1994

The one- and two-generator bracket (GENERIC) formalisms, both in its bracket as well as in its operator form, have been used in several modeling applications. From these, we will report here some examples of either the one-generator formalism or the structural properties of the Poisson bracket. There are at least three applications that we can cite belonging to this last category. In the first one [50], use was made of the requirement that, as we move from distributions to moment equations, the

Concluding remarks

The bracket formulation of the dynamic equations in nonequilibrium thermodynamics has been reviewed in its two forms, involving one or two generators. That brought the theory developed in [21] as a one-generator form in close comparison to a two-generator form, which corresponds to the GENERIC formalism [22], [23]. In most cases the equations produced by either one of these forms are identical, in some (those arising from kinetic theory) there are some differences but they are small

Acknowledgements

Enlightening discussions with B.J. Edwards, H.C. Öttinger, R. Keunings as well as the audience of a short course, offered within the framework of advanced graduate courses in mechanics ‘GRASCOM 1999-2000’ in the Division of Applied Mechanics, at the Université catholique de Louvain, Louvain-la-Neuve, Belgium, are gratefully acknowledged. Also acknowledged are the partial support by the ARC 97/02-210 project of the Communauté Française de Belgique and a Fulbright Scholar Award in Belgium for

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