General Equation For Nonequilibrium Reversible-irreversible Coupling Research Articles

Comparison of the symmetric hyperbolic thermodynamically compatible framework with Hamiltonian mechanics of binary mixtures

AbstractHow to properly describe continuum thermodynamics of binary mixtures where each constituent has its own momentum? The Symmetric Hyperbolic Thermodynamically Consistent (SHTC) framework and Hamiltonian mechanics in the form of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) provide two answers, which are similar but not identical, and are compared in this article. They are compared both analytically and numerically on several levels of description, varying in the amount of detail. Namely, a reduction to a more common one-momentum setting is shown, where the effects of the second momentum translate into diffusive fluxes. Both SHTC and GENERIC can thus be interpreted as a method specifying diffusive flux in standard theory. The GENERIC equations, stemming from the Liouville equation, contain terms expressing self-advection of the relative velocity by itself, which lead to a vorticity-dependent diffusion matrix after the reduction. The SHTC equations, on the other hand, do not contain such terms. We also discuss the possibility to formulate a theory of mixtures with two momenta and only one temperature that is compatible with the Liouville equation and possesses the Hamiltonian structure, including Jacobi identity.

On the role of geometry in statistical mechanics and thermodynamics. II. Thermodynamic perspective

The General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) provides the structure of mesoscopic multiscale dynamics that guarantees the emergence of equilibrium states. Similarly, a lift of the GENERIC structure to iterated cotangent bundles, called a rate GENERIC, guarantees the emergence of the vector fields that generate the approach to equilibrium. Moreover, the rate GENERIC structure also extends Onsager’s variational principle. The maximum entropy principle in the GENERIC structure becomes the Onsager variational principle in the rate GENERIC structure. In the absence of external forces, the rate entropy is a potential that is closely related to the entropy production. In the presence of external forces when the entropy does not exist, the rate entropy still exists. While the entropy at the conclusion of the GENERIC time evolution gives rise to equilibrium thermodynamics, the rate entropy at the conclusion of the rate GENERIC time evolution gives rise to rate thermodynamics. Both GENERIC and rate GENERIC structures are put into the geometrical framework in the first paper of this series. The rate GENERIC is also shown to be related to Grad’s hierarchy analysis of reductions of the Boltzmann equation. Chemical kinetics and kinetic theory provide illustrative examples. We introduce rate GENERIC extensions (and thus also Onsager-variational-principle formulations) of both chemical kinetics and the Boltzmann kinetic theory.

On the role of geometry in statistical mechanics and thermodynamics. I. Geometric perspective

This paper contains a fully geometric formulation of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). Although GENERIC, which is the sum of Hamiltonian mechanics and gradient dynamics, is a framework unifying a vast range of models in non-equilibrium thermodynamics, it has unclear geometric structure due to the diverse geometric origins of Hamiltonian mechanics and gradient dynamics. The difference can be overcome by cotangent lifts of the dynamics, which leads, for instance, to a Hamiltonian form of gradient dynamics. Moreover, the lifted vector fields can be split into their holonomic and vertical representatives, which provides a geometric method of dynamic reduction. The lifted dynamics can be also given physical meaning, here called the rate-GENERIC. Finally, the lifts can be formulated within contact geometry, where the second law of thermodynamics is explicitly contained within the evolution equations.

On compatibility of the natural configuration framework with general equation for non-equilibrium reversible–irreversible coupling (GENERIC): Derivation of anisotropic rate-type models

Within the framework of natural configurations developed by Rajagopal and Srinivasa, evolution within continuum thermodynamics is formulated as evolution of a natural configuration linked with the current configuration. On the other hand, withing the General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) framework, the evolution is split into Hamiltonian mechanics and (generalized) gradient dynamics. These seemingly radically different approaches have actually a lot in common and we show their compatibility on a wide range of models. Both frameworks are illustrated on isotropic and anisotropic rate-type fluid models. We propose an interpretation of the natural configurations within GENERIC and vice versa (when possible). • Equilibrium thermodynamics is considered to be a closed theory • Non-equilibrium thermodynamics, on the contrary, is not • In many frameworks an intensive research has been conducted separately • However, less attention is paid to their relation and general understanding • We compare the Natural Configuration framework and GENERIC on particular models

On the relations between large-scale models of superfluid helium-4

Superfluid helium-4 is characterized by extremely small values of kinematic viscosity, and its thermal conductivity can be huge, orders of magnitude larger than that of water or air. Additionally, quantum vortices may exist within the fluid. Therefore, its behavior cannot be explained by using the classical tools of Newtonian fluid mechanics, and, over the years, a few alternative models have been proposed. In order to highlight similarities and differences between these models, we recast them within a unifying framework, the general equation for non-equilibrium reversible-irreversible coupling (GENERIC). We begin by comparing the original two-fluid model, developed by Tisza and Landau, with the Hall–Vinen–Bekarevich–Khalatnikov model, both prescribing two types of fluid motion and two fluid densities, at flow scales appreciably larger than the typical distance between quantum vortices. We find from the geometrical structure of the models that only one fluid density plays the role of state variable, which should be taken into account when choosing an adequate expression for the free energy. We also recast within the GENERIC framework the one-fluid model of superfluid helium-4, where the inviscid component of two-fluid models is replaced by a caloric quantity, such as entropy. We find that the corresponding geometrical structures are analogous, with the roles of density and entropy swapped. In short, our work demonstrates that the studied models are compatible with each other, at least when focusing on the reversible parts of the models.

Thermomass Theory in the Framework of GENERIC

Thermomass theory was developed to deal with the non-Fourier heat conduction phenomena involving the influence of heat inertia. However, its structure, derived from an analogy to fluid mechanics, requires further mathematical verification. In this paper, General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework, which is a geometrical and mathematical structure in nonequilibrium thermodynamics, was employed to verify the thermomass theory. At first, the thermomass theory was introduced briefly; then, the GENERIC framework was applied in the thermomass gas system with state variables, thermomass gas density ρh and thermomass momentum mh, and the time evolution equations obtained from GENERIC framework were compared with those in thermomass theory. It was demonstrated that the equations generated by GENERIC theory were the same as the continuity and momentum equations in thermomass theory with proper potentials and eta-function. Thermomass theory gives a physical interpretation to the GENERIC theory in non-Fourier heat conduction phenomena. By combining these two theories, it was found that the Hamiltonian energy in reversible process and the dissipation potential in irreversible process could be unified into one formulation, i.e., the thermomass energy. Furthermore, via the framework of GENERIC, thermomass theory could be extended to involve more state variables, such as internal source term and distortion matrix term. Numerical simulations investigated the influences of the convective term and distortion matrix term in the equations. It was found that the convective term changed the shape of thermal energy distribution and enhanced the spreading behaviors of thermal energy. The distortion matrix implies the elasticity and viscosity of the thermomass gas.

Energy‐momentum‐entropy consistent numerical methods for large‐strain thermoelasticity relying on the GENERIC formalism

SummaryIn the present paper, structure‐preserving numerical methods for finite strain thermoelastodynamics are proposed. The underlying variational formulation is based on the general equation for nonequilibrium reversible‐irreversible coupling (GENERIC) formalism and makes possible the free choice of the thermodynamic state variable. The notion “GENERIC consistent space discretization” is introduced, which facilitates the design of Energy‐Momentum‐Entropy (EME) consistent schemes. In particular, three alternative EME schemes result from the present approach. These schemes are directly linked to the respective choice of the thermodynamic variable. Numerical examples confirm the structure‐preserving properties of the newly developed EME schemes, which exhibit superior numerical stability.

A general framework for the thermodynamically consistent time integration of open nonlinear thermoelastic systems

AbstractThis work deals with Energy‐Momentum‐Entropy (EME) consistent methods of open thermoelastic systems. While Energy‐Momentum (EM) preserving integrators are well‐known for conservative mechanical systems, Romero introduced in [1] the class of thermodynamically consistent (TC) integrators for coupled thermomechanical systems, which further respect symmetries of the underlying coupled system and are capable of conserving associated momentum maps and can therefore also be termed EME schemes in analogy to EM schemes for conservative systems. As mathematical framework for the geometric structure of the non‐equilibrium thermodynamics the GENERIC (General Equation for Non‐Equilibrium Reversible‐Irreversible Coupling) framework, originally introduced for complex fluids in [2], is used. Since the GENERIC framework does not depend on a specific choice of the thermodynamical state variables [3], we explore the structure of the GENERIC framework using the entropy density (see e.g. [4]), the absolute temperature (see e.g. [5]) and further the internal energy density as thermodynamical state variable. Applying the notion of a discrete gradient in the sense of [6] leads to an EME integrator. As boundary conditions rely on the specific choice of the thermodynamical state variable we extend the GENERIC framework to be suitable for open systems following the procedure in [7].

Beyond equilibrium thermodynamics in the low temperature plasma processor

Low temperature plasmas are open driven thermodynamic systems capable of increasing the free energy of the mass that flows through them. An interesting thing about low temperature plasmas is that different species have different temperatures at the same location in space. Since thermal equilibrium cannot be assumed, many of the familiar results of equilibrium thermodynamics cannot be applied in their familiar form to predict, e.g., the direction of a chemical reaction. From the perspective of classical processing governed by thermal equilibrium, examples of highly unexpected gas-phase chemical reactions (CO2 dissociation, NO, N2H4, O3 synthesis) and solid material transformations (surface activation, size-focusing, and hyperdoping) promoted by low temperature plasmas are presented. The lack of a known chemical reaction equilibrium criterion prevents assessment of predictive kinetics models of low temperature plasmas, to ensure that they comply with the laws of thermodynamics. There is a need for a general method to predict chemical reaction equilibrium in low temperature plasmas or an alternative method to establish the thermodynamic admissibility of a proposed kinetics model. Toward those ends, two ideas are explored in this work. The first idea is that chemical reactions in low temperature plasmas proceed toward a thermal equilibrium state at an effective temperature intermediate between the neutral gas temperature and the electron temperature. The effective temperature hypothesis is simple, and surprisingly is adequate for elucidation in some systems, but it lacks generality. The general equation for nonequilibrium reversible–irreversible coupling (GENERIC) is a general beyond equilibrium thermodynamics framework that can be used to rigorously establish the thermodynamic admissibility of a set of dynamic modeling equations, such as a kinetic model, without knowledge of the final state that the system is tending toward. The use of GENERIC is described by way of example using a two-temperature hydrodynamic model from the literature. The conclusion of the GENERIC analysis presented in this work is that the concept of superlocal equilibrium is thermodynamically admissible and may be applied to describe low temperature plasmas, provided that appropriate terms are included for exchange of internal energy and momentum between different species that may have different temperatures and bulk velocities at the same location in space. The concept of superlocal equilibrium is expected to be of utility in future work focused on deriving equilibrium criteria for low temperature plasmas.

A temperature‐based GENERIC approach for the thermodynamically consistent integration of thermoelastic solids

AbstractThis work deals with the thermodynamically consistent (TC) time integration of thermoelastic systems with polyconvex density functions using the notion of the tensor‐cross‐product. While energy‐momentum preserving integrators are well‐known for conservative (isothermal) mechanical systems, Romero introduced in [7, 8] the new class of TC integrators. While [8] dealt with the sample application of thermo‐elastodynamics, the scope of application was extended in [2] to coupled thermo‐viscoelastodynamics in temperature form. A first step towards the systematic design of a TC integrator is to cast the evolution equations into the GENERIC (General Equation for Non‐Equilibrium Reversible‐Irreversible Coupling) framework [6] which reveals additional underlying physical structures of the system. Relying on a polyconvex density function and using the notion of the tensor‐cross‐product [1] we arrive at a polyconvex version of the GENERIC framework. Further applying the notion of a discrete gradient leads to a TC integrator.Using the entropy as the thermodynamical state variable as in [5, 8] the GENERIC framework possesses an easy structure. However, this choice of thermodynamical state variable only allows to prescribe entropy Dirichlet boundary conditions directly. This drawback can be compensated by using Lagrange‐multipliers to be able to handle temperature Dirichlet boundary conditions leading to an extended system of algebraic equations to be solved, see [5]. Alternatively, the present work uses the temperature as the thermodynamical state variable, see also [2, 3] and the use of an energy‐based Newton‐Raphson termination criterion. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Thermodynamic Model Formulations for Inhomogeneous Solids with Application to Non-isothermal Phase Field Modelling

Abstract The purpose of the current work is the comparison of thermodynamic model formulations for chemically and structurally inhomogeneous solids at finite deformation based on “standard” non-equilibrium thermodynamics [SNET: e. g. S. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North Holland, 1962] and the general equation for non-equilibrium reversible–irreversible coupling (GENERIC) [H. C. Öttinger, Beyond Equilibrium Thermodynamics, Wiley Interscience, 2005]. In the process, non-isothermal generalizations of standard isothermal conservative [e. g. J. W. Cahn and J. E. Hilliard, Free energy of a non-uniform system. I. Interfacial energy. J. Chem. Phys. 28 (1958), 258–267] and non-conservative [e. g. S. M. Allen and J. W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979), 1085–1095; A. G. Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, 1983] diffuse interface or “phase-field” models [e. g. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Modern Phys. 49 (1977), 435–479; N. Provatas and K. Elder, Phase Field Methods in Material Science and Engineering, Wiley-VCH, 2010.] for solids are obtained. The current treatment is consistent with, and includes, previous works [e. g. O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D 43 (1990), 44–62; O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model. Phys. D 69 (1993), 107–113] on non-isothermal systems as a special case. In the context of no-flux boundary conditions, the SNET- and GENERIC-based approaches are shown to be completely consistent with each other and result in equivalent temperature evolution relations.

Formulation of the relativistic heat equation and the relativistic kinetic Fokker–Planck equations using GENERIC

In this paper, we formulate the relativistic heat equation and the relativistic kinetic Fokker–Planck equations into the GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) framework. We also show that the relativistic Maxwellian distribution is the stationary solution of the latter. The GENERIC formulation provides an alternative justification that the two equations are meaningful relativistic generalizations of their non-relativistic counterparts.

An energy‐entropy‐consistent time stepping scheme for nonlinear thermo‐viscoelastic continua

This paper deals with an energy‐entropy‐consistent time integration of a thermo‐viscoelastic continuum in Poissonian variables. The four differential evolution equations of first‐order are transformed by a new General Equation for Non‐Equilibrium Reversible‐Irreversible Coupling (GENERIC) format into a matrix‐vector notation. Since the entropy is a primary variable, we include thermal constraints to affect the temperatures at the boundary of the body. This enhanced GENERIC format with thermal constraints yields with the related degeneracy conditions structure preservation properties for a system with thermal constraints. The properties of an isolated system are in addition to a constant total linear and angular momentum, the constant total energy, an increasing total entropy and a decreasing Lyapunov function. The last one is a stability criterion for thermo‐viscoelastic systems and also for unisolated systems without loads valid. The discretization in time is done with a new TC (Thermodynamically Consistent) integrator. This ETC integrator is constructed such, that the algorithmic structural properties after the space‐time discretization reflect the underlying enhanced GENERIC format with thermal constraints. As discretization in space the finite element method is used. A projection of the test function of the thermal evolution equation is necessary for an energy‐consistent discretization in space. The enhanced GENERIC format with thermal constraints, which is here given in the strong evolution equations, contains the external loads. The consistency properties are discussed for representative numerical examples with different boundary conditions. The coupled mechanical system is solved with a multi‐level Newton‐Raphson method based on a consistent linearization.

Nonequilibrium thermodynamics of the shear-transformation-zone model.

The shear-transformation-zone (STZ) model has been applied numerous times to describe the plastic deformation of different types of amorphous systems. We formulate this model within the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework, thereby clarifying the thermodynamic structure of the constitutive equations and guaranteeing thermodynamic consistency. We propose natural, physically motivated forms for the building blocks of the GENERIC, which combine to produce a closed set of time evolution equations for the state variables, valid for any choice of free energy. We demonstrate an application of the new GENERIC-based model by choosing a simple form of the free energy. In addition, we present some numerical results and contrast those with the original STZ equations.

Dynamics of multiphase systems with complex microstructure. II. Particle-stabilized interfaces

In this paper we use the GENERIC (general equation for nonequilibrium reversible-irreversible coupling) nonequilibrium thermodynamics framework to derive constitutive equations for the surface extra stress tensor of an interface stabilized by a two-dimensional suspension of anisotropic colloidal particles. The dependence of the surface stress tensor on the microstructure of the interface is incorporated through a dependence on a single tensorial structural variable, characterizing the average orientation of the particles. The constitutive equation for the stress tensor is combined with a time-evolution equation describing the changes in the orientation tensor as a result of the applied deformation field. We examine the predictions of the model in in-plane steady shear flow, in-plane oscillatory shear flow, and oscillatory dilatational flow. The model is able to predict the experimentally observed shear thinning behavior in surface shear flow, and also the experimentally observed emergence of even harmonics in the frequency spectrum of the surface stress in oscillatory dilatational flow. Our results show that the highly nonlinear stress-deformation behavior of interfaces with a complex microstructure can be modeled well using simple structural models like the one presented here.

Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Thermodynamic models for viscoplastic solids are often formulated in the context of continuum thermodynamics and the dissipation principle. The purpose of the current work is to show that models for such material behavior can also be formulated in the form of a General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC), see, e.g., Grmela and Ottinger (Phys Rev E, 56:6620–6632, 1997), Ottinger and Grmela (Phys Rev E, 56:6633–6655, 1997), Grmela (J Non-Newtonian Fluid Mech, 165:980–986, 2010). A GENERIC combines Hamiltonian-dynamics-based modeling of time-reversible processes with Onsager–Casimir-based modeling of time-irreversible processes. The result is a model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to model complex fluids, it has recently been applied to anisotropic inelastic solids in Eulerian (Hutter and Tervoort, in J Non-Newtonian Fluid Mech, 152:45–52, 2008; Hutter and Tervoort, in J Non-Newtonian Fluid Mech, 152:53–65, 2008; Hutter and Tervoort, in Adv Appl Mech, 42:254–317, 2008) and Lagrangian (Hutter and Svendsen, in J Elast 104:357–368, 2011) settings, as well as to damage mechanics. For simplicity, attention is focused in the current work on the case of thermoelastic viscoplasticity. Central to this formulation is a GENERIC-based form for the viscoplastic flow rule. A detailed comparison with the formulation based on continuum thermodynamics and the dissipation principle is given.

Rheological properties and microstructural evolution of semi-flexible fiber suspensions under shear flow

The mechanical properties of fiber-reinforced polymers are highly dependent on fiber orientation. A key factor is fiber flexibility, which varies with the intrinsic properties of the fiber, its aspect ratio, and the force of the flow field. In this work, the rheological behavior of model suspensions consisting of a Newtonian silicone oil and fibers with different flexibilities has been studied in transient shear flow. The viscous and elastic properties of the fiber suspensions were shown to increase markedly with fiber flexibility. The suspensions exhibited a viscosity overshoot in stress growth experiments and the magnitude and width of the overshoot became larger as the fiber flexibility increased. The evolution of the suspensions microstructure has also been monitored at different shear rates. The components of the second-order orientation tensor at different times (or strain) were calculated using the images obtained from the visualization experiments. Those experiments showed that fiber orientation was dependent on shear rate and magnitude of applied strain. At low shear rates, rigid fibers oriented to a larger extent in the flow direction as compared to more flexible fibers and their orientation was also faster. Finally, a mesoscopic model in the framework of the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) has been used to predict the rheological behavior of semi-flexible fiber suspensions as well as their microstructure. The model takes into account fiber-fiber interactions and also the semi-flexible nature of the fibers. The predictions of the model are in good qualitative agreement with the rheological and visualization data.

Smoothed particle hydrodynamic model for viscoelastic fluids with thermal fluctuations

We present a fluid-particle model for a polymer solution in nonisothermal situations. The state of the fluid particles is characterized by the thermodynamic variables and a configuration tensor that describes the underlying molecular orientation of the polymer molecules. The specification of very simple physical mechanisms inspired by the dynamics of single polymer molecules allows one, with the help of the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) formalism, to derive the equations of motion for a set of fluid particles carrying polymer molecules in suspension. In the simplest case of Hookean dumbbells we recover a fluid-particle version of the Oldroyd-B model in which thermal fluctuations are included consistently. Generalization to more complex viscoelastic models, such as finitely extensible nonlinear elastic Peterlin (FENE-P) model, with the proper introduction of thermal fluctuations is straightforward.

Thermodynamic admissibility of the extended Pom-Pom model for branched polymers

The thermodynamic consistency of the eXtended Pom-Pom (XPP) model for branched polymers of Verbeeten et al. [W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model, J. Rheol. 45 (4) (2001) 823–843; W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model (vol 45, pg 823–843, 2001), J. Rheol. 45 (6) (2001) 1489] as well as its modified version [J. van Meerveld, Note on the thermodynamic consistency of the integral pom-pom model, J. Non-Newtonian Fluid Mech. 108 (1–3) (2002) 291–299] is investigated from the perspective of non-equilibrium thermodynamics, namely the General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) framework. The thermodynamic admissibility of the XPP model is shown for both its original and modified form. According to the GENERIC formalism, the parameter α introduced by Verbeeten et al. to predict non-zero second normal stress in shear flows must fulfill the condition 0 ≤ α ≤ 1.