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. 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 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.

Reexamination of multi-component non-ideal polymer solution based on the general equation for nonequilibrium reversible-irreversible coupling

We reexamine nonequilibrium thermodynamics of multi-component fluids that undergo chemical reactions and reveal how to describe it in the context of GENERIC (general equation for the nonequilibrium reversible-irreversible coupling) framework. In the former part of this paper, we study monomer solutions. The independent variables are chosen so that the symmetry in the thermodynamic description of the mixed solution is kept, while avoiding the redundancy in the independent variables that describe the system. In the latter part of this paper, we apply this approach to polymer solutions. One of the species of the solution is the unreactive polymer chain represented by the bead-spring model. The polymer solution is neither dilute nor ideal. We construct the solution entropy so that the contributions from mixing and chain conformation are fully separated. The pressure tensor derived from such entropy with the help of the degeneracy requirement is slightly different from that derived previously for a dilute polymer solution. We discuss how the difference occurs.

Nonlinear coupled equations for electrochemical cells as developed by the general equation for nonequilibrium reversible-irreversible coupling.

We show how the Butler-Volmer and Nernst equations, as well as Peltier effects, are contained in the general equation for nonequilibrium reversible and irreversible coupling, GENERIC, with a unique definition of the overpotential. Linear flux-force relations are used to describe the transport in the homogeneous parts of the electrochemical system. For the electrode interface, we choose nonlinear flux-force relationships. We give the general thermodynamic basis for an example cell with oxygen electrodes and electrolyte from the solid oxide fuel cell. In the example cell, there are two activated chemical steps coupled also to thermal driving forces at the surface. The equilibrium exchange current density obtains contributions from both rate-limiting steps. The measured overpotential is identified at constant temperature and stationary states, in terms of the difference in electrochemical potential of products and reactants. Away from these conditions, new terms appear. The accompanying energy flux out of the surface, as well as the heat generation at the surface are formulated, adding to the general thermodynamic basis.

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.

On the Formulation of Continuum Thermodynamic Models for Solids as General Equations for Non-equilibrium Reversible-Irreversible Coupling

The purpose of this work is the comparison of some aspects of the formulation of material models in the context of continuum thermodynamics (e.g., Silhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997) with their formulation in the form of a General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC: e.g., Grmela and Ottinger in Phys. Rev. E 56: 6620–6632, 1997; Ottinger and Grmela in Phys. Rev. E 56: 6633–6655, 1997; Ottinger in Beyond equilibrium thermodynamics, Wiley, New York, 2005; Grmela in J. Non-Newton. Fluid Mech. 165: 980–998, 2010). A GENERIC represents a generalization of the Ginzburg-Landau model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to formulate non-equilibrium thermodynamic models for complex fluids, it has recently been applied to anisotropic inelastic solids in a Eulerian setting (Hutter and Tervoort in J. Non-Newton. Fluid Mech. 152: 45–52, 2008; 53–65, 2008; Adv. Appl. Mech. 42: 254–317, 2009) as well as to damage mechanics (Hutter and Tervoort in Acta Mech. 201: 297–312, 2008). In the current work, attention is focused for simplicity on the case of thermoelastic solids with heat conduction and viscosity in a Lagrangian setting (e.g., Silhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997, Chaps. 9–12). In the process, the relation of the two formulations to each other is investigated in detail. A particular point in this regard is the concept of dissipation and its model representation in both contexts.

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.