Thermal Non-Equilibrium Flows in Three Space Dimensions

We study the equations describing the motion of a thermal non-equilibrium gas in three space dimensions. It is a hyperbolic system of six equations with a relaxation term. The dissipation mechanism induced by the relaxation is weak in the sense that the Shizuta-Kawashima criterion is violated. This implies that a perturbation of a constant equilibrium state consists of two parts: one decays in time while the other stays. In fact, the entropy wave grows weakly along the particle path as the process is irreversible. We study thermal properties related to the well-posedness of the nonlinear system. We also obtain a detailed pointwise estimate on the Green’s function for the Cauchy problem when the system is linearized around an equilibrium constant state. The Green’s function provides a complete picture of the wave pattern, with an exact and explicit leading term. Comparing with existing results for one dimensional flows, our results reveal a new feature of three dimensional flows: not only does the entropy wave not decay, but the velocity also contains a non-decaying part, strongly coupled with its decaying one. The new feature is supported by the second order approximation via the Chapman-Enskog expansions, which are the Navier-Stokes equations with vanished shear viscosity and heat conductivity.     

Entropy in self-similar shock profiles

In this paper, we study the structure of a gaseous shock, and in particular the distribution of entropy within, in both a thermodynamics and a statistical mechanics context. The problem of shock structure has a long and distinguished history that we review. We employ the Navier–Stokes equations to construct a self-similar version of Becker’s solution for a shock assuming a particular (physically plausible) Prandtl number; and that solution reproduces the well-known result of Morduchow & Libby that features a maximum of the equilibrium entropy inside the shock profile. We then construct an entropy profile, based on gas kinetic theory, that is smooth and monotonically increasing. The extension of equilibrium thermodynamics to irreversible processes is based in part on the assumption of local thermodynamic equilibrium. We show that this assumption is not valid except for the weakest shocks. We conclude by hypothesizing a thermodynamic nonequilibrium entropy and demonstrating that it closely estimates the gas kinetic nonequilibrium entropy within a shock.     

Viscous regularization of the full set of nonequilibrium‐diffusion Grey Radiation‐Hydrodynamic equations

A viscous regularization technique, based on the local entropy residual, was proposed by Delchini et al. (2015) to stabilize the nonequilibrium‐diffusion Grey Radiation‐Hydrodynamic equations using an artificial viscosity technique. This viscous regularization is modulated by the local entropy production and is consistent with the entropy minimum principle. However, Delchini et al. (2015) only based their work on the hyperbolic parts of the Grey Radiation‐Hydrodynamic equations and thus omitted the relaxation and diffusion terms present in the material energy and radiation energy equations. Here, we extend the theoretical grounds for the method and derive an entropy minimum principle for the full set of nonequilibrium‐diffusion Grey Radiation‐Hydrodynamic equations. This further strengthens the applicability of the entropy viscosity method as a stabilization technique for radiation‐hydrodynamic shock simulations. Radiative shock calculations using constant and temperature‐dependent opacities are compared against semi‐analytical reference solutions, and we present a procedure to perform spatial convergence studies of such simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

An entropy fixed cell‐centered Lagrangian scheme

On the basis of the work [P.‐H. Maire, R. Abgrall, J. Breil, J. Ovadia, SIAM J. Sci. Comput. 29 (2007), 1781–1824], we present an entropy fixed cell‐centered Lagrangian scheme for solving the Euler equations of compressible gas dynamics. The scheme uses the fully Lagrangian form of the gas dynamics equations, in which the primary variables are cell‐centered. And using the nodal solver, we obtain the nodal viscous‐velocity, viscous‐pressures, antidissipation velocity, and antidissipation pressures of each node. The final nodal velocity is computed as a weighted sum of viscous‐velocity and antidissipation velocity, so do nodal pressures, whereas these weights are calculated through the total entropy conservation for isentropic flows. Consequently, the constructed scheme is conservative in mass, momentum, and energy; preserves entropy for isentropic flows, and satisfies a local entropy inequality for nonisentropic flows. One‐ and two‐dimensional numerical examples are presented to demonstrate theoretical analysis and performance of the scheme in terms of accuracy and robustness.Copyright © 2013 John Wiley & Sons, Ltd.     

Some relations of nonequilibrium thermodynamics for a two-temperature and two-velocity gas with phase transitions

We study the nonequilibrium thermodynamics of two-phase media with nonequilibrium phase transitions. Assuming local (point) equilibrium within the limits of the phase and assuming mass additivity of the mixture entropy with respect to the phases, we obtain an expression for the mixture entropy production.We consider the motion of such media and derive formulas for the thermodynamic friction forces, heat transfer, condensation, and evaporation.In conclusion the author wishes to thank V. N. Nikolaevskii and V. V. Gogosov for helpful discussions.     

On thermodynamic closures for two-phase flow with interfacial area concentration transport equation

The present work is a part of a modelling of forest fires fighting by aerial means. In this paper, we study different kind of closures for modelling two-phase flows with an almost “infinite range” of scales. Since theories like homogenization are not, in this case, relevant for obtaining the equivalent medium equations, the averaging method has been preferred. The variables are averaged by convolution with a smooth kernel with compact support, as the equations are non-linear, new quantities are defined in order to obtain the equations satisfied by averaged quantities; the entropy production is determined and closures or phenomenological equations are obtained using the second principle of thermodynamics. Main features of this work are, firstly a derivation in this framework of a balance equation for the interfacial area concentration and secondly, since this introduces a new unclosed variable: the mean velocity of interfaces, extended irreversible thermodynamics is used to obtain the general form of the appropriate closures equations.     

Thermodynamically consistent coarse graining the non-equilibrium dynamics of unentangled polymer melts

A thermodynamically consistent strategy of coarse-graining microscopic models for complex fluids is illustrated for low-molecular polymeric melts subjected to homogeneous flow fields. The systematic coarse-graining method is able to efficiently bridge the time- and length scale gap between microscopic and macroscopic dynamics. A projection operator derivation is employed within the framework of nonequilibrium thermodynamics. From an alternating Monte-Carlo-molecular dynamics iteration scheme we obtain the thermodynamic building blocks of the macroscopic model. We investigate a number of imposed shear and elongational flows of interest and find that the coarse-grained model predicts structural as well as material functions beyond the regime of linear response. The elimination of fast degrees of freedom gives rise to dissipation, which we analyse in terms of the Rouse model. The results are in quantitative agreement with those obtained via standard nonequilibrium molecular dynamics (NEMD) simulations for planar shear and elongation. The coarse-graining method is able to deal with more general flows, which are not accessible by standard NEMD simulations.     

Extended thermodynamics, viscoelasticity, and strain of solids

In this paper will be presented a formulation of extended thermodynamics for viscoelastic materials with heat conduction. The application of the Galilean invariance of the balance equations and the principle of entropy lead to the introduction of Lagrange multipliers, which provide constitutive equations for the flows. A condition of hyperbolicity system of equations is achieved by the concavity of the entropy density. The balance equations are linearized.     

Ricci Curvature of Finite Markov Chains via Convexity of the Entropy

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry–Émery and Otto–Villani. Further, we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.     

Two-dimensional mixed flows of a supersaturated and two-phase medium with nonequilibrium phase transitions

The singularities of two-dimensional interchannel flows of a condensing and damp vapor with nonequilibrium phase transitions which contain gas-dynamic discontinuities are investigated. A through-computation difference method is constructed for such flows. The results of the numerical investigation of steam flows with spontaneous condensation in supersonic plane nozzles containing a break in the walls and flow around a wedge are represented. It is shown that nonequilibrium condensation can result in a qualitative rearrangement of the wave structure of the flow which is impossible to obtain within the framework of the one-dimensional approach.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 87–93, July–August, 1978.     

Gas Dynamics in Thermal Nonequilibrium¶and General Hyperbolic Systems¶with Relaxation

We study gas flow in vibrational nonequilibrium. The model is a 4Ǹ nonlinear hyperbolic system with relaxation. Under physical assumptions, properties of thermodynamic variables relevant to stability are obtained, global existence for Cauchy problems with smooth and small data is established, and large time behavior is studied in the pointwise sense. We formulate the fundamental solution in a systematic way for a general linear system with relaxation. The fundamental solution provides insights to the behavior of the nonlinear system, and is crucial to obtain our pointwise asymptotic picture for the nonequilibrium flow. We also clarify in a general setting the relation between subcharacteristic conditions and a dissipative criterion that was originally proposed for hyperbolic-parabolic systems and has now proved to be important also for hyperbolic systems with relaxation.     

Properties of thermocapillary fluids and symmetrization of motion equations

The equations of fluid motions are considered in the case of internal energy depending on mass density, volume entropy and their spatial derivatives. The model corresponds to domains with large density gradients in which the temperature is not necessarily uniform. The new general representation is written in symmetric form with respect to the mass and entropy densities. For conservative motions of perfect thermocapillary fluids, Kelvin's circulation theorems are always valid. Dissipative cases are also considered; we obtain the balance of energy and we prove that equations are compatible with the second law of thermodynamics. The internal energy form allows to obtain a Legendre transformation inducing a quasi-linear system of conservation laws which can be written in a divergence form and the stability near equilibrium positions can be deduced. The result extends classical hyperbolicity theory for governing-equations' systems in hydrodynamics, but symmetric matrices are replaced by Hermitian matrices.     

Relative Entropies, Suitable Weak Solutions, and Weak-Strong Uniqueness for the Compressible Navier–Stokes System

We introduce the notion of relative entropy for the weak solutions to the compressible Navier–Stokes system. In particular, we show that any finite energy weak solution satisfies a relative entropy inequality with respect to any couple of smooth functions satisfying relevant boundary conditions. As a corollary, we establish the weak-strong uniqueness property in the class of finite energy weak solutions, extending thus the classical result of Prodi and Serrin to the class of compressible fluid flows.     

Existence and Stability of Supersonic Euler Flows Past Lipschitz Wedges

It is well known that, when the vertex angle of a straight wedge is less than the critical angle, there exists a shock-front emanating from the wedge vertex so that the constant states on both sides of the shock-front are supersonic. Since the shock-front at the vertex is usually strong, especially when the vertex angle of the wedge is large, then a global flow is physically required to be governed by the isentropic or adiabatic Euler equations. In this paper, we systematically study two-dimensional steady supersonic Euler (i.e. nonpotential) flows past Lipschitz wedges and establish the existence and stability of supersonic Euler flows when the total variation of the tangent angle functions along the wedge boundaries is suitably small. We develop a modified Glimm difference scheme and identify a Glimm-type functional, by naturally incorporating the Lipschitz wedge boundary and the strong shock-front and by tracing the interaction not only between the boundary and weak waves, but also between the strong shock-front and weak waves, to obtain the required BV estimates. These estimates are then employed to establish the convergence of both approximate solutions to a global entropy solution and corresponding approximate strong shock-fronts emanating from the vertex to the strong shock-front of the entropy solution. The regularity of strong shock-fronts emanating from the wedge vertex and the asymptotic stability of entropy solutions in the flow direction are also established.     

Integration of linear and some model non-linear equations of motion of incompressible fluids

We suggest a new exact method that allows one to construct solutions to a wide class of linear and some model non-linear hydrodynamic-type systems. The method is based on splitting a system into a few simpler equations; two different representations of solutions (non-symmetric and symmetric) are given. We derive formulas that connect solutions to linear three-dimensional stationary and non-stationary systems (corresponding to different models of incompressible fluids in the absence of mass forces) with solutions to two independent equations, one of which being the Laplace equation and the other following from the equation of motion for any velocity component at zero pressure. To illustrate the potentials of the method, we consider the Stokes equations, describing slow flows of viscous incompressible fluids, as well as linearized equations corresponding to Maxwell's and some other viscoelastic models. We also suggest and analyze a differential-difference fluid model with a constant relaxation time. We give examples of integrable non-linear hydrodynamic-type systems. The results obtained can be suitable for the integration of linear hydrodynamic equations and for testing numerical methods designed to solve non-linear equations of continuum mechanics.     

Thermodynamics of the activated state of materials

The thermodynamics of irreversible processes is extended to deformable materials whose state and behavior under nonequilibrium conditions are determined by the value and evolution of the additional parameter — the activation parameter. General thermodynamic relations are presented. The concept of the time of existence of a nonequilibrium state is introduced, and the phase coexistence conditions are generalized taking into account the properties of the interface. Methods are described to generalize the relations for irreversible flows, thermodynamic forces, and the equations of state. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 141–152, January–February, 2009.