Local Equilibrium Hypothesis Research Articles

RELATIVISTIC VISCOELASTIC FLUID MECHANICS

We explain the relativistic theory of viscoelasticity which we have recently constructed on the basis of Onsager's linear nonequilibrium thermodynamics. This theory universally reduces to the standard relativistic Navier-Stokes fluid mechanics in the long time limit. Since effects of elasticity are taken into account, the dynamics at short time scales is modified from that given by the Navier-Stokes equations, so that acausal problems intrinsic to relativistic Navier-Stokes fluids are significantly remedied. We then present conformal higher-order viscoelastic fluid mechanics with strain allowed to take arbitrarily large values. We particularly show that a conformal second-order fluid with all possible parameters in the constitutive equations can be obtained without breaking the hypothesis of local thermodynamic equilibrium, if the conformal fluid is defined as the long time limit of a conformal second-order viscoelastic system.

Comparisons of Two Models for the Simulation of a DC Arc Plasma Torch

The hypothesis of local thermal equilibrium (LTE) in thermal plasma has been widely accepted. Most of the simulation models for the arc plasma torch are based on the hypothesis of LTE and its results indicate good validity to mimic the pattern of plasma flow inside a plasma torch. However, according to the LTE hypothesis, electrical conductivity near electrodes is significantly lower because of the low gas temperature. Consequently, it is difficult for electrical current flows to pass between the anode and cathode. Therefore, the key subject for a model concentrating on the LTE assumption is to deal with the low electrical conductivity near the electrodes. In this study, two models determining the electrical conductivity at the vicinity of the electrodes with two different assumptions were used to calculate the flow patterns inside a non-transferred DC arc plasma torch. Gas temperature, velocity, voltage drop, and heat energy of the plasma arc were compared between the two models. The results indicated that the plasma arc inside the plasma torch fluctuates, as simulated by both models. It seems that the model can obtain comparable accuracy with the experimental results if the plasma gas electrical conductivity is determined by nominal electron temperature.

Mean absorption coefficient of H2O–air–MgCl2/CaCl2/NaCl thermal plasmas

Under the local thermodynamic equilibrium hypothesis, the mean absorption coefficients (MACs) were calculated for H2O–air–MgCl2/CaCl2/NaCl thermal plasmas in a temperature range from 300 to 30 000 K and at atmospheric pressure. The MACs were computed under the hypothesis of isothermal plasmas which allows a good description of the radiation absorbed in cold regions. In this study, we took into account the absorption radiation resulting from the atomic continuum, molecular continuum, atomic lines and some molecular bands. Free–free transitions (bremsstrahlung) and free–bound (electron–ion recombination and electron attachment) or bound–free transitions in terms of absorption were considered for the calculation of atomic continuum. For bound–bound transitions, natural, resonance, van der Waals, Stark and Doppler effects were taken into account for the line broadenings while the escape factors were used to treat the self-absorption of the resonance lines. Molecular continuum was considered for the main molecules (H2, O2, N2, OH, NO, H2O, N2O, NO2, O3, NO3 and N2O5) whereas we studied only diatomic systems O2, N2, NO and for the absorption of molecular bands. The influence of the proportion of MgCl2, CaCl2 or NaCl in a water–air mixture was analysed as the effect of the strong self-absorbed resonance lines of the alkaline salts (Ca, Ca+, Na, Na+, Mg, Mg+, Cl and Cl+). Our results show that a low concentration of alkaline salts (less than 1% in molar proportions) in the plasma increased the MACs at low temperatures (T < 10 000 K) due to the resonance lines mainly localized in the near-UV and visible spectral regions in opposition to hydrogen, oxygen or nitrogen species for which 90% of them exist in ultraviolet. In addition to the atomic and molecular continuum, the absorption radiation of molecular bands is important at low temperatures.

Macrostatistics and Fluctuating Hydrodynamics

We extend our earlier macrostatistical treatment of hydrodynamical fluctuations about nonequilibrium steady states to viscous fluids. Since the scale dependence of the Navier-Stokes equations precludes the applicability of any infinite scale (hydrodynamical) limit, this has to be based on the generic model of a large but finite system, rather than an infinite one. On this basis, together with assumptions of Onsager's regression hypothesis and conditions of local equilibrium and chaoticity, we show that the hydrodynamical fluctuations of a reservoir driven fluid about a nonequilibrium steady state execute a Gaussian Markov process that constitutes a mathematical structure for a generalised version of Landau's fluctuating hydrodynamics and generically carries long range spatial correlations.

Modelling CO2-induced fluid–rock interactions in the Altensalzwedel gas reservoir. Part I: from experimental data to a reference geochemical model

Modelling fluid–rock interactions induced by CO2 is a key issue when evaluating the technical feasibility and long-term safety assessment of CO2 storage projects in deep formations. The German R&D programme CLEAN (CO2 Large-Scale Enhanced Gas Recovery in the Altmark Natural Gas Field) investigated the almost depleted onshore gas reservoir located in the Rotliegend sandstone at over 3,000-m depth. The high salinity of the formation fluids and the elevated temperature in the reservoir exceed the validity limits of commonly available thermodynamic databases needed for predictive geochemical modelling. In particular, it is shown that the activity model of Pitzer has to be applied, even if necessary input data for this model are incomplete or inconsistent for complex systems and for the considered temperatures. Simulations based on Debye-Hückel activity model lead to severe, systematic discrepancies already in the simple proposed reference case where experimental data could be used for comparison. A simplified geochemical model, consistent with the average measured composition of formation fluids and the prevailing mineralogical assemblage of the host rock, identifies the mineral phases most likely to be considered at equilibrium with the formation fluid. The simulated reactions due to CO2 injection, under the hypothesis of local thermodynamical equilibrium, result in a moderate reactivity of the system, with the dissolution of anhydrite cementation and haematite being the most relevant expected mineral reactions. This is compensated, at equilibrium, by the precipitation of new carbonates, like calcite and siderite, for an overall very small loss of porous space. The simulated rather small effect of mineral alteration is also due to the scarce amount of water available for reactions in the reservoir. The results of the model are qualitatively in line with observations from batch experiments and from natural analogues.

Conformal higher-order viscoelastic fluid mechanics

We present a generally covariant formulation of conformal higher-order viscoelastic fluid mechanics with strain allowed to take arbitrarily large values. We give a general prescription to determine the dynamics of a relativistic viscoelastic fluid in a way consistent with the hypothesis of local thermodynamic equilibrium and the second law of thermodynamics. We then elaborately study the transient time scales at which the strain almost relaxes and becomes proportional to the gradients of velocity. We particularly show that a conformal second-order fluid with all possible parameters in the constitutive equations can be obtained without breaking the hypothesis of local thermodynamic equilibrium, if the conformal fluid is defined as the long time limit of a conformal second-order viscoelastic system. We also discuss how local thermodynamic equilibrium could be understood in the context of the fluid/gravity correspondence.

Non-linear finite element formulation applied to thermoelectric materials under hyperbolic heat conduction model

In the present work, a three-dimensional, dynamic and non-linear finite element to simulate thermoelectric behavior under a hyperbolic heat conduction model is presented. The transport equations, which couple electric and thermal energies by the Seebeck, Peltier and Thomson effects, are analytically obtained through extended non-equilibrium thermodynamics, since the local equilibrium hypothesis is not valid under the hyperbolic model. In addition, unidimensional analytical solutions are obtained to validate the finite element formulation. Numerically, isoparametric eight-node elements with two degrees of freedom (voltage and temperature) per node are used. Non-linearities due to the temperature-dependence on the transport properties and the Joule effects are addressed with the Newton–Raphson algorithm. For the dynamic problem, HHT and Newmark-β algorithms are compared to obtain accurate results, since numerical oscillations (Gibbs phenomena) are present when the initial boundary conditions are discontinuous. The last algorithm, which is regularized by relating time steps and element sizes, provides the best results. Finally, the finite element implementation is validated, comparing the analytical and the numerical solutions, and a three-dimensional example is presented.

An Improved Local Thermal Equilibrium Model of DC Arc Plasma Torch

Most of the simulation models about arc plasma are based on the hypothesis of local thermal equilibrium (LTE). The nonequilibrium model is very complicated due to the calculation of electron temperature. In this paper, an improved LTE model is developed and applied to the 3-D simulation of the flow patterns inside a nontransferred dc arc plasma torch. Numerical calculations on the distributions of gas temperature and velocity in the plasma torch were carried out using argon as the plasma gas. The electric current density and potential are also discussed. The results indicate that the temperature and velocity distributions of the arc are almost axisymmetrical. The results of voltage drop agree well with the experimental observations. It seems that anode erosion is located on the internal surface of the anode, where the largest number of electrons is injected.

Mesoscopic transport equations and contemporary thermodynamics: an introduction

The local equilibrium hypothesis is a very successful basis for non-equilibrium thermodynamics over a wide range of phenomena and physical situations. However, the increasing interest in small systems in nanotechnology, in rarefied gases in high-altitude aeronautics, in high-frequency behaviour in information processing, or the search for new materials with sophisticated internal microstructures and tailored thermal properties have led one to ask about the limits of validity of this hypothesis, and to go beyond it. Here we do so in a constructive way, i.e. not only pointing out at these limits, but also embedding the local-equilibrium theory in a more general framework which explicitly exhibits these limits and suggests how to go beyond them, in search for a wider range of applications and a deeper understanding of the foundations.

Beyond the Cahn-Hilliard Equation: a Vacancy-Based Kinetic Theory

A Self-Consistent Mean Field (SCMF) kinetic theory including an explicit description ofthe vacancy diffusion mechanism is developed. The present theory goes beyond the usual local equi-librium hypothesis. It is applied to the study of the early time spinodal decomposition in alloys. Theresulting analytical expression of the structure function highlights the contribution of the vacancydiffusion mechanism. Instead of the single amplification rate of the Cahn-Hillard linear theory, thelinearized SCMF kinetic equations involve three constant rates, first one describing the vacancy re-laxation kinetics, second one related to the kinetic coupling between local concentrations and paircorrelations and the third one representing the spinodal amplification rate. Starting from the same va-cancy diffusion model, we perform kineticMonte Carlo simulations of a Body Centered Cubic (BCC)demixting alloy. The resulting spherically averaged structure function is compared to the SCMF pre-dictions. Both qualitative and quantitative agreements are satisfying.

Gradient stability of numerical algorithms in local nonequilibrium problems of critical dynamics

The critical dynamics of a spatially inhomogeneous system are analyzed with allowance for local nonequilibrium, which leads to a singular perturbation in the equations due to the appearance of a second time derivative. An extension is derived for the Eyre theorem, which holds for classical critical dynamics described by first-order equations in time and based on the local equilibrium hypothesis. It is shown that gradient-stable numerical algorithms can also be constructed for second-order equations in time by applying the decomposition of the free energy into expansive and contractive parts, which was suggested by Eyre for classical equations. These gradient-stable algorithms yield a monotonically nondecreasing free energy in simulations with an arbitrary time step. It is shown that the gradient stability conditions for the modified and classical equations of critical dynamics coincide in the case of a certain time approximation of the inertial dynamics relations introduced for describing local nonequilibrium. Model problems illustrating the extended Eyre theorem for critical dynamics problems are considered.

Phase equilibria, diffusion growth and diffusivities in Ni-Al-Pt system using Pt/β-NiAl diffusion couples

AbstractThe phase equilibria, diffusion growth and diffusivities in the Ni-Al-Pt system at 1 150, 1 200 and 1 250 °C were studied using Pt/β-NiAl diffusion couples. Based on the measured concentration profiles coupled with the local equilibrium hypothesis, the tie-lines between neighboring phases were determined. Two intermediate phases, Pt3Al and α-NiPt(Al), are found to develop between the Pt and β-NiAl couples. The thicknesses of Pt3Al and α-NiPt(Al) layers varies linearly with the square of annealing time, indicating that the growth of Pt3Al and α-NiPt(Al) are diffusion-controlled. The calculated Arrhenius equations of the parabolic growth constants, k, for the Pt3Al and β-NiPt(Al) phases are obtained, respectively. The average ternary inter-diffusivities of the Pt3Al and β-NiAl phases at different temperatures are determined from an integration of inter-diffusion fluxes calculated directly from the experimental concentration profiles.

NLTE strontium abundance in a sample of extremely metal poor stars and the Sr/Ba ratio in the early Galaxy

Heavy element abundances in extremely metal-poor stars provide strong constraints on the processes of forming these elements in the first stars. We attempt to determine precise abundances of strontium in a homogeneous sample of extremely metal-poor stars. The abundances of strontium in 54 very or extremely metal-poor stars, was redetermined by abandoning the local thermodynamic equilibrium (LTE) hypothesis, and fitting non-LTE (NLTE) profiles to the observed spectral lines. The corrected Sr abundances and previously obtained NLTE Ba abundances are compared to the predictions of several hypothetical formation processes for the lighter neutron-capture elements. Our NLTE abundances confirm the previously determined huge scatter of the strontium abundance in low metallicity stars. This scatter is also found (and is even larger) at very low metallicities (i. e. early in the chemical evolution). The Sr abundance in the extremely metal-poor (EMP) stars is compatible with the main r-process involved in other processes (or by variations of the r-process), as we briefly discuss.

Unconditionally gradient-stable computational schemes in problems of fast phase transitions

Equations of fast phase transitions, in which the phase boundaries move with velocities comparable with the atomic diffusion speed or with the speed of local structural relaxation, are analyzed. These equations have a singular perturbation due to the second derivative of the order parameter with respect to time, which appears due to phenomenologically introduced local nonequilibrium. To develop unconditionally stable computational schemes, the Eyre theorem [D. J. Eyre, unpublished] proved for the classical equations, based on hypotheses of local equilibrium, is used. An extension of the Eyre theorem for the case of equations for fast phase transitions is given. It is shown that the expansion of the free energy on contractive and expansive parts, suggested by Eyre for the classical equations of Cahn-Hilliard and Allen-Cahn, is also true for the equations of fast phase transitions. Grid approximations of these equations lead to gradient-stable algorithms with an arbitrary time step for numerical modeling, ensuring monotonic nonincrease of the free energy. Special examples demonstrating the extended Eyre theorem for fast phase transitions are considered.

Relaxation time for the temperature in a dilute binary mixture from classical kinetic theory

The system of our interest is a dilute binary mixture, in which we consider that the species have different temperatures as an initial condition. To study their time evolution, we use the full version of the Boltzmann equation, under the hypothesis of partial local equilibrium for both species. Neither a diffusion force nor mass diffusion appears in the system. We also estimate the time in which the temperatures of the components reach the full local equilibrium. In solving the Boltzmann equation, we imposed no assumptions on the collision term. We work out its solution by using the well known Chapman-Enskog method to first order in the gradients. The time in which the temperatures relax is obtained following Landau's original idea. The result is that the relaxation time for the temperatures is much smaller than the characteristic hydrodynamical times but greater than a collisional time. The main conclusion is that there is no need to study binary mixtures with different temperatures when hydrodynamical properties are sought.

Simulation of laminar impinging jet on a porous medium with a thermal non-equilibrium model

This work shows numerical simulations of an impinging jet on a flat plate covered with a layer of a porous material. Macroscopic equations for mass and momentum are obtained based on the volume-average concept. Two macroscopic models are employed for analyzing energy transport, namely the one-energy equation model, based on the Local Thermal Equilibrium assumption (LTE), and the two-energy equation closure, where distinct transport equations for the fluid and the porous matrix follow the Local Non-Thermal Equilibrium hypothesis (LNTE). The numerical technique employed for discretizing the governing equations was the finite volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure–velocity coupling. Parameters such as porosity, porous layer thickness, material permeability and thermal conductivity ratio were varied in order to analyze their effects on flow and heat transport. Results indicate that for low porosities, low permeabilities, thin porous layers and for high thermal conductivity ratios, a different distribution of local Nusselt number at the wall is calculated depending on the energy model applied. The use of the LNTE model indicates that it is advantageous to use a layer of highly conducting and highly porous material attached to the hot wall.

On the thermodynamic derivation of differential equations for turbulent flow transfer in a compressible heat-conducting fluid

This paper considers the modern approach to the thermodynamic modeling of developed turbulent flows of a compressible fluid based on the systematic application of the formalism of extended irreversible thermodynamics (EIT) that goes beyond the local equilibrium hypothesis, which is an inseparable attribute of classical nonequilibrium thermodynamics (CNT). In addition to the classical thermodynamic variables, EIT introduces new state parameters—dissipative flows and the means to obtain the respective evolutionary equations consistent with the second law of thermodynamics. The paper presents a detailed discussion of a number of physical and mathematical postulates and assumptions used to build a thermodynamic model of turbulence. A turbulized liquid is treated as an indiscrete continuum consisting of two thermodynamic sub-systems: an averaged motion subsystem and a turbulent chaos subsystem, where turbulent chaos is understood as a conglomerate of small-scale vortex bodies. Under the above formalism, this representation enables the construction of new models of continual mechanics to derive cause-and-effect differential equations for turbulent heat and impulse transfer, which describe, together with the averaged conservations laws, turbulent flows with transverse shear. Unlike gradient (noncausal) relationships for turbulent flows, these differential equations can be used to investigate both hereditary phenomena, i.e., phenomena with history or memory, and nonlocal and nonlinear effects. Thus, within EIT, the second-order turbulence models underlying the so-called invariant modeling of developed turbulence get a thermodynamic explanation. Since shear turbulent flows are widespread in nature, one can expect the given modification of the earlier developed thermodynamic approach to developed turbulence modeling (see Kolesnichenko, 1980; 1998; 2002–2004; Kolesnichenko and Marov, 1985; Kolesnichenko and Marov, 2009) to be used in research on a broad class of dissipative phenomena in various astro- and geophysical applications. In particular, a major application of the proposed approach is the reconstruction of the processes in the preplanetary circumsolar disk, which might help solve the fundamental problems of stellar-planetary cosmogony.

A modulated gradient model for large-eddy simulation: Application to a neutral atmospheric boundary layer

The subgrid-scale (SGS) parametrization represents a critical component of a successful large-eddy simulation (LES). It is known that in LES of high-Reynolds-number atmospheric boundary layer turbulence, standard eddy-viscosity models poorly predict mean shear in the near-wall region and yield erroneous velocity profiles. In this paper, a modulated gradient model is proposed. This approach is based on the Taylor expansion of the SGS stress and uses local equilibrium hypothesis to evaluate the SGS kinetic energy. To ensure numerical stability, a clipping procedure is used to avoid local kinetic energy transfer from unresolved to resolved scales. Two approaches are considered to specify the model coefficient: a constant value of 1 and a simple correction to account for the effects of the clipping procedure on the SGS energy production rate. The model is assessed through a systematic comparison with well-established empirical formulations and theoretical predictions of a variety of flow statistics in a neutral atmospheric boundary layer. Overall, the statistics of the simulated velocity field obtained with the new model show good agreement with reference results and a significant improvement compared to simulations with standard eddy-viscosity models. For instance, the new model is capable of reproducing the expected log-law mean velocity profile and power-law energy spectra. Simulations also yield streaky structures and near-Gaussian probability density functions of velocity in the near-wall region. It is found that using a constant coefficient of 1 yields a slightly excessive SGS dissipation, which is corrected when the coefficient is modified using the above mentioned correction.

Long-range correlations in a simple stochastic model of coupled transport

We study coupled transport in the nonequilibrium stationary state of a model consisting of independent random walkers, moving along a one-dimensional channel, which carry a conserved energy-like quantity, with density and temperature gradients imposed by reservoirs at the ends of the channel. In our model, walkers interact with other walkers at the same site by sharing energy at each time step, but the amount of energy carried does not affect the motion of the walkers. We find that already in this simple model long-range correlations arise in the nonequilibrium stationary state which are similar to those observed in more realistic models of coupled transport. We derive an analytical expression for the source of these correlations, which we use to obtain semi-analytical results for the correlations themselves assuming a local-equilibrium hypothesis. These are in very good agreement with results from direct numerical simulations.

A Numerical Study on the Validity of the Local Equilibrium Hypothesis in Modeling Hydrogen Thermal Desorption Spectra

We present a systematic benchmark study on different numerical models for analyzing hydrogen thermal desorption spectra, by focusing on the adoption of the local equilibrium hypothesis in these models. We find that the direct numerical method of the full set of the extended mass conservation equations is only able to predict the experimental behavior of thermal desorption spectra for pure iron in the thin specimen limit, while other models incorporating the local equilibrium hypothesis fail to predict this behavior.