Статті в журналах: "Partial local Saha equilibrium"

1

Timmermans, C. J., R. J. Rosado, and D. C. Schram. "An Investigation of Non-Equilibrium Effects in Thermal Argon Plasmas." Zeitschrift für Naturforschung A 40, no. 8 (August 1, 1985): 810–25. http://dx.doi.org/10.1515/zna-1985-0804.

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The parameters and transport properties of a wall stabilized argon arc (40-200 A) at atmospheric pressure with diameters of 5 and 8 mm are studied by spectroscopy and interferometry. The plasma is assumed to be partial local thermal equilibrium and this assumption is verified with the aid of a collisional-radiative model. The departures from Saha-equilibrium of the argon neutral ground state are found to be associated with particle diffusion and the escape of recombination radiation. The measurement of the total excitation rate, from the ground level, including direct ionization, of neutral argon is in reasonable agreement with the literature value.

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2

Anders, A. "Plasma fluctuations, local partial Saha equilibrium, and the broadening of vacuum-arc ion charge state distributions." IEEE Transactions on Plasma Science 27, no. 4 (1999): 1060–67. http://dx.doi.org/10.1109/27.782282.

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3

Cherevko, A. S., and A. A. Morozova. "On the violation of local thermodynamic equilibrium in the plasma jet of an argon arc two-jet plasmatron." Industrial laboratory. Diagnostics of materials 85, no. 1II) (February 15, 2019): 145–50. http://dx.doi.org/10.26896/1028-6861-2019-85-1-ii-145-150.

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The degree and nature of the violation of local thermodynamic equilibrium (LTE) in the analytical zone of a plasma jet generated by an argon arc two-jet plasmatron (TJP) was estimated using an unconventional method based on determination of the nonequilibrium parameterbiequal to the ratio of the experimentally determined actual population of the energy level (ni) of the element to the population of the same level calculated from the Saha equation (nis). Partial ionizing deviation of plasma under study from the equilibrium state takes place only when low-lying atomic levels are overpopulated. The distinct dependence ofbion the ionization potential of the considered element (e.g., Ca, Mg, and Be) is shown. The results were interpreted in the light of the increasing role of radiation processes upon excitation of spectra in the argon arc two-jet plasmatron.

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4

RUBIANO, J. G., R. RODRÍGUEZ, J. M. GIL, P. MARTEL, and E. MÍNGUEZ. "Calculation of the ionization state for LTE plasmas using analytical potentials." Laser and Particle Beams 17, no. 4 (October 1999): 635–47. http://dx.doi.org/10.1017/s0263034699174081.

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In this work, the Saha equation is solved using atomic data provided by means of analytical potentials to calculate the ionization state and ion abundances for local thermodynamic equilibrium (LTE) plasmas of Al, Fe, and Au. The plasma effects are taking into account using an analytical potential which includes plasma effects. The problem of the cut off partition functions in the Saha equation is also analyzed using three different criteria. Finally, some opacity calculations are performed.

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5

Kochuev, D. A., A. F. Galkin, A. S. Chernikov, R. V. Chkalov, A. A. Voznesenskaya, K. S. Khorkov, and A. V. Egorova. "Local thermodynamic equilibrium in a laser-induced plasma channel formed by interaction of femtosecond laser radiation with argon." Journal of Physics: Conference Series 2131, no. 5 (December 1, 2021): 052090. http://dx.doi.org/10.1088/1742-6596/2131/5/052090.

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Abstract In this paper, we estimate the possibility of applying local thermodynamic equilibrium conditions for a laser-induced plasma channel formed by femtosecond laser radiation in an argon medium at different pressures. The presence of a local thermodynamic equilibrium was determined on the basis of the time of heat exchange of electrons with argon atoms. The Saha equation is used to estimate the concentration of free electrons, the temperature of the laser-induced plasma channel, and its conductivity. A necessary condition for using this ratio was the presence of a state of local thermodynamic equilibrium in the plasma under study.

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6

White, J. R., W. Johns, C. J. Fontes, N. M. Gill, N. R. Shaffer, and C. E. Starrett. "Charge state distributions in dense plasmas." Physics of Plasmas 29, no. 4 (April 2022): 043301. http://dx.doi.org/10.1063/5.0084109.

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Charge state distributions in hot, dense plasmas are a key ingredient in the calculation of spectral quantities like the opacity. However, they are challenging to calculate, as models like Saha–Boltzmann become unreliable for dense, quantum plasmas. Here, we present a new variational model for the charge state distribution, along with a simple model for the energy of the configurations that includes the orbital relaxation effect. Comparison with other methods reveals generally good agreement with average atom-based calculations, the breakdown of the Saha–Boltzmann method, and mixed agreement with a chemical model. We conclude that the new model gives a relatively inexpensive, but reasonably high fidelity method of calculating the charge state distribution in hot dense plasmas, in local thermodynamic equilibrium.

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7

Barthélemy, Olivier, Joëlle Margot, Stéphane Laville, François Vidal, Mohamed Chaker, Boris Le Drogoff, Tudor W. Johnston, and Mohamad Sabsabi. "Investigation of the State of Local Thermodynamic Equilibrium of a Laser-Produced Aluminum Plasma." Applied Spectroscopy 59, no. 4 (April 2005): 529–36. http://dx.doi.org/10.1366/0003702053641531.

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In this work, the assumption of local thermodynamic equilibrium (LTE) for a laser-induced plasma in ambient air is examined experimentally using two different laser systems, namely an infrared short-pulse Ti:Sapphire laser and an ultraviolet long-pulse XeCl excimer laser. The LTE assumption is investigated by examining the plasma produced at a laser fluence of 10 J/cm2 from aluminum targets containing iron and magnesium impurities. The excitation temperature is deduced from Boltzmann diagrams built from a large number of spatially integrated neutral iron lines distributed from 3.21 to 6.56 eV. It is shown that at any time after the end of the laser pulse, the neutral excited states are in excellent Boltzmann equilibrium. Detailed investigation of Boltzmann equilibrium further validates previous temperature measurements using less accurate diagrams. However, observations of ion lines provide some evidence that the ionized species do not obey Saha equilibrium, thereby indicating departure from LTE. This could be explained by the fact that the plasma cannot be considered as stationary for these species.

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8

Hey, J. D., C. C. Chu, and J. P. S. Rash. "Partial local thermal equilibrium in a low-temperature hydrogen plasma." Journal of Quantitative Spectroscopy and Radiative Transfer 62, no. 3 (June 1999): 371–87. http://dx.doi.org/10.1016/s0022-4073(98)00110-1.

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9

Novakovic, R. N. V., B. S. Milic, S. M. Stojilkovic, and Dragan Gajic. "Some transport properties in plasmas containing argon and fluorine." Facta universitatis - series: Physics, Chemistry and Technology 2, no. 5 (2003): 285–91. http://dx.doi.org/10.2298/fupct0305285n.

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In this paper some results of numerical evaluation of transport coefficients in plasmas in the mixtures of argon and fluorine are presented. These transport characteristics are given in the function of the temperature for low pressures ranging from 0,1 kPa to 1,0 kPa and for low temperatures between 500 K and 5 000 K in argon plasmas with 20% and 30% of the fluorine added. It is assumed that the system is kept under constant pressure and that a corresponding state of local thermodynamical equilibrium (LTE) is attained in it. The equilibrium plasma composition, necessary for the evaluations, was determined on the ground of the Saha equations for ionization processes and the law of mass action for the thermal dissociation of F2, combined with the charge conservation relation and the assumption that the pressure remained constant in the course of temperature variations. The ionization energy lowering, required in conjunction with the Saha equations, was obtained with the aid of a modified expression for the plasma Debye radius proposed previously. A previously derived expression for the modified Debye radius, offering the possibility to treat the plasmas considered as weakly non-ideal in the whole temperature range, is used. The cut-off at the Landau length rather than of the smallest of ionic radii is introduced. This alteration in the evaluation procedure brings different considerable changes in the final numerical results for the all relevant quantities.

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10

Khater, Mohamed A. "Vacuum ultraviolet assessment of ionization temperature by space-resolved emission spectroscopy." Canadian Journal of Physics 91, no. 9 (September 2013): 733–36. http://dx.doi.org/10.1139/cjp-2012-0550.

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In this article, the electron ionization temperature in plasmas generated by 1064 nm laser pulses in the vacuum ultraviolet spectral range is evaluated as a function of the axial distance from a steel target surface using emission spectroscopy. The temperature was determined using the relative line intensities ratio of C II 90.41 nm and C III 97.7 nm spectral lines, applied to the Saha–Boltzmann equation. Ionization temperatures determined in this way changed from 33 900 K at the target surface to 26 800 K (2.92–2.31 eV) at 4.0 mm away from it. Large differences between the measured excitation and ionization temperatures suggest nonthermal equilibrium conditions between electrons and heavier ionic species. Based upon the results obtained from this and a previous study under the same operating conditions, the validity of the local thermal equilibrium condition in the plasmas investigated is presented and discussed.

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11

Kroesen, G. M. W., D. C. Schram, C. J. Timmermans, and J. C. M. de Haas. "The energy balance of a plasma in partial local thermodynamic equilibrium." IEEE Transactions on Plasma Science 18, no. 6 (1990): 985–91. http://dx.doi.org/10.1109/27.61514.

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12

Mlodik, M. E., E. J. Kolmes, I. E. Ochs, T. Rubin, and N. J. Fisch. "Partial-ionization deconfinement effect in magnetized plasma." Physics of Plasmas 29, no. 11 (November 2022): 112111. http://dx.doi.org/10.1063/5.0114967.

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In partially ionized plasma, where ions can be in different ionization states, each charge state can be described as a different fluid for the purpose of multi-ion collisional transport. In the case of two charge states, transport pushes plasma toward equilibrium, which is found to be a combination of local charge-state equilibrium and generalized pinch relations between ion fluids representing different charge states. Combined, these conditions lead to a dramatic deconfinement of ions. This deconfinement happens on the timescale similar but not identical to the multi-ion cross-field transport timescale, as opposed to electron–ion transport timescale in fully ionized plasma. Deconfinement occurs because local charge-state equilibration enforces the disparity in diamagnetic drift velocities of ion fluid components, which in turn leads to the cross-field transport due to ion–ion friction.

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13

Wyttenbach, A., P. Mollière, D. Ehrenreich, H. M. Cegla, V. Bourrier, C. Lovis, L. Pino, et al. "Mass-loss rate and local thermodynamic state of the KELT-9 b thermosphere from the hydrogen Balmer series." Astronomy & Astrophysics 638 (June 2020): A87. http://dx.doi.org/10.1051/0004-6361/201937316.

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KELT-9 b, the hottest known exoplanet, with Teq ~ 4400 K, is the archetype of a new planet class known as ultra-hot Jupiters. These exoplanets are presumed to have an atmosphere dominated by neutral and ionized atomic species. In particular, Hα and Hβ Balmer lines have been detected in the KELT-9 b upper atmosphere, suggesting that hydrogen is filling the planetary Roche lobe and escaping from the planet. In this work, we detected δ Scuti-type stellar pulsation (with a period Ppuls = 7.54 ± 0.12 h) and studied the Rossiter-McLaughlin effect (finding a spin-orbit angle λ = −85.01° ± 0.23°) prior to focussing on the Balmer lines (Hα to Hζ) in the optical transmission spectrum of KELT-9 b. Our HARPS-N data show significant absorption for Hα to Hδ. The precise line shapes of the Hα, Hβ, and Hγ absorptions allow us to put constraints on the thermospheric temperature. Moreover, the mass loss rate, and the excited hydrogen population of KELT-9 b are also constrained, thanks to a retrieval analysis performed with a new atmospheric model. We retrieved a thermospheric temperature of T = 13 200−720+800 K and a mass loss rate of Ṁ = 1012.8±0.3 g s−1 when the atmosphere was assumed to be in hydrodynamical expansion and in local thermodynamic equilibrium (LTE). Since the thermospheres of hot Jupiters are not expected to be in LTE, we explored atmospheric structures with non-Boltzmann equilibrium for the population of the excited hydrogen. We do not find strong statistical evidence in favor of a departure from LTE. However, our non-LTE scenario suggests that a departure from the Boltzmann equilibrium may not be sufficient to explain the retrieved low number densities of the excited hydrogen. In non-LTE, Saha equilibrium departure via photo-ionization, is also likely to be necessary to explain the data.

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14

Sevougian, S. David, Robert S. Schechter, and Larry W. Lake. "Effect of partial local equilibrium on the propagation of precipitation/dissolution waves." Industrial & Engineering Chemistry Research 32, no. 10 (October 1993): 2281–304. http://dx.doi.org/10.1021/ie00022a013.

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15

Lotfi, El Mehdi, Mehdi Maziane, Khalid Hattaf, and Noura Yousfi. "Partial Differential Equations of an Epidemic Model with Spatial Diffusion." International Journal of Partial Differential Equations 2014 (February 10, 2014): 1–6. http://dx.doi.org/10.1155/2014/186437.

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The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.

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16

Kryzhevich, Dmitrij Sergeevich, Aleksandr Vyacheslavovich Korchuganov, Konstantin Petrovich Zolnikov, and Sergey Grigorievich Psakhie. "Role of Localized Non-Equilibrium States in Nucleation of Plastic Deformation in Nanocrystalline Materials." Solid State Phenomena 258 (December 2016): 21–24. http://dx.doi.org/10.4028/www.scientific.net/ssp.258.21.

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A molecular dynamics simulation of the behavior of nanocrystalline materials in the fields of external influences was carried out. Crystallites of the fcc copper and bcc iron under different schemes of mechanical loading were investigated. Revealed specific localized non-equilibrium states served as the mechanism of formation and evolution of partial dislocations in fcc materials and twin growth in bcc materials. These non-equilibrium states were realized on the basis of local transformation of the martensitic type when the nearest surrounding of atoms – the centers of local rearrangements – changed according to the A-B(C) scheme, where A, B and C are types of crystal lattice. The bcc-fcc-bcc local rearrangements during twin growth were typical for bcc iron. The fcc-bcc-hcp and hcp-bcc-fcc local rearrangements during the partial dislocation movement were typical for fcc copper.

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17

GIOVANGIGLI, VINCENT, and LIONEL MATUSZEWSKI. "MATHEMATICAL MODELING OF SUPERCRITICAL MULTICOMPONENT REACTIVE FLUIDS." Mathematical Models and Methods in Applied Sciences 23, no. 12 (August 18, 2013): 2193–251. http://dx.doi.org/10.1142/s0218202513500309.

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We investigate a system of partial differential equations modeling supercritical multicomponent reactive fluids. These equations involve nonideal fluid thermodynamics, nonideal chemistry as well as multicomponent diffusion fluxes driven by chemical potential gradients. Only local symmetrization of the resulting system of partial differential equations may be achieved because of thermodynamic instabilities even though the entropy function is globally defined. Local symmetrized forms are explicitly evaluated in terms of the inverse of the chemical potential Hessian and local normal forms lead to global existence and asymptotic stability of equilibrium states as well as decay estimates. We also discuss the deficiency of the resulting system of partial differential equations at thermodynamically unstable states typically associated with nonideal fluids.

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18

Bai, Mei, and Lishun Ren. "An SEIV Epidemic Model for Childhood Diseases with Partial Permanent Immunity." Computational and Mathematical Methods in Medicine 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/420952.

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An SEIV epidemic model for childhood disease with partial permanent immunity is studied. The basic reproduction numberR0has been worked out. The local and global asymptotical stability analysis of the equilibria are performed, respectively. Furthermore, if we take the treated rateτas the bifurcation parameter, periodic orbits will bifurcate from endemic equilibrium whenτpasses through a critical value. Finally, some numerical simulations are given to support our analytic results.

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19

Novakovic, Nadezda, S. M. Stojilkovic, and Dragan Gajic. "Wiedemann-Franz ratio in plasmas in the mixtures of xenon, argon and cesium." Facta universitatis - series: Physics, Chemistry and Technology 3, no. 1 (2004): 1–6. http://dx.doi.org/10.2298/fupct0401001n.

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In the present paper, the xenon plasmas containing argon and cesium as additive and pertaining to the above mentioned type (high-pressure low-temperature, LIE) are studied, and a theoretical evaluation of their basic transport parameters is presented. The numerical calculations were based on the assumption that the system is kept under constant pressure and temperature and that it attained local thermo dynamical equilibrium. Some results of a theoretical study which embraced numerical evaluations of the plasma composition, electron collision frequency, electrical and thermal conductivities and Wiedemann-Franz ratio, all regarded as functions of pressure and temperature, are given for high-pressure (from 0,1 MPa to 1,5 MPa) and low-temperature (from 2 000 K to 20 000 K). A previously derived modified expression for the Debye radius, offering the possibility to treat the plasmas considered as weakly non-ideal in the whole temperature range of interest, is used in the evaluations, in combination with the Saha equations for the relevant ionization stages of xenon, argon and cesium. .

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20

Garrett, Daniel G., Eric Ohrn, and Juan Carlos Suárez Serrato. "Tax Policy and Local Labor Market Behavior." American Economic Review: Insights 2, no. 1 (March 1, 2020): 83–100. http://dx.doi.org/10.1257/aeri.20190041.

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Since 2002, the US government has encouraged business investment using accelerated depreciation policies that significantly reduce investment costs. We provide the first in-depth analysis of this stimulus on employment and earnings. Our local labor markets approach exploits cross-industry variation in policy generosity interacted with county-level industry location data. This strategy identifies the partial equilibrium effects of accelerated depreciation. Places that experience larger decreases in investment costs see an increase in employment and earnings. In contrast, the policy does not have positive effects on earnings-per-worker. Overall, our findings suggest federal corporate tax policy has large effects on local labor markets. (JEL D25, G31, H25, H32, J23, J31, R23)

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21

Meng, Xin-You, and Jiao-Guo Wang. "Analysis of a delayed diffusive model with Beddington–DeAngelis functional response." International Journal of Biomathematics 12, no. 04 (May 2019): 1950047. http://dx.doi.org/10.1142/s1793524519500475.

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In this paper, a delayed diffusive phytoplankton-zooplankton model with Beddington–DeAngelis functional response and toxins is investigated. Existence of equilibria of the system are solved. The global asymptotic stability of the zooplankton-free equilibrium is obtained. The local stability of the coexistent equilibrium and existence of Hopf bifurcation are discussed. In addition, the properties of the Hopf bifurcation are studied based on the center manifold and normal form theory for partial differential equations. Finally, some numerical simulations are also carried out to confirm our theoretical analysis.

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22

Embacher, Peter, Nicolas Dirr, Johannes Zimmer, and Celia Reina. "Computing diffusivities from particle models out of equilibrium." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2212 (April 2018): 20170694. http://dx.doi.org/10.1098/rspa.2017.0694.

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A new method is proposed to numerically extract the diffusivity of a (typically nonlinear) diffusion equation from underlying stochastic particle systems. The proposed strategy requires the system to be in local equilibrium and have Gaussian fluctuations but it is otherwise allowed to undergo arbitrary out-of-equilibrium evolutions. This could be potentially relevant for particle data obtained from experimental applications. The key idea underlying the method is that finite, yet large, particle systems formally obey stochastic partial differential equations of gradient flow type satisfying a fluctuation–dissipation relation. The strategy is here applied to three classic particle models, namely independent random walkers, a zero-range process and a symmetric simple exclusion process in one space dimension, to allow the comparison with analytic solutions.

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23

De la Sen, Manuel, Santiago Alonso-Quesada, and Asier Ibeas. "On a Discrete SEIR Epidemic Model with Exposed Infectivity, Feedback Vaccination and Partial Delayed Re-Susceptibility." Mathematics 9, no. 5 (March 2, 2021): 520. http://dx.doi.org/10.3390/math9050520.

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A new discrete Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model is proposed, and its properties of non-negativity and (both local and global) asymptotic stability of the solution sequence vector on the first orthant of the state-space are discussed. The calculation of the disease-free and the endemic equilibrium points is also performed. The model has the following main characteristics: (a) the exposed subpopulation is infective, as it is the infectious one, but their respective transmission rates may be distinct; (b) a feedback vaccination control law on the Susceptible is incorporated; and (c) the model is subject to delayed partial re-susceptibility in the sense that a partial immunity loss in the recovered individuals happens after a certain delay. In this way, a portion of formerly recovered individuals along a range of previous samples is incorporated again to the susceptible subpopulation. The rate of loss of partial immunity of the considered range of previous samples may be, in general, distinct for the various samples. It is found that the endemic equilibrium point is not reachable in the transmission rate range of values, which makes the disease-free one to be globally asymptotically stable. The critical transmission rate which confers to only one of the equilibrium points the property of being asymptotically stable (respectively below or beyond its value) is linked to the unity basic reproduction number and makes both equilibrium points to be coincident. In parallel, the endemic equilibrium point is reachable and globally asymptotically stable in the range for which the disease-free equilibrium point is unstable. It is also discussed the relevance of both the vaccination effort and the re-susceptibility level in the modification of the disease-free equilibrium point compared to its reached component values in their absence. The influences of the limit control gain and equilibrium re-susceptibility level in the reached endemic state are also explicitly made viewable for their interpretation from the endemic equilibrium components. Some simulation examples are tested and discussed by using disease parameterizations of COVID-19.

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24

Zhang, Xuebing, and Hongyong Zhao. "Harvest control for a delayed stage-structured diffusive predator–prey model." International Journal of Biomathematics 10, no. 01 (November 15, 2016): 1750004. http://dx.doi.org/10.1142/s1793524517500048.

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In this paper, we have considered a delayed stage-structured diffusive prey–predator model, in which predator is assumed to undergo exploitation. By using the theory of partial functional differential equations, the local stability of an interior equilibrium is established and the existence of Hopf bifurcations at the interior equilibrium is also discussed. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Finally, the complex dynamics are obtained and numerical simulations substantiate the analytical results.

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25

Hackl, Klaus, Franz Dieter Fischer, and Jiri Svoboda. "Constraints in thermodynamic extremal principles for non-local dissipative processes." Continuum Mechanics and Thermodynamics 32, no. 5 (November 15, 2019): 1337–45. http://dx.doi.org/10.1007/s00161-019-00846-3.

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AbstractPhenomena treated by non-equilibrium thermodynamics can be very effectively described by thermodynamic variational principles. The remarkable advantage of such an approach consists in possibility to account for an arbitrary number of constraints among state or kinetic variables stemming, e.g., from conservation laws or balance equations. As shown in the current paper, the variational principles can provide original evolution equations for the state variables implicitly respecting the constraints. Moreover, the variational approach allows formulating the problem directly in discrete state variables and deriving their evolution equations without the necessity to solve partial differential equations. The variational approach makes it also possible to use different kinetic variables in formulation of dissipation and dissipation function.

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26

Leal, Allan M. M., Dmitrii A. Kulik, William R. Smith, and Martin O. Saar. "An overview of computational methods for chemical equilibrium and kinetic calculations for geochemical and reactive transport modeling." Pure and Applied Chemistry 89, no. 5 (May 1, 2017): 597–643. http://dx.doi.org/10.1515/pac-2016-1107.

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AbstractWe present an overview of novel numerical methods for chemical equilibrium and kinetic calculations for complex non-ideal multiphase systems. The methods we present for equilibrium calculations are based either on Gibbs energy minimization (GEM) calculations or on solving the system of extended law of mass-action (xLMA) equations. In both methods, no a posteriori phase stability tests, and thus no tentative addition or removal of phases during or at the end of the calculations, are necessary. All potentially stable phases are considered from the beginning of the calculation, and stability indices are immediately available at the end of the computation to determine which phases are actually stable at equilibrium. Both GEM and xLMA equilibrium methods are tailored for computationally demanding applications that require many rapid local equilibrium calculations, such as reactive transport modeling. The numerical method for chemical kinetic calculations we present supports both closed and open systems, and it considers a partial equilibrium simplification for fast reactions. The method employs an implicit integration scheme that improves stability and speed when solving the often stiff differential equations in kinetic calculations. As such, it requires compositional derivatives of the reaction rates to assemble the Jacobian matrix of the resultant implicit algebraic equations that are solved at every time step. We present a detailed procedure to calculate these derivatives, and we show how the partial equilibrium assumption affects their computation. These numerical methods have been implemented in Reaktoro (reaktoro.org), an open-source software for modeling chemically reactive systems. We finish with a discussion on the comparison of these methods with others in the literature.

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27

Yakovleva, Svetlana A., Andrey K. Belyaev, and Lyudmila I. Mashonkina. "Inelastic Processes in Strontium-Hydrogen Collisions and Their Impact on Non-LTE Calculations." Atoms 10, no. 1 (March 17, 2022): 33. http://dx.doi.org/10.3390/atoms10010033.

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Inelastic processes rate coefficients for low-energy Sr + H, Sr+ + H−, Sr+ + H, and Sr2+ + H− collisions are calculated using the multichannel quantum model approach. A total of 31 scattering channels of SrH+ and 17 scattering channels of SrH are considered. The partial cross sections and the partial rate coefficients are hence calculated for 1202 partial processes in total. Using new quantum data for Sr ii + H i collisions, we updated the model atom of Sr ii and performed the non-local thermodynamic equilibrium (non-LTE) calculations. We provide the non-LTE abundance corrections for the Sr ii resonance lines in two grids of model atmospheres, which are applicable to very metal-poor ([Fe/H] ≤−2) dwarfs and giants.

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28

Shurygin, Vladimir A. "Nonlinear Problems of Equilibrium Charge State Transport in Hot Plasmas." Symmetry 13, no. 2 (February 16, 2021): 324. http://dx.doi.org/10.3390/sym13020324.

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The general coupling between particle transport and ionization-recombination processes in hot plasma is considered on the key concept of equilibrium charge state (CS) transport. A theoretical interpretation of particle and CS transport is gained in terms of a two-dimensional (2D) Markovian stochastic (random) processes, a discrete 2D Fokker-Plank-Kolmogorov equation (in charge and space variables) and generalized 2D coronal equilibrium between atomic processes and particle transport. The basic tool for analysis of CS equilibrium and transport is the equilibrium cell (EC) (two states on charge and two on space), which presents simultaneously a unit phase volume, the characteristic scales (in space and time) of local equilibrium, and a comprehensive solution for the simplest nonlinear relations between transport and atomic processes. The space-time relationships between the equilibrium constant, transport rates, density distributions, and impurity confinement time are found. The subsequent direct calculation of the total and partial density profiles and the transport coefficients of argon impurity showed a strong dependence of the 2D CS equilibrium and transport on the atomic structure of ions. A model for recovering the recombination rate profiles of carbon impurity was developed basing on the CS equilibrium conditions, the derived relationships, the data about density profiles, plasma parameters and ionization rates.

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29

He, Ze-Rong, Qiang-Jun Xie, and Hai-Tao Wang. "Stability analysis for a size-structured model of species in a space-limited habitat." International Journal of Biomathematics 09, no. 06 (August 2, 2016): 1650093. http://dx.doi.org/10.1142/s1793524516500935.

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We investigate the stability of steady states of a size- and stage-structured population model, which is a hybrid system of ordinary and partial differential equations with global integral feedbacks. After the formulation of a criterion by spectrum method, we derive conditions for global stability of the trivial state and local stability of the positive equilibrium via the basic reproduction rate. Furthermore, some examples and simulations are presented to illustrate the obtained results.

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30

Sommeria, J., C. Staquet, and R. Robert. "Final equilibrium state of a two-dimensional shear layer." Journal of Fluid Mechanics 233 (December 1991): 661–89. http://dx.doi.org/10.1017/s0022112091000642.

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We test a new statistical theory of organized structures in two-dimensional turbulence by direct numerical stimulations of the Navier–Stokes equations, using a pseudo-spectral method. We apply the theory to the final equilibrium state of a shear layer evolving from a band of uniform vorticity: a relationship between vorticity and stream function is predicted by maximizing an entropy with the constraints due the constants of the motion. A partial differential equation for the stream function is then obtained. In the particular case of a very thin initial vorticity band, the Stuart's vortices appear to be a family of solutions for this equation. In more general cases we do not solve the equation, but we test the theory by inspecting the relationship between stream function and vorticity in the final equilibrium state of the numerical computation. An excellent agreement is obtained in regions with strong vorticity mixing. However, local equilibrium is obtained before a complete mixing can occur in the whole fluid domain.

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31

Straughan, B. "Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2157 (September 8, 2013): 20130187. http://dx.doi.org/10.1098/rspa.2013.0187.

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There is increasing interest in convection in local thermal non-equilibrium (LTNE) porous media. This is where the solid skeleton and the fluid may have different temperatures. There is also increasing interest in thermal wave motion, especially at the microscale and nanoscale, and particularly in solids. Much of this work has been based on the famous model proposed by Carlo Cattaneo in 1948. In this paper, we develop a model for thermal convection in a fluid-saturated Darcy porous medium allowing the solid and fluid parts to be at different temperatures. However, we base our thermodynamics for the fluid on Fourier's law of heat conduction, whereas we allow the solid skeleton to transfer heat by means of the Cattaneo heat flux theory. This leads to a novel system of partial differential equations involving Darcy's law, a parabolic fluid temperature equation and effectively a hyperbolic solid skeleton temperature equation. This system leads to novel physics, and oscillatory convection is found, whereas for the standard LTNE Darcy model, this does not exist. We are also able to derive a rigorous nonlinear global stability theory, unlike work in thermal convection in other second sound systems in porous media.

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32

Darmawati and Wahyudin Nur. "Stability, cost-effectiveness, and global sensitivity analysis of COVID-19 model incorporating non-pharmaceutical interventions and indirect transmission." Data Analytics and Applied Mathematics (DAAM) 3, no. 1 (March 30, 2022): 28–41. http://dx.doi.org/10.15282/daam.v3i1.7594.

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Covid-19 is an ongoing pandemic caused by SARS-CoV-2. Some interventions are implemented to control the spread of the disease. In Indonesia, there is a campaign related to non-pharmaceutical approach called 3M. This campaign is carried out so that people use masks, wash their hands, and keep their distance. In this paper, we propose a mathematical model considering non-pharmaceutical interventions and indirect transmission. The non-pharmaceutical interventions studied are the implementation of mask-wearing, handwashing, and social distancing. The model is presented as a system of first-order differential equations. The basic reproduction number is determined. The system has two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point. The local stability condition of the disease-free equilibrium point is proved using the Lienard-Chipart criterion. Center manifold theory is used to prove the local stability condition of the endemic equilibrium point. We also study the optimal control strategy related to mask-wearing, handwashing, and social distancing. Furthermore, cost-effectiveness analysis of intervention strategies is also conducted by studying the average cost-effectiveness ratio of each intervention strategy. Our results show that the most effective strategy to control covid-19 spread is the combination of mask-wearing, handwashing, and social distancing. Moreover, the most cost-effective strategy is mask-wearing intervention. Global sensitivity analysis is performed by studying the partial rank correlation coefficient. The results show that mask-wearing intervention is the most influential intervention on basic reproduction number compared to social distancing and handwashing

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33

Pittel, Boris. "On random quadratic forms: supports of potential local maxima." Journal of Applied Probability 55, no. 4 (December 2018): 1113–30. http://dx.doi.org/10.1017/jpr.2018.74.

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Abstract The selection model in population genetics is a dynamic system on the set of the probability distributions 𝒑=(p1,…,pn) of the alleles A1…,An, with pi(t+1) proportional to pi(t) multiplied by ∑jfi,jpj(t), and fi,j=fj,i interpreted as a fitness of the gene pair (Ai,Aj). It is known that 𝒑̂ is a locally stable equilibrium if and only if 𝒑̂ is a strict local maximum of the quadratic form 𝒑T𝒇𝒑. Usually, there are multiple local maxima and lim𝒑(t) depends on 𝒑(0). To address the question of a typical behavior of {𝒑(t)}, John Kingman considered the case when the fi,j are independent and [0,1]-uniform. He proved that with high probability (w.h.p.) no local maximum may have more than 2.49n1∕2 positive components, and reduced 2.49 to 2.14 for a nonbiological case of exponentials on [0,∞). We show that the constant 2.14 serves a broad class of smooth densities on [0,1] with the increasing hazard rate. As for a lower bound, we prove that w.h.p. for all k≤2n1∕3, there are many k-element subsets of [n] that pass a partial test to be a support of a local maximum. Still, it may well be that w.h.p. the actual supports are much smaller. In that direction, we prove that w.h.p. a support of a local maximum, which does not contain a support of a local equilibrium, is very unlikely to have size exceeding ⅔log2n and, for the uniform fitnesses, there are super-polynomially many potential supports free of local equilibriums of size close to ½log2n.

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34

Wu, Haijiang, and Stéphan Marette. "Local and Global Welfare When Regulating Organic Products: Should Local Regulation Target Production or Consumption?" Sustainability 12, no. 14 (July 8, 2020): 5492. http://dx.doi.org/10.3390/su12145492.

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Based on a welfare approach using a partial equilibrium model coming from microeconomics, this paper analyzes whether a local regulation aimed at reducing risks due to pesticides should be imposed at the production level or the consumption level. This paper characterizes the economic impact of these possible regulations from a theoretical point of view. Local and nonlocal producers compete only in the local market regarding selling conventional and organic products. Local producers incur variable costs related to reducing the risk of damage resulting from their new organic production methods. A local policymaker maximizing local welfare chooses either a regulation that is imposed on its local producers via production requirements or on all local and nonlocal producers via retail requirements that directly affect consumption. We show that local regulation is selected for relatively large values of damage. In this case, the organic regulation is influenced by whether the damage is incurred by residents and the environment close to the production site or by consumers. If the damage is incurred by residents and the environment close to the production site, only regulations targeting the local producers are selected, which improves the profits for nonlocal producers. Concerning damage incurred by consumers after their consumption, each type of regulation is selected depending on the cost of the safe technology, but the regulation targeting the consumption level harms nonlocal producers.

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35

Abboubakar, Hamadjam, Rubin Fandio, Brandon Satsa Sofack, and Henri Paul Ekobena Fouda. "Fractional Dynamics of a Measles Epidemic Model." Axioms 11, no. 8 (July 26, 2022): 363. http://dx.doi.org/10.3390/axioms11080363.

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In this work, we replaced the integer derivative with Caputo derivative to model the transmission dynamics of measles in an epidemic situation. We began by recalling some results on the local and global stability of the measles-free equilibrium point as well as the local stability of the endemic equilibrium point. We computed the basic reproduction number of the fractional model and found that is it equal to the one in the integer model when the fractional order ν = 1. We then performed a sensitivity analysis using the global method. Indeed, we computed the partial rank correlation coefficient (PRCC) between each model parameter and the basic reproduction number R0 as well as each variable state. We then demonstrated that the fractional model admits a unique solution and that it is globally stable using the Ulam–Hyers stability criterion. Simulations using the Adams-type predictor–corrector iterative scheme were conducted to validate our theoretical results and to see the impact of the variation of the fractional order on the quantitative disease dynamics.

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36

Ma, Xinxing, Xianwen Li, Shouwen Zhang, Yanming Zhang, Xiangie Hao, and Jishan Liu. "Impact of Local Effects on the Evolution of Unconventional Rock Permeability." Energies 12, no. 3 (February 1, 2019): 478. http://dx.doi.org/10.3390/en12030478.

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When gas is extracted from unconventional rock, local equilibrium conditions between matrixes and fractures are destroyed and significant local effects are introduced. Although the interactions between the matrix and fracture have a strong influence on the permeability evolution, they are not understood well. This may be the reason why permeability models in commercial codes do not include the matrix-fracture interactions. In this study, we introduced the local force to define the interactions between the matrix and the fracture and derived a set of partial differential equations to define the full coupling of rock deformation and gas flow both in the matrix and in the fracture systems. The full set of cross-coupling formulations were solved to generate permeability evolution profiles during unconventional gas extraction. The results of this study demonstrate that the contrast between the matrix and fracture properties controls the processes and their evolutions. The primary reason is the gas diffusion from fractures to matrixes. The diffusion changes the force balance, mass exchange and deformation.

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37

Lascola, Robert, Patrick E. O’Rourke, and Edward A. Kyser. "A Piecewise Local Partial Least Squares (PLS) Method for the Quantitative Analysis of Plutonium Nitrate Solutions." Applied Spectroscopy 71, no. 12 (October 5, 2017): 2579–94. http://dx.doi.org/10.1177/0003702817734000.

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We have developed a piecewise local (PL) partial least squares (PLS) analysis method for total plutonium measurements by absorption spectroscopy in nitric acid-based nuclear material processing streams. Instead of using a single PLS model that covers all expected solution conditions, the method selects one of several local models based on an assessment of solution absorbance, acidity, and Pu oxidation state distribution. The local models match the global model for accuracy against the calibration set, but were observed in several instances to be more robust to variations associated with measurements in the process. The improvements are attributed to the relative parsimony of the local models. Not all of the sources of spectral variation are uniformly present at each part of the calibration range. Thus, the global model is locally overfitting and susceptible to increased variance when presented with new samples. A second set of models quantifies the relative concentrations of Pu(III), (IV), and (VI). Standards containing a mixture of these species were not at equilibrium due to a disproportionation reaction. Therefore, a separate principal component analysis is used to estimate of the concentrations of the individual oxidation states in these standards in the absence of independent confirmatory analysis. The PL analysis approach is generalizable to other systems where the analysis of chemically complicated systems can be aided by rational division of the overall range of solution conditions into simpler sub-regions.

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38

Gugat, Martin, Falk M. Hante, and Li Jin. "Closed loop control of gas flow in a pipe: stability for a transient model." at - Automatisierungstechnik 68, no. 12 (November 18, 2020): 1001–10. http://dx.doi.org/10.1515/auto-2020-0071.

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AbstractThis contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p=\frac{3}{2}. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow.We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an {L^{2}}-sense. The analysis is done both for the PDE system and an ODE system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions.

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39

Wang, Yanqin, Ling Yang, and Jie Yan. "Dynamics in a delayed diffusive cell cycle model." Nonlinear Analysis: Modelling and Control 23, no. 5 (November 21, 2018): 691–709. http://dx.doi.org/10.15388/na.2018.5.4.

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In this paper, we construct a delayed diffusive model to explore the spatial dynamics of cell cycle in G2/M transition. We first obtain the local stability of the unique positive equilibrium for this model, which is irrelevant to the diffusion. Then, through investigating the delay-induced Hopf bifurcation in this model, we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions. Applying the normal form and center manifold theorem of functional partial differential equations, we also determine the stability and direction of these bifurcating periodic solutions. Finally, numerical simulations are presented to validate our theoretical results.

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40

Scaburri, Raffaele, Agostino Desalvo, and Roberta Nipoti. "Simulation of the Incomplete Ionization of the n-Type Dopant Phosphorus in 4H-SiC, Including Screening by Free Carriers." Materials Science Forum 679-680 (March 2011): 397–400. http://dx.doi.org/10.4028/www.scientific.net/msf.679-680.397.

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The simulation of the incomplete ionization of substitutional dopants in Silicon Carbide (SiC) is often performed using Boltzmann statistics and ionization energy values that do not depend on free carrier concentrations. But in the case of heavy doping Fermi-Dirac statistics is needed, while the case of an inhomogeneous dopants distribution or that of an excess carrier injection requires local free carrier concentration-dependent impurity ionization energies. Here a model for describing partial ionization from diluted to high homogeneous doping densities in SiC and in thermal equilibrium is presented and compared with results on Phosphorus doped 4H-SiC.

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41

Al-Musa, Abdullah, Stanislav Shabunya, Vladimir Martynenko, and Vladimir Kalinin. "Modeling of natural gas partial oxidation in an in-house developed pilot scale catalytic reactor based on local thermodynamic equilibrium concept." Applied Thermal Engineering 113 (February 2017): 238–45. http://dx.doi.org/10.1016/j.applthermaleng.2016.11.035.

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42

Nalewajski, Roman F. "Phase Equalization, Charge Transfer, Information Flows and Electron Communications in Donor–Acceptor Systems." Applied Sciences 10, no. 10 (May 23, 2020): 3615. http://dx.doi.org/10.3390/app10103615.

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Subsystem phases and electronic flows involving the acidic and basic sites of the donor (B) and acceptor (A) substrates of chemical reactions are revisited. The emphasis is placed upon the phase–current relations, a coherence of elementary probability flows in the preferred reaction complex, and on phase-equalization in the equilibrium state of the whole reactive system. The overall and partial charge-transfer (CT) phenomena in alternative coordinations are qualitatively examined and electronic communications in A—B systems are discussed. The internal polarization (P) of reactants is examined, patterns of average electronic flows are explored, and energy changes associated with P/CT displacements are identified using the chemical potential and hardness descriptors of reactants and their active sites. The nonclassical (phase/current) contributions to resultant gradient information are investigated and the preferred current-coherence in such donor–acceptor systems is predicted. It is manifested by the equalization of equilibrium local phases in the entangled subsystems.

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43

Chen, Hao, Benoît Appolaire, and Sybrand van der Zwaag. "Interface Motion and Interface Mobility of the Partitioning Phase Transformations in Fe-Mn-C and Fe-C Alloys: A Cyclic Phase Transformation Approach." Materials Science Forum 706-709 (January 2012): 1367–72. http://dx.doi.org/10.4028/www.scientific.net/msf.706-709.1367.

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A cyclic phase transformation concept has been proposed to investigate the growthkinetics of the austenite (γ) to ferrite (α ) and vice versa in Fe-Mn-C and Fe-C alloys. In the caseof cyclic partial transformations in Fe-Mn-C alloys, two new and special stages are observed:a stagnant stage in which the degree of transformation does not vary while the temperaturechanges and an inverse phase transformation stage, during which the phase transformationproceeds in a direction contradictory to the temperature change. The local equilibrium (LE)and paraequilibrium (PE) are both applied to analyzing the new observations. The stagnantstage was found to be caused by the Mn partitioning, while the inverse phase transformationstage was due to equilibrium conditions not being reached at the transition temperatures.A mixed-mode model is applied to simulating the cyclic phase transformation in Fe-C alloy,and it is found that the cyclic phase transformation concept is a very promising method forinvestigating the interface mobility.

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44

Roze, Denis, and François Rousset. "Selection and Drift in Subdivided Populations: A Straightforward Method for Deriving Diffusion Approximations and Applications Involving Dominance, Selfing and Local Extinctions." Genetics 165, no. 4 (December 1, 2003): 2153–66. http://dx.doi.org/10.1093/genetics/165.4.2153.

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AbstractPopulation structure affects the relative influence of selection and drift on the change in allele frequencies. Several models have been proposed recently, using diffusion approximations to calculate fixation probabilities, fixation times, and equilibrium properties of subdivided populations. We propose here a simple method to construct diffusion approximations in structured populations; it relies on general expressions for the expectation and variance in allele frequency change over one generation, in terms of partial derivatives of a “fitness function” and probabilities of genetic identity evaluated in a neutral model. In the limit of a very large number of demes, these probabilities can be expressed as functions of average allele frequencies in the metapopulation, provided that coalescence occurs on two different timescales, which is the case in the island model. We then use the method to derive expressions for the probability of fixation of new mutations, as a function of their dominance coefficient, the rate of partial selfing, and the rate of deme extinction. We obtain more precise approximations than those derived by recent work, in particular (but not only) when deme sizes are small. Comparisons with simulations show that the method gives good results as long as migration is stronger than selection.

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45

Kočiřík, Milan, Arlette Zikánová, Josef Dubský, and Petr Kroček. "Molecular Transport in a Two-Dimensional Lattice of Sorption Sites." Collection of Czechoslovak Chemical Communications 59, no. 5 (1994): 1001–17. http://dx.doi.org/10.1135/cccc19941001.

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It is known from experimental study of the sorption of some aromatic compounds (such as p-xylene) in zeolites with the MFI lattice that the adsorption kinetics in such systems exhibits appreciable deviations from the 2nd Fick law. To account for those deviations, whose origin lies in the crystal bulk, the existence of two phases which are not in local chemical equilibrium is postulated: a mobile phase is assumed to occur in the straight zeolite channels whereas an immobile phase is assumed in the zig-zag channels. A microkinetic model is proposed assuming localization of the sorbed molecules on two kinds of centres arranged in a regular two-dimensional lattice, along which migration takes place in agreement with the Langmuir kinetics of exchange of molecules between the occupied and unoccupied centres. For an exact description of the time evolution of the considered physical system the model results in a set of ordinary differential equations which are transformed into a system of two partial differential equations providing a spatially continuous description of the physical system. This approach enables the phenomenological kinetic parameters to be interpreted in terms of the jump frequency of molecules between the various kinds of adsorption centres. For small perturbations in the concentration of the sorbing substance the mathematical model is linearized, and the behaviour of the system is illustrated using the numerical solution of the model equations. In dependence on changes in the model parameters, the transition from the case of pure diffusion along a one-dimensional lattice in the mobile phase, with the zig-zag channels inaccessible, via the case of a slowly establishing local equilibrium between the two sub-lattices, to the limiting case of instantaneously establishing local equilibrium between the sub-lattices is discussed.

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46

GIOVANGIGLI, VINCENT, and BINH TRAN. "MATHEMATICAL ANALYSIS OF A SAINT-VENANT MODEL WITH VARIABLE TEMPERATURE." Mathematical Models and Methods in Applied Sciences 20, no. 08 (August 2010): 1251–97. http://dx.doi.org/10.1142/s0218202510004593.

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We investigate the derivation and the mathematical properties of a Saint-Venant model with an energy equation and with temperature-dependent transport coefficients. These equations model shallow water flows as well as thin viscous sheets over fluid substrates like oil slicks, atlantic waters in the Strait of Gilbraltar or float glasses. We exhibit an entropy function for the system of partial differential equations and by using the corresponding entropic variable, we derive a symmetric conservative formulation of the system. The symmetrized Saint-Venant quasilinear system of partial differential equations is then shown to satisfy the nullspace invariance property and is recast into a normal form. Upon establishing the local dissipative structure of the linearized normal form, global existence results and asymptotic stability of equilibrium states are obtained. We finally derive the Saint-Venant equations with an energy equation taking into account the temperature-dependence of transport coefficients from an asymptotic limit of a three-dimensional model.

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47

Adler, P. M., and J. F. Thovert. "Real Porous Media: Local Geometry and Macroscopic Properties." Applied Mechanics Reviews 51, no. 9 (September 1, 1998): 537–85. http://dx.doi.org/10.1115/1.3099022.

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The random geometry of real porous media is analyzed with the objective of reproducing it numerically; adequate algorithms are proposed for consolidated materials which may be statistically homogeneous or not, and may possess more than one solid phase; random packings of star-shape grains are built to mimic non-consolidated materials which are usually obtained by settling processes. The macroscopic properties of all these media can be deduced by solving the local partial differential equations which govern the phenomena; finite difference schemes are used most of the time. A number of physical situations have been already addressed. Elementary transport phenomena such as convection, diffusion and convection-diffusion provide the basic illustrations of our methodology. Multiphase flows is an exception in the sense that the resolution is achieved by means of a lattice-Boltzmann algorithm. The electrokinetic phenomena associated with the motion of an electrolyte through a charged medium are addressed close to equilibrium, in the limit of small dzeta potentials and thick double layers. Industrial processes may involve deposition and/or dissolution of a solute; first-order reactions of a single solute could be successfully analyzed and rationalized with the help of the Pe´clet and the Damko¨hler numbers. Similarly, the macroscopic mechanical properties of the solid matrix of a porous medium can be obtained by solving the elastostatic equations; macroscopic coefficients such as the equivalent Young’s modulus were derived for a number of structures. Some tentative remarks conclude this review. This article contains 289 references.

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48

Son, K. E. "Reduction of the Complete System of Chemical Kinetics Equations for Multicomponent High Temperature Gas Flows Based on the Partial Local Equilibrium Method." High Temperature 58, no. 1 (January 2020): 83–91. http://dx.doi.org/10.1134/s0018151x20010198.

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49

Anders, A., S. Anders, and E. Hantzsche. "Validity conditions for complete and partial local thermodynamic equilibrium of nonhydrogenic level systems and their application to copper vapor arcs in vacuum." IEEE Transactions on Plasma Science 17, no. 5 (1989): 653–56. http://dx.doi.org/10.1109/27.41175.

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50

Groll, R. "Statistical Eulerian Diffusion Approach of Four-Way-Coupled Multiphase Systems." Defect and Diffusion Forum 297-301 (April 2010): 832–37. http://dx.doi.org/10.4028/www.scientific.net/ddf.297-301.832.

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Volume-fraction weighted and Reynolds averaged momentum transport equations are solved in an Euler/Euler approach to simulate numerically the turbulent, dispersed two- phase °ow in a two-dimensional channel and a three-dimensional conic di®user °ow. Particular attention is given to the modelling of turbulent di®usion and particle wall interaction, assuming local equilibrium but introducing individual terms for particle/°uid drag interaction, particle collisions and trajectory crossings. These in°uences have been quanti¯ed in terms of partial viscosities, a restitution power and a turbulence structure parameter. Boussinesq approxima- tions have been used for each phase and their interaction, whose formulation was provided in the framework of the eddy-viscosity modelling concept.

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