Cross-diffusion Coefficients Research Articles

Precipitation loss of Van Allen radiation belt electrons by hiss waves outside the plasmasphere

Plasmaspheric hiss waves have been frequently invoked to explain the slow loss of the radiation belt electrons. However, the effect of hiss waves outside the plasmasphere on the radiation belt electrons remains unclear. Here, on the basis of Van Allen Probes observations and quasilinear simulations, we show that the hiss waves outside the plasmasphere are able to cause the significant precipitation loss of energetic electrons on a timescale of 1 day. In the event of interest, the hiss wave power spectra density reached up to \(10^{-6}~\mbox{nT}^{2}/\mbox{Hz}\), and the obtained pitch-angle diffusion coefficients are found to be \(10^{2}\)–\(10^{4}\) times larger than the momentum and cross diffusion coefficients. During a period of 1 day, the modeled hiss waves caused the depletion of 300–500 keV electrons by up to 10 times. These results suggest that the hiss waves outside the plasmasphere should be taken into account in the future radiation belt modeling.

Spatiotemporal Patterns in a Predator–Prey Model with Cross-Diffusion Effect

The present research deals with the emergence of spatiotemporal patterns of a two-dimensional (2D) continuous predator–prey system with cross-diffusion effect. First, we work out the critical lines of Hopf and Turing bifurcations of the current model system in a 2D spatial domain by means of bifurcation theory. More specifically, the exact Turing region is specified in a two-parameter space. In effect, by choosing the cross-diffusion coefficient as one of the momentous parameter, we demonstrate that the model system undergoes a sequence of spatiotemporal patterns in a homogeneous environment through diffusion-driven instability. Our results via numerical simulation authenticate that cross-diffusion be able to create stationary patterns which enrich the findings of pattern formation in an ecosystem.

Effects of density dependent cross-diffusion on the chaotic patterns in a ratio-dependent prey-predator model

Spatio-temporal chaos is an intriguing part of the spatio-temporal pattern formation, observed in many interacting population models when their heterogeneous distributions within their habitats and movement from one location to the other are taken care of within the modeling approach. When the homogeneous steady-states become unstable, the solutions of the corresponding reaction–diffusion systems never approach a stationary state rather exhibit an irregular nature with respect to both space and/or time. Cross-diffusion terms are incorporated in a system of reaction–diffusion equations to model the situation where presence, absence, abundance of one species influence the movement of another species and vice-versa. In this work, cross-diffusion is considered in a prey–predator model with ratio-dependent functional response along with the self-diffusion terms. After deriving the Turing instability conditions in terms of cross-diffusion parameters, extensive numerical simulations are carried out to study the effect of cross-diffusion on the chaotic dynamics and stationary Turing patterns generated in the system containing self-diffusion terms only. Appropriate numerical tools are used to characterize the spatio-temporal chaos. Route to spatio-temporal chaos and its disappearance are discussed in detail. The chaotic dynamics of the self-diffusion model may be suppressed leading to a stationary state or preserved depending on the cross-diffusion coefficients. The stationary patches of both the species generated in the Turing domain remain stationary but their configurations may change due to the effect of density-dependent cross-diffusion. On the other hand, stationary patches generated in the Turing-Hopf domain change to spatio-temporal chaos for higher dispersal rates of predator avoidance by prey.

Influence of cross-diffusion on the fecally-orally epidemic model with spatial heterogeneity

A strongly coupled cooperative parabolic system, which describes fecally-orally epidemic model with cross-diffusion in a heterogeneous environment, was formulated and analyzed. The basic reproduction number R_{0} ^{D}, which serves as a threshold parameter that predicts whether the coexistence will exist or not, is introduced by the next infection operator and the related eigenvalue problems. By applying upper and lower solutions method, we present the sufficient conditions for the existence of the coexistence solution. The true positive solutions can also be obtained by monotone iterative method. Our results imply that the fecally-orally epidemic model with cross-diffusion admits at least one coexistence solution when the basic reproduction number exceeds one and the cross-diffusion coefficient is sufficiently small, while no coexistence exists when the basic reproduction number is smaller than one or the cross-diffusion coefficient is large enough. Finally, some numerical simulations are exhibited to confirm our analytical findings.

Cross-diffusion-driven Turing instability and weakly nonlinear analysis of Turing patterns in a uni-directional consumer-resource system

Spatiotemporal patterns driven by cross-diffusion of a uni-directional consumer-resource (C-R) system with Holling-II type functional response are investigated in this paper. The existence of a unique positive steady state of the considered system is studied first. The linear stability analysis shows that the cross-diffusion is the key mechanism for the formation of spatiotemporal patterns through Turing bifurcation. We choose the cross-diffusion coefficient as bifurcation parameter and discuss three different types of Turing bifurcations, corresponding to simple, double non-resonant and double resonant cases. Based on weakly nonlinear analysis with the multiple scale method and the adjoint system theory, we derive the amplitude equations of the Turing patterns near the Turing bifurcation point and obtain the analytical approximation solutions of the patterns for each case. Specially, some qualitative results of amplitude equations of the resonant case are given in detail. Finally, numerical simulations are performed to illustrate the weakly nonlinear theoretical predictions and through these simulations some patterns (single mode pattern, mixed-mode pattern, super-squares pattern, roll pattern, hexagonal pattern) are found. Simultaneously, numerical simulations show that the resource supplying rate has an important impact on the direction of Turing bifurcation.

Interaction of human migration and wealth distribution

Dynamics of human populations depends on various economical and social factors. Their migration is partially determined by the economical conditions and it can also influence these conditions. This work is devoted to the analysis of the interaction of human migration and wealth distribution. The model consists of a system of equations for the population density and for the wealth distribution with conventional diffusion terms and with cross diffusion terms describing human migration determined by the wealth gradient and wealth flux determined by human migration. Wealth production and consumption depend on the population density while the natality and mortality rates depend on the level of wealth. In the absence of cross diffusion terms, dynamics of solutions is described by travelling wave solutions of the corresponding reaction–diffusion systems of equations. We show persistence of such solutions for sufficiently small cross diffusion coefficients. This result is based on the perturbation methods and on the spectral properties of the linearized operators.

Uncertainty analysis of assembly and core-level calculations with application to CASMO-4E and SIMULATE-3

The topic of this paper is the development of sensitivity and uncertainty analysis capability to the CASMO-4/CASMO-4E – SIMULATE-3 code sequence in the context of the OECD/NEA benchmark ‘Uncertainty Analysis in Best-Estimate Modelling for Design, Operation and Safety Analysis of LWRs’ (UAM). The developed capability uses a two-step approach. In the first step, Uncertainties in nuclear data are propagated to two-group cross sections, diffusion coefficients, and assembly discontinuity factors. This is carried out using deterministic, perturbation-theory-based uncertainty analysis methodology. In the second step, a global covariance matrix, characterizing the uncertainties of the group constants, is formed, and the uncertainties are propagated through a full core SIMULATE calculation using a stochastic approach. This system enables the analysis of nuclear data related uncertainties in assembly-homogenized group constants, assembly discontinuity factors, and pin powers, as well as full core results such as multiplication factor and power distribution. The mathematical background of the deterministic uncertainty analysis methodology is reviewed and the main conclusions related to the implementation are summarized. Numerical results are presented for the full core Three Mile Island model in exercise I-3 of the UAM benchmark at hot zero power with all rods out and inserted. The computational efficiency of the calculations is discussed.

Stability Control in a Binary Fluid Mixture Subjected to Cross Diffusive Coefficients

The effect of feedback control on the onset of double-diffusive convection in a horizontal binary fluid layer is studied analytically subjected to cross diffusion coefficients which are the Soret and Dufour parameters. The confined boundaries of the binary fluid layer are considered to be free-free, rigid-free and rigid-rigid which described the lower and upper surfaces respectively. The linear stability theory is applied and the eigenvalue solution is obtained numerically using Galerkin technique. Focusing on the stationary convection, it is shown that there is a positive thermal resistance in the presence of feedback control on the onset of double-diffusive convection in binary fluid mixture.

Positive solutions for Lotka–Volterra competition system with large cross-diffusion in a spatially heterogeneous environment

In this paper, we study the stationary problem for the Lotka–Volterra competition system with cross-diffusion in a spatially heterogeneous environment. Although some sufficient conditions for the existence of positive solutions are obtained by using global bifurcation theory, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we focus on the asymptotic behaviour of positive solutions and derive two shadow systems as the cross-diffusion coefficient tends to infinity, moreover, the structure of positive solutions of the limiting system is analysed. The result of asymptotic behaviour also reveals different phenomena from that studied in Wang and Li (2013).

Does Oxygen Transport Affect the Cell Voltages of Metal/Air Batteries?

Simultaneous transport of an electrolyte and dissolved oxygen is analyzed with Newman’s concentrated-solution theory to assess how nonuniform oxygen distributions might impact the voltages of metal/air batteries. For a solution comprising a neutral solvent, a simple salt, and oxygen, the Onsager–Stefan–Maxwell transport equations are inverted, yielding flux-explicit laws for oxygen, anion, and cation transport that distinguish the effects of individual diffusion and migration driving forces. Along with the ionic conductivity, electrolyte diffusivity, oxygen diffusivity, and cation transference number, a migration coefficient and a cross-diffusion coefficient are identified, which respectively account for the effects of electro-osmotic drag on oxygen and diffusional drag between salt and oxygen. A derived current/voltage relation reveals how oxygen gradients can in principle affect the cell potential; significant diffusion potential can arise from oxygen if it experiences electro-osmotic drag. Prior models are proven to follow from an assumption that cross-diffusion and electro-osmosis are both negligible, or, equivalently, that oxygen/ion interactions are weak. Experiments to quantify the novel transport properties are discussed, along with quantitative estimates of the cross diffusivity and migration coefficient.

Direct measurement of the plasma loss width in an optimized, high ionization fraction, magnetic multi-dipole ring cusp

The loss width of plasma in the WiPAL multi-dipole magnetic ring cusp [Cooper et al., Phys. Plasmas 21, 13505 (2014); Forest et al., J. Plasma Phys. 81, 345810501 (2015)] has been directly measured using a novel array of probes embedded in the insulating plasma limiters. The large plasma volume (∼10 m3), small loss area associated with strong rare earth permanent magnets (Bo∼2.23 kG at face), and large heating power (≤200 kW) produces a broad range of electron temperatures (2<Te<15 eV), ion temperatures (0.03<Ti<2 eV), plasma densities (3×1010<ne<2×1012 cm−3), and ionization fractions (0.05<ne/(ne+nn)<1), in both argon and helium, all of which were accurately measured. This plasma regime, accessible with high magnetic fields, differs from previous devices: the cusp loss width is much larger than the Debye length and electron gyroradius and comparable to the collision length. Plasma parameters measured at the surface of ceramic limiter tiles covering the magnets and along radial chords in the cusp magnetic field indicate that electron density and temperature are nearly constant on magnetic field lines and that the mirror forces play little role in confining the plasma other than to constrict the loss area. Particle balance modeling is used to determine the cross field diffusion coefficient base on the measured losses to the limiters. The experimentally determined cross field diffusion coefficient (which determines the cusp loss width) is consistent with ambipolar diffusion across five orders of magnitude. The ambipolar diffusion across a given field line is set primarily by the electron-neutral collisions in the region where the magnetic field is the weakest, even though these plasmas can have ionization fractions near 1.

Cross-diffusion-driven gravitational instability in a Hele-Shaw cell saturated with a ternary solution

The effect of cross diffusion on the onset of the gravitational instabilities in a Hele-Shaw cell saturated with a ternary solution is analyzed. Based on the linear stability theory, new stability equations are derived in the global domain and then transformed into the similar domain. These stability equations are solved by employing various methods such as an initial value problem approach and quasi-steady state approximations (QSSA’s). Through the initial growth rate analysis without the QSSA, we prove that initially the system is unconditionally stable. However, the QSSA in the global domain showed that the system can be initially unstable for a certain condition. Based on the QSSA in the similar domain (QSSAζ), we obtain the critical time for the onset of instability motion. As expected, the higher δ21β makes the system more unstable, i.e., accelerates the onset of instability motion; here δ21 and β represent the normalized cross diffusion coefficient and the ratio of densification coefficients, respectively. Based on the linear analysis, fully nonlinear analyses are also conducted by using the Fourier spectral method. The present nonlinear analyses show that the double-diffusive and diffusive-layer convection-type of instabilities are possible for the positive and negative δ21β-values, respectively. From the present nonlinear analysis, the system having δ22 > 1 prefers the instabilities with a larger wavelength than the system having δ22 < 1. Here δ22 is the normalized normal diffusion coefficient of component B.

Thermal gravitational separation of ternary mixture n-dodecane/isobutylbenzene/tetralin components in a porous medium.

Our present work focuses on the coupling between thermal diffusion and convection in order to improve the thermal gravitational separation of mixture components. The separation phenomenon was studied in a porous medium contained in vertical columns. We performed analytical and numerical simulations to corroborate the experimental measurements of the thermal diffusion coefficients of ternary mixture n-dodecane, isobutylbenzene, and tetralin obtained in microgravity in the international space station. Our approach corroborates the existing data published in the literature. The authors show that it is possible to quantify and to optimize the species separation for ternary mixtures. The authors checked, for ternary mixtures, the validity of the "forgotten effect hypothesis" established for binary mixtures by Furry, Jones, and Onsager. Two complete and different analytical resolution methods were used in order to describe the separation in terms of Lewis numbers, the separation ratios, the cross-diffusion coefficients, and the Rayleigh number. The analytical model is based on the parallel flow approximation. In order to validate this model, a numerical simulation was performed using the finite element method. From our new approach to vertical separation columns, new relations for mass fraction gradients and the optimal Rayleigh number for each component of the ternary mixture were obtained.

Overview of methodology for spatial homogenization in the Serpent 2 Monte Carlo code

This paper describes the methods used in the Serpent 2 Monte Carlo code for producing homogenized group constants for nodal diffusion and other deterministic reactor simulator calculations. The methodology covers few-group reaction cross sections, scattering matrices, diffusion coefficients and poison cross sections condensed in infinite and B1 leakage-corrected critical spectra, as well as the calculation of discontinuity factors, pin-power form factors, delayed neutron parameters and total and partial albedos. Also included is a description of an automated burnup sequence, which was recently implemented for the handling of restart calculations with branch variations. This capability enables covering the full range of local operating conditions required for the parameterization of group constants within a single run. The purpose of this paper is to bring the methodological description provided in earlier publications up to date, and provide insight into the developed methods and capabilities, including their limitations and known flaws.

Mass and heat diffusion in ternary polymer solutions: A classical irreversible thermodynamics approach

Governing equations for evolution of concentration and temperature in three-component systems were derived in the framework of classical irreversible thermodynamics using Onsager’s variational principle and were presented for solvent/solvent/polymer and solvent/polymer/polymer systems. The derivation was developed from the Gibbs equation of equilibrium thermodynamics using the local equilibrium hypothesis, Onsager reciprocal relations and Prigogine’s theorem for systems in mechanical equilibrium. It was shown that the details of mass and heat diffusion phenomena in a ternary system are completely expressed by a 3 × 3 matrix whose entries are mass diffusion coefficients (4 entries), thermal diffusion coefficients (2 entries) and three entries that describe the evolution of heat in the system. The entries of the diffusion matrix are related to the elements of Onsager matrix that are bounded by some constraints to satisfy the positive definiteness of entropy production in the system. All the elements of diffusion matrix were expressed in terms of derivatives of exchange chemical potentials of the components with respect to concentration and temperature. The spinodal curves of ternary polymer solutions were derived from the governing equations and their correctness was checked by the Hessian of free energy density. Moreover, it was proved that setting cross-diffusion coefficients to zero results in a contradiction, and the governing equations without cross-diffusion coefficients do not express the actual phase behavior of the system.

Assessment of rabbit hemorrhagic disease in controlling the population of red fox: A measure to preserve endangered species in Australia

Predator's management requires a detailed understanding of the ecological circumstances associated with predation. Predation by foxes has been a significant contributor to the Australian native animal reduction. This paper mainly focuses on the dissemination of rabbit hemorrhagic disease in the rabbit population and its subsequences on red fox (Vulpes vulpes) population, by qualitative and quantitative analyses of a designed eco-epidemiological model with simple law of mass action and sigmoid functional response.Existence of solution has been analyzed and shown to be uniformly bounded. The basic reproduction number (R0) is obtained and the occurrence of a backward bifurcation at R0=1 is shown to be possible using central manifold theory. Global stability of endemic equilibrium is established by geometric approach. Criteria for diffusion-driven ecological instability caused by local random movements of European rabbits and red fox are obtained. Detailed analyses of Turing patterns formation selected by reaction-diffusion system under zero flux boundary conditions are presented. We found that transmission rate, self and cross-diffusion coefficients have appreciable influence on spatial spread of epidemics. Numerical simulation results confirm the analytical finding and generate patterns which indicate that population of red foxes might be controlled if rabbit hemorrhagic disease (RHD) is introduced into the rabbit population and thus ecological balance can be maintained.

Investigation on fluctuations in full-size Molten Salt Reactor with coupled neutronic/thermal-hydraulic model

In this paper, the fluctuations in neutron fluxes, delayed neutron precursors, fuel salt temperature and velocity in a full-size MSR are investigated by coupling the neutronic and thermal-hydraulic models. The neutronic model is established based on two-group diffusion theory with one averaged group of delayed neutron precursors. The balance equations for mass and energy are chosen to build the thermal-hydraulic model, while the momentum equation is neglected in the assumption of constant pressure. The group constants, including the cross sections and diffusion coefficients, are generated by means of the HELIOS code and fitted as functions of temperature to build the thermal feedback. The equations for fluctuations are derived based on linear perturbation theory, assuming the perturbation is small enough. All the equations are discretized and numerically solved with the help of a developed code. The static results show that the neutron flux distributions are almost independent of the fuel recirculation, and the distribution of precursors is not that sensitive to the fuel velocity change, due to the small delayed neutron fraction and large decay constant of the fuel material. In the dynamic case, it is found that the thermal feedback effect is important at low frequencies, i.e. when the point kinetic effect is prominent and the neutronic power is significantly perturbed. Similar to the static case, the fuel recirculation has apparent influence on the precursor noise, whereas its effect on the neutron noise is very small. Moreover, due to the interference among the effects of fluctuations in various group constants, the fast and thermal neutron noise are in opposite phases. Such out-phase behavior has been first observed in the case of traditional reactors and reconfirmed in the present study for a closed-loop MSR system.

Effect of 2-Hydroxypropyl-β-cyclodextrin on the diffusion behaviour of L-dopa in aqueous solutions

Ternary mutual diffusion coefficients (D11, D22, D12 and D21) measured by the Taylor dispersion method are reported for aqueous solutions of levodopa (L-dopa) (1) in the presence of 2-Hydroxypropyl-β-cyclodextrin (HP-β-CD) (2) at 25°C and 37°C and concentrations up to 0.005mol·dm−3. The effects of HP-β-CD (HP-β-CD) on the transport of L-dopa are assessed by comparing the binary diffusion coefficient of aqueous L-dopa solutions with the main diffusion coefficient (D11) measured for ternary {L-dopa (1)+HP-β-CD (2)} solutions. 1H NMR spectroscopy suggests no significant complexation between HP-β-CD and L-dopa. However, coupled diffusion of L-dopa and HP-β-CD occurs, as indicated by non-zero values of the cross-diffusion coefficients, D12 and D21. HP-β-CD concentration gradients produce significant co-current coupled flows of L-dopa.

An elliptic cross-diffusion system describing two-species models on a bounded domain with different natural conditions

This paper is concerned with an elliptic cross-diffusion system describing two-species models on a bounded domain Ω, where Ω consists of a finite number of subdomains Ωi (i=1,…,m) separated by interfaces Γj (j=1,…,m−1) and the natural conditions of the subdomains Ωi are different. This system is strongly coupled and the coefficients of the equations are allowed to be discontinuous on interfaces Γj. The main goal is to show the existence of nonnegative solutions for the system by Schauder's fixed point theorem. Furthermore, as applications, the existence of positive solutions for some Lotka–Volterra models with cross-diffusion, self-diffusion and discontinuous coefficients are also investigated.

Fickian Diffusion in Ternary Mixtures Composed by 1,2,3,4-Tetrahydronaphthalene, Isobutylbenzene, and n-Dodecane

We report the Fickian diffusion coefficients in 20 ternary mixtures formed by 1,2,3,4-tetrahydronaphthalene (THN), isobutylbenzene (IBB) and n-dodecane (nC12) measured by the Taylor dispersion technique at 298.1 K and atmospheric pressure. Four diffusion coefficients of the ternary mixtures were measured along six concentration paths starting on one binary subsystem and moving toward the other one. We found expressions for the diffusion matrix of a ternary mixture approaching to the binary limits. The measured diffusion coefficients were thoroughly verified by comparison with the theoretical asymptotic behavior. The main diffusion coefficients vary smoothly over the entire concentration space and D11 is always larger than D22. One of the two cross-diffusion coefficients is of the same order of magnitude as the main ones and, hence, not negligible, whereas the other one is close to zero. The investigated mixtures also comprise compositions that were examined in microgravity experiments in the ESA DCMIX1 project.