Full Transport Equation Research Articles

Effects of non-continuous inverse Compton cooling in blazars

Blazar flares provide a window onto the extreme physical processes occurring in relativistic outflows. Most numerical codes used for modelling blazar emission during flares use a simplified continuous-loss description of particle cooling due to the inverse Compton (IC) process, neglecting non-continuous (discrete) effects that arise in the Klein-Nishina (KN) regime. The significance of such effects has not yet been explored in detail. In this study, we investigate the importance of non-continuous Compton cooling losses and their impact on the electron spectrum and spectral energy distribution (SED) of blazars during high flux states (flares), as well as in the low state. We solve the full transport equation numerically, accounting for large relative jumps in energy by extending our existing blazar flare modelling code EMBLEM. We perform a detailed physical modelling of the brightest gamma -ray flare of the archetypal flat-spectrum radio quasar (FSRQ) 3C\,279 detected in June 2015. We then compare results obtained using the full cooling term and using the continuous-loss approximation. We show that during flaring states of FSRQs characterised by high Compton dominance, the non-continuous cooling can lead to significant modification of the electron spectrum, introducing a range of distinct features, such as low-energy tails, hardening and/or softening, narrow and extended particle excesses, and shifts in the cooling break position . Such distortion translates to differences in the associated SED of up to sim 50<!PCT!>. This highlights the importance of non-continuous effects and the need to consider them in blazar emission models, particularly applied to extreme gamma -ray flares.

Laminar drag reduction in surfactant-contaminated superhydrophobic channels

Although superhydrophobic surfaces (SHSs) show promise for drag reduction applications, their performance can be compromised by traces of surfactant, which generate Marangoni stresses that increase drag. This question is addressed for soluble surfactant in a three-dimensional laminar channel flow, with periodic SHSs made of long finite-length longitudinal grooves located on both walls. We assume that bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. Exploiting long-wave theory and accounting for the difference between the rapid transverse and slower longitudinal Marangoni flows, we derive a one-dimensional model for surfactant transport from the full three-dimensional transport equations. Our one-dimensional model allows us to predict the drag reduction and surfactant distribution across the parameter space. The system exhibits multiple regimes, involving competition between Marangoni effects, bulk and interfacial diffusion, bulk and interfacial advection, shear dispersion and surfactant exchange between the bulk and the interface. We map out asymptotic regions in the high-dimensional parameter space, and derive explicit closed-form approximations of the drag reduction, without any fitting or empirical parameters. The physics underpinning the drag reduction effect and the negative effect of surfactant is discussed through analysis of the velocity field and surfactant concentrations, which show both uniform and non-uniform stress distributions. Our theoretical predictions of the drag reduction compare well with results from the literature solving numerically the full three-dimensional transport problem. Our atlas of maps provides a comprehensive analytical guide for designing surfactant-contaminated channels with SHSs, to maximise the drag reduction in applications.

Effects of Different Closure Choices in Core-collapse Supernova Simulations

The two-moment method is widely used to approximate the full neutrino transport equation in core-collapse supernova (CCSN) simulations, and different closures lead to subtle differences in the simulation results. In this paper, we compare the effects of closure choices on various physical quantities in 1D and 2D time-dependent CCSN simulations with our multigroup radiation hydrodynamics code Fornax. We find that choices of the third-order closure relations influence the time-dependent simulations only slightly. Choices of the second-order closure relation have larger consequences than choices of the third-order closure, but these are still small compared to the remaining variations due to ambiguities in some physical inputs such as the nuclear equation of state. We also find that deviations in Eddington factors are not monotonically related to deviations in physical quantities, which means that simply comparing the Eddington factors does not inform one concerning which closure is better.

Charge–velocity correlation transport equations in gas–solid flow with triboelectric effects

This paper is dedicated to the development of particle charge and velocity second-order moment transport equations for monodisperse particles in gas flow with a tribocharging effect. The full transport equations for the particle charge–velocity covariance and the charge variance are derived in the framework of the kinetic theory of granular flow assuming that the electrostatic interaction does not modify the collision dynamics. The collision integrals are solved without presuming the form of the electric part for the particle probability density function. The full second-order transport equation model is tested in a one-dimensional periodic domain. The results show that this model is able to capture more important physical mechanisms that are neglected by simple algebraic models proposed in the past. An in-depth analysis of the transport equations is also performed. This study reveals that, for sufficiently small covariance characteristic destruction time scales, the transient and third-order moments terms can be safely neglected. In addition, two different reduced-order models are proposed: a more general algebraic model that takes into account the variance effect and a semi-algebraic model that only resolves a transport equation for the charge variance coupled with an algebraic model for the covariance. The former could, however, lead to non-physical predictions in many cases, while the latter can be a suitable alternative only for a sufficiently small interparticle collision time. Finally, a simple chart based on test case simulations is proposed to show under which conditions a semi-algebraic model could be considered as a suitable alternative.

Comprehensive modulation potential for the solar modulation of Galactic cosmic rays

The concept of the modulation potential being utilized to describe the solar modulation of galactic cosmic rays is calculated based on the full three-dimensional Parker transport equation. This is in sharp contrast to the force-field approach, which is based on a series of approximations of this equation and where traditionally all the physical processes are condensed into a single parameter without knowing their relative contribution. In our comprehensive approach to the modulation potential, the contribution of all the different physical processes is given explicitly as derived directly from the parameters of Parker's transport equation. We use the stochastic differential equations approach to study the effects of each of these different physical modulation processes thoroughly and then also determine the rigidity and mass-to-charge-ratio ($A/Z$) of these processes for the first time in the context of a modulation potential. It is found that: (1) This comprehensive modulation potential for galactic cosmic particles with a given rigidity at the Earth is a random variable, which can be fit by an inverse Gaussian distribution; (2) the rigidity and $A/Z$ dependence of this modulation potential can be divided into three categories based on the index value of power law diffusion; (3) the modulation caused by convection becomes weaker with increased rigidity and eventually dissipates, leading to a larger modulation potential for particles with a larger $A/Z$; (4) particle drift significantly reduces this modulation potential and its variation with changes in the tilt angle of the heliospheric current sheet has a distinct pattern for each of the two solar magnetic field polarities.

Quasilinear theory of general electromagnetic fluctuations including discrete particle effects for magnetized plasmas: General analysis

The general quasilinear Fokker–Planck kinetic equation for the gyrophase-averaged plasma particle distribution functions in magnetized plasmas is derived, making no restrictions on the energy of the particles and on the frequency of the electromagnetic fluctuations and avoiding the often made Coulomb approximation of the electromagnetic interactions. The inclusion of discrete particle effects breaks the dichotomy of nonlinear kinetic plasma theory divided into the test particle and the test fluctuation approximation because it provides expression of both the non-collective and collective electromagnetic fluctuation spectra in terms of the plasma particle distribution functions. Within the validity of the quasilinear approach, the resulting full quasilinear transport equation can be regarded as a determining nonlinear equation for the time evolution of the plasma particle distribution functions.

Destabilization of binary mixing layer in supercritical conditions

Compressible mixing layer instabilities are of importance to a wide range of environmental and industrial applications. Past studies have focused on either ideal-gas or real-fluid thermodynamic regimes of single-species mixing layers. However, mixing layers of binary mixtures at supercritical conditions, commonly encountered in fuel injection systems, introduce additional complexities due to the added compositional degree-of-freedom. Moreover, the effect of strong variations in thermodynamic response functions across the Widom line on the binary mixing layer stability remains poorly understood. Thus, the objective of this study is to examine the coupling between the hydrodynamic instability and the real-fluid thermodynamics across the Widom line and its effects on the overall binary mixing layer dynamics. To this end, we develop a linear stability analysis of the full binary-species compressible transport equations coupled with the PC-SAFT equation of state. Analysis shows the existence of a novel instability mechanism that arises from juxtapositioning of the Widom-line transition and the hydrodynamic inflection point. This novel thermodynamically induced instability mechanism has the net effect of destabilizing the binary mixing layer at lowering supercritical conditions towards the critical pressure point. This is in contrast to previous stability analyses of supercritical single-species mixing layers, where increasing pressure destabilizes the flow due to its effect on reducing the density stratification. The discovered thermodynamically induced instability mechanism of binary mixing flows highlights the need for an extension of classical instability criteria to incorporate the effect of strong variations in the thermodynamic response functions across the Widom line on mixing layer instability.

Modeling of deuterium and carbon radiation transport in MAST-U tokamak advanced divertors

Modest effects of deuterium and carbon radiation opacity in the Super-X and snowflake divertor plasmas are predicted for MAST Upgrade tokamak with core plasma input power 2.5–5 MW and plasma current 1 MA. The radiation transport modeling is based on the SOLPS-EIRENE and UEDGE code divertor plasma predictions. Two radiation transport models are used: one is based on a full radiation transport equation implemented in the radiation transport and collisional-radiative code CRETIN (without feedback on the background plasma), and another is an internal self-consistent UEDGE model with ionization, recombination, and heating rates corrected for Ly α line trapping based on the escape probability model implemented in CRETIN. In MAST-U, the Super-X and snowflake divertor plasmas are predicted to reach detached regimes at lower upstream densities than the standard divertor, and the conclusion still holds with radiation transport effects included. At neutral densities m−3, modest Ly α deuterium line trapping with optical depths 10–15 is predicted in the Super-X divertor. Divertor plasmas are optically thin to other Lyman and Balmer lines, as well as to strong C III and C IV lines that are responsible for most of divertor radiated power. Insignificant changes (within a few percent) to divertor deuterium ionization and recombination rates are found. Radiation fluxes on outer divertor target are modified within a factor of 2–3 when the radiation transport is accounted for, and a similar variation is found due to the line shape models that define the absorption and emission line profiles in the radiation transport modeling. The predicted Lyman and Balmer spectral intensities are significantly modified due to radiation trapping. A measurement of divertor radiation transport effects is discussed using the Ly β /Ba α line ratio. In the snowflake divertor configuration, divertor plasmas are found to be optically thin to Lyman series lines within a large range of parameter variations that include upstream density, divertor transport coefficients, and magnetic configurations. Modest radiation transport effects are only found in a few cases with strongest divertor transport and magnetic configurations closest to the ideal snowflake configuration, however, the plasma background models that were used are yet to be validated with an experiment.

Dynamical conditions and causal transport of dissipative spherical collapse in f(R, T) gravity

In this paper, we have investigated the non-adiabatic spherical gravitational collapse in the framework of the $f(R,T)$ theory of gravity with a locally anisotropic fluid that undergoes dissipation in the form of heat flux, free-streaming radiation, and shearing viscosity. The dynamical equations are analyzed in detail, both in the Newtonian and post-Newtonian regimes. Finally we couple the dynamical equations to the full causal transport equation in the context of Israel-Stewart theory of dissipative systems. This yields us a better understanding of the collapse dynamics and may be connected to various astrophysical consequences.

Stochastic re-acceleration and magnetic-field damping in Tycho’s supernova remnant

Context. Tycho’s supernova remnant (SNR) is associated with the historical supernova (SN) event SN 1572 of Type Ia. The explosion occurred in a relatively clean environment, and was visually observed, providing an age estimate. This SNR therefore represents an ideal astrophysical test-bed for the study of cosmic-ray acceleration and related phenomena. A number of studies suggest that shock acceleration with particle feedback and very efficient magnetic-field amplification combined with Alfvénic drift are needed to explain the rather soft radio spectrum and the narrow rims observed in X-rays. Aims. We show that the broadband spectrum of Tycho’s SNR can alternatively be well explained when accounting for stochastic acceleration as a secondary process. The re-acceleration of particles in the turbulent region immediately downstream of the shock should be efficient enough to impact particle spectra over several decades in energy. The so-called Alfvénic drift and particle feedback on the shock structure are not required in this scenario. Additionally, we investigate whether synchrotron losses or magnetic-field damping play a more profound role in the formation of the non-thermal filaments. Methods. We solved the full particle transport equation in test-particle mode using hydrodynamic simulations of the SNR plasma flow. The background magnetic field was either computed from the induction equation or follows analytic profiles, depending on the model considered. Fast-mode waves in the downstream region provide the diffusion of particles in momentum space. Results. We show that the broadband spectrum of Tycho can be well explained if magnetic-field damping and stochastic re-acceleration of particles are taken into account. Although not as efficient as standard diffusive shock acceleration, stochastic acceleration leaves its imprint on the particle spectra, which is especially notable in the emission at radio wavelengths. We find a lower limit for the post-shock magnetic-field strength ∼330 μG, implying efficient amplification even for the magnetic-field damping scenario. Magnetic-field damping is necessary for the formation of the filaments in the radio range, while the X-ray filaments are shaped by both the synchrotron losses and magnetic-field damping.

Dynamics of dissipative gravitational collapse with full causal approach in f(R) gravity

The aim of this outline is to investigate the consequences of viscous dissipative self-gravitational collapse in non-static spherically symmetric geometry by adopting a Misner–Sharp approach in the context of f(R) metric formalism. In this outline, we propose a viable f(R) model for the physical significance of a self-gravitating system. We determine the junction conditions between interior and exterior space–times. Furthermore, we extend our work for the dissipative dark source case in both forms of heat transformation and free radiation streaming. Finally, the dissipative variables such as heat flux, bulk, and shear viscosity are evaluated in the dynamical equation and then coupled with full causal transport equations in the background of the Israel–Stewart notion. The present analysis discusses the whole study of self-gravitational collapse and recognizes the form of a reduced collapse due to decreasing gravitational mass density, which depends on thermodynamical collapsing variables.

Study of non-spherical bubble oscillations under acoustic irradiation in viscous liquid

The effect of dissipation on the shape stability of a harmonically excited bubble is investigated. The employed liquid is the highly viscous glycerine. The rate of the dissipation is controlled through the alteration of viscosity of the liquid by varying its temperature. The mean radius of the bubble during its radial oscillation is described by the Keller-Miksis equation. Two approaches are used to describe the surface oscillations. The first model solves the surface dynamics equations of each mode together with the transport equation of the vorticity in the liquid domain. The second model approximates the transport equation, which is a partial differential equation, with a boundary layer approximation reducing the required computational resources significantly. The comparison of the surface models shows qualitative agreement at low dissipation rate; however, at high viscosity the application of the full transport equation is mandatory. The results show that an increasing rate of dissipation can significantly extend the shape stable domains in the excitation frequency-pressure amplitude parameter plane. Nevertheless, the collapse strength is decreasing due to the highly damped oscillations. It has been found that an optimal range of dissipation rate in terms of temperature can be defined expressing a good compromise between the collapse strength and surface stability. The computations are carried out by an in-house GPU accelerated initial value problem solver.

The impact of different neutrino transport methods on multidimensional core-collapse supernova simulations

Neutrinos play a crucial role in the core-collapse supernova (CCSN) explosion mechanism. The requirement of accurately calculating the transport of neutrinos makes simulations of the CCSN mechanism extremely challenging and computationally expensive. Historically, this stiff challenge has been met by making approximations to the full transport equation. In this work, we compare CCSN simulations in one- and two-dimensions with three approximate neutrino transport schemes, each implemented in the FLASH simulation framework. We compare a two-moment M1 scheme with an analytic closure (M1), the isotropic diffusion source approximation (IDSA), and the advanced spectral leakage method. We identify and discuss the advantages and disadvantages of each scheme. For each approximate transport scheme, we use identical grid setups, hydrodynamics, and gravity solvers to investigate the transport effects on supernova shock dynamics and neutrino quantities. We find that the transport scheme has a small effect on the evolution of protoneutron star (PNS) radius, PNS mass, and the mass accretion rate. The neutrino luminosities, mean energies, and shock radii have a ∼10%–20% quantitative difference but the overall qualitative trends are fairly consistent between all three approximations. We find larger differences in the gain region properties, including the gain region mass and the net heating rate in the gain region, as well as the strength of PNS convection in the core. We investigate the progenitor, nuclear equation of state, and stochastic perturbation dependence of our simulations and find similar magnitudes of impact on key quantities. We also compare the computational expense of the various approximations.

Modeling of the spatially resolved nonthermal emission from the Vela Jr. supernova remnant

Vela Jr. (RX J0852.0−4622) is one of just a few known supernova remnants (SNRs) with a resolved shell across the whole electromagnetic spectrum from radio to very-high-energy (>100 GeV; VHE) gamma-rays. Its proximity and large size allow for detailed spatially resolved observations of the source, making Vela Jr. one of the primary sources used for the study of particle acceleration and emission mechanisms in SNRs. High-resolution X-ray observations reveal a steepening of the spectrum toward the interior of the remnant. In this study we aim for a self-consistent radiation model of Vela Jr. which at the same time would explain the broadband emission from the source and its intensity distribution. We solve the full particle transport equation combined with the high-resolution one-dimensional (1D) hydrodynamic simulations (using Pluto code) and subsequently calculate the radiation from the remnant. The equations are solved in the test particle regime. We test two models for the magnetic field profile downstream of the shock: damped magnetic field, which accounts for the damping of strong magnetic turbulence downstream, and transported magnetic field. Neither of these scenarios can fully explain the observed radial dependence of the X-ray spectrum under spherical symmetry. We show, however, that the softening of the spectrum and the X-ray intensity profile can be explained under the assumption that the emission is enhanced within a cone.

Phonon backscatter, trapping, and misalignment effects on microscale thermal conductance below the Casimir limit

At nanometer to micron length scales, there exists a strong competition between intrinsic and extrinsic scattering mechanisms that can curtail the free flight of phonons and ultimately affect the thermal transport. Despite significant progress in showing the ability to reach behaviors significantly below the Casimir limit, little appears to be understood about the competition between these scattering sources. In this investigation, we propose a simple one-parameter geometry that simultaneously modulates backscattering and trapping effects to enable directed study of these different means of controlling phonons. The geometry is a simple sequence of chambers offset from one another by a defined distance. We use the geometry to study the effects of phonon backscatter, trapping, and corner-turning on the thermal conductance in Si nanowires (NWs). We employ a full Brillouin zone Boltzmann Transport Equation (BTE) method to determine spatially-varying phonon densities in the geometry. Significantly greater impact is seen due to backscatter than any other means of arresting phonon flow. By creating a geometry that maximizes backscatter, a roughly 8-fold reduction in thermal conductance below the Casimir limit can be achieved at room temperature which is a factor of four smaller than the nearest reported value in the literature. The geometry is also useful for systematic investigation of other means of controlling phonons and affecting thermal transport; particularly, we investigate diffuse versus specular boundary scattering and the induced misalignment between the phonon flow and thermal flux due to the shape of the geometry. These effects combine to offer new insights into fundamental phonon behaviors and possible routes to phonon control.

Monte Carlo simulation of single-crystalline PbSe nanowire thermal conductivity using first-principle phonon properties

Prior experimental studies showed that nanowires are promising structures for improving the thermoelectric performance of practical thermoelectric materials due to the strongly induced phonon-boundary scattering. However, few studies examined the impact of phonon-boundary scattering on the thermal conductivity of thermoelectric nanowires from a first-principle approach. In this work, we systematically study the role of phonon-boundary scattering with different boundary specularities on the thermal conductivity of PbSe nanowires by rigorously solving the full phonon Boltzmann transport equation without any adjustable parameters. We observe significant thermal conductivity reduction for rough PbSe nanowires with diameters less than a few hundred nanometers. The reduction reaches ∼40% for 10 nm thick, rough PbSe nanowires at room temperature. The diameter-dependent thermal conductivities are found to contain important information about the phonon mean free path distribution from a standard reconstruction algorithm. The simulation results are important for fundamental understanding of nanoscale thermal transport in thermoelectric materials and will guide the future design of thermoelectric devices to achieve better energy conversion efficiency.

Analytic closures for M1 neutrino transport

Carefully accounting for neutrino transport is an essential component of many astrophysical studies. Solving the full transport equation is too expensive for most realistic applications, especially those involving multiple spatial dimensions. For such cases, resorting to approximations is often the only viable option for obtaining solutions. One such approximation, which recently became popular, is the M1 method. It utilizes the system of the lowest two moments of the transport equation and closes the system with an ad hoc closure relation. The accuracy of the M1 solution depends on the quality of the closure. Several closures have been proposed in the literature and have been used in various studies. We carry out an extensive study of these closures by comparing the results of M1 calculations with precise Monte Carlo calculations of the radiation field around spherically symmetric protoneutron star models. We find that no closure performs consistently better or worse than others in all cases. The level of accuracy that a given closure yields depends on the matter configuration, neutrino type and neutrino energy. Given this limitation, the maximum entropy closure by Minerbo on average yields relatively accurate results in the broadest set of cases considered in this work.

Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity

Consider the dynamics of a thin film flowing down a heated substrate. The substrate heating generates a temperature distribution on the free surface, which in turn induces surface-tension gradients and corresponding thermocapillary stresses that affect the free surface and therefore the fluid flow. We study here the effect of finite substrate thermal diffusivity on the film dynamics. Linear stability analysis of the full Navier-Stokes and heat transport equations indicates if the substrate diffusivity is sufficiently small, the film becomes unstable at a finite wavelength and at a Reynolds number smaller than that predicted in the long-wavelength limit. This property is captured in a reduced-order system of equations derived using a weighted-residual integral-boundary-layer method. This reduced-order model is also used to compute the bifurcation diagrams of solution branches connecting the trivial flat film to travelingwaves including solitary pulses. The effect of finite diffusivity is to separate a simultaneous Hopf-transcritical bifurcation into its individual component bifurcations. The appropriate Hopf bifurcation then connects only to the solution branch of negative-hump pulses, with wave speed less than the linear wave speed, while the branch of positive-single-hump pulses merges with the branch of positive-two-hump pulses at a supercritical Reynolds number. In the regime where finite-wavelength instability occurs, there exists a Hopf-bifurcation pair connected by a branch of periodic solutions, whose period cannot be increased indefinitely. Numerical simulation of the reduced-order system shows the development of a train of coherent structures, each of which resembles a stationary positive-hump pulse, and, in the regime of finite-wavelength instability, wavelength selection and saturation to periodic traveling waves.

CFD modeling the hydrodynamics of binary particle mixture in pseudo-2D bubbling fluidized bed: Effect of model parameters

The hydrodynamics of binary coal-sand mixture in a pseudo-2D rectangular bubbling fluidized bed (0.385m×0.005m×0.128m) was simulated using the multi-fluid Eulerian model incorporating the kinetic theory of granular flow. Parametric studies of the boundary wall condition, particle-particle restitution coefficient, friction packing limit, as well as transport equation for granular temperature were performed to investigate their influences on the predicted mixing/segregation behavior. The CFD simulation results demonstrated that the predicted mixing behavior was closely related to the expression for granular temperature transport equation and specularity coefficient. When the full transport equation for granular temperature was adopted, the predicted mixing degree decreased with the increase of specularity coefficient. And the best agreement between simulation results and experimental data was achieved when specularity coefficient was equal to 1.0. Nevertheless, when the algebraic transport equation for granular temperature was adopted, the system was always predicted in well-mixing rather than segregation state. Under the full transport equation for granular temperature and the no slip boundary wall condition, the predicted mixing degree decreased with the increase of the particle-particle restitution coefficient and frictional packing limit. The supplementary simulations indicated that for the considered gas-solid system there exist a critical bed thickness larger than which the system was in well-mixing state and the simulation results were independent from the investigated parameters. The hydrodynamic analysis indicated that the reduction of bubble size and the solid axial movement could be the mechanism responsible for the occurrence of axial segregation.

Modeling pH variation in reverse osmosis

The transport of hydronium and hydroxide ions through reverse osmosis membranes constitutes a unique case of ionic species characterized by uncommonly high permeabilities. Combined with electromigration, this leads to complex behavior of permeate pH, e.g., negative rejection, as often observed for monovalent ions in nanofiltration of salt mixtures. In this work we employed a rigorous phenomenological approach combined with chemical equilibrium to describe the trans-membrane transport of hydronium and hydroxide ions along with salt transport and calculate the resulting permeate pH. Starting from the Nernst-Planck equation, a full non-linear transport equation was derived, for which an approximate solution was proposed based on the analytical solution previously developed for trace ions in a dominant salt. Using the developed approximate equation, transport coefficients were deduced from experimental results obtained using a spiral wound reverse osmosis module operated under varying permeate flux (2–11 μm/s), NaCl feed concentrations (0.04–0.18 M) and feed pH values (5.5–9.0). The approximate equation agreed well with the experimental results, corroborating the finding that diffusion and electromigration, rather than a priori neglected convection, were the major contributors to the transport of hydronium and hydroxide. The approach presented here has the potential to improve the predictive capacity of reverse osmosis transport models for acid-base species, thereby improving process design/control.