Closure Relations Research Articles

Ionization of H(2p) atom by 30 keV proton and antiproton impact

Ionization process of the hydrogen atom, being initially in the excited 2p-state, projected by 30 keV proton and antiproton impact has been investigated. The second-Born approximation (SBA) is used to measure the triple differential cross sections (TDCSs). The closure relation is employed on the second-Born term to come over consideration of the intermediate states. The triple differential cross-sections are represented graphically against the ejection angle, for two scattering angles and ejected energies into both scattering and perpendicular geometric planes. The single differential cross-section (SDCS) is calculated in the framework of the first-Born approximation.

Situating the Vector Density Approach Among Contemporary Continuum Theories of Dislocation Dynamics

Abstract For the past century, dislocations have been understood to be the carriers of plastic deformation in crystalline solids. However, their collective behavior is still poorly understood. Progress in understanding the collective behavior of dislocations has primarily come in one of two modes: the simulation of systems of interacting discrete dislocations and the treatment of density measures of varying complexity that are considered as continuum fields. A summary of contemporary models of continuum dislocation dynamics is presented. These include, in order of complexity, the two-dimensional statistical theory of dislocations, the field dislocation mechanics treating the total Kröner–Nye tensor, vector density approaches that treat geometrically necessary dislocations on each slip system of a crystal, and high-order theories that examine the effect of dislocation curvature and distribution over orientation. Each of theories contain common themes, including statistical closure of the kinetic dislocation transport equations and treatment of dislocation reactions such as junction formation. An emphasis is placed on how these common themes rely on closure relations obtained by analysis of discrete dislocation dynamics experiments. The outlook of these various continuum theories of dislocation motion is then discussed.

Importance of considering interstitial fluid effects in the kinetic theory of granular flow for raceway formation prediction

Owing to its computational efficiency, the two-fluid model is widely used in computational fluid dynamics (CFD) to describe gas–solid flows in large-scale industrial processes. The motion of the solid phase is commonly modeled by the kinetic theory of granular flow (KTGF). The main conceptual difference between the two commonly used KTGF closure relations, namely the closure models by Lun e al. (1984) and Agrawal et al. (2001), is the treatment of the interstitial fluid. We theoretically compare both models and investigate their effects on simulation results. The interstitial fluid effects are essential during the raceway shape prediction but can be neglected in the fluidized bed regime. The correct prediction of the raceway size and shape is essential for predicting the processes in the raceway zone of ironmaking blast furnaces. Our results show that considering the interstitial fluid effects in the KTGF enables more accurate predictions in the raceway regime.

Convergence of Coupling-Parameter Expansion-Based Solutions to Ornstein–Zernike Equation in Liquid State Theory

The objective of this paper is to investigate the convergence of coupling-parameter expansion-based solutions to the Ornstein–Zernike equation in liquid state theory. The analytically solved Baxter’s adhesive hard sphere model is analyzed first by using coupling-parameter expansion. It was found that the expansion provides accurate approximations to solutions—including the liquid-vapor phase diagram—in most parts of the phase plane. However, it fails to converge in the region where the model has only complex solutions. Similar analysis and results are obtained using analytical solutions within the mean spherical approximation for the hardcore Yukawa potential. However, numerical results indicate that the expansion converges in all regions in this model. Next, the convergence of the expansion is analyzed for the Lennard-Jones potential by using an accurate density-dependent bridge function in the closure relation. Numerical results are presented which show convergence of correlation functions, compressibility versus density profiles, etc., in the single as well as two-phase regions. Computed liquid-vapor phase diagrams, using two independent schemes employing the converged profiles, compare excellently with simulation data. The results obtained for the generalized Lennard-Jones potential, with varying repulsive exponent, also compare well with the simulation data. Solution-spaces and the bifurcation of the solutions of the Ornstein–Zernike equation that are relevant to coupling-parameter expansion are also briefly discussed. All of these results taken together establish the coupling-parameter expansion as a practical tool for studying single component fluid phases modeled via general pair-potentials.

The WAF scheme for the isentropic drift-flux model of compressible two-phase flows

This paper focuses on the extension of the Weighted Average Flux (WAF) scheme for the numerical simulation of two-phase gas–liquid flow by imposing velocity equilibrium and without mechanical equilibrium of the transient drift-flux model. The model becomes a hyperbolic system of conservation laws with realistic closure relations where both phases are strongly coupled during their motion. Exploiting this, the drift-flux model discretization, simulation and investigation becomes very fast, simple and robust. The efficiency of the WAF scheme as being a second order in space and time without data reconstruction have been demonstrated in the published literature for compressible single-phase flows. However, the scheme is rarely applied to compressible two-phase flows. Based on a recent and complete exact Riemann solver for the drift-flux model, the model is numerically solved by the WAF scheme. The numerical algorithm accuracy and ability are validated through different published test cases. It is shown that the proposed scheme can be effectively employed to simulate two-phase flows involving discontinuities such as shocks and interfaces. The proposed WAF scheme is also compared with other numerical methods. Simulation results show appropriate agreement of WAF scheme even with the exact solutions. Comparisons of the presented simulations demonstrate that the behaviour of WAF scheme is encouraging, more accurate and fast than other numerical methods.

Effective hierarchical clustering based on structural similarities in nearest neighbor graphs

Hierarchical clustering allows better performance in grouping heterogeneous and non-spherical datasets than the center-based clustering, at the expense of increased time complexity. Meanwhile, the bottom-up approach of hierarchical clustering methods often tend to be sensitive or vulnerable to datasets containing obscure cluster boundaries. This paper presents an effective method for hierarchical clustering, called HCNN, which utilizes two types of structural similarities in nearest neighbor graph of a dataset to group similar data into clusters. In particular, the first metric is used to identify those pairs of data with maximal similarity in their nearest neighborhoods that can serve as local centers of every initial cluster. This can contribute to observing the boundaries and detecting clusters, hubs and outliers, thereby alleviating remarkably the influence of obscure boundaries between clusters. The initial clusters will be merged recursively in accordance with the second similarity metric measured in terms of the connectivity between clusters in the nearest neighbor graph. In this case, rather than combining two clusters with highest similarity during each iteration, we consider the maximum similarity as a transitive and closure relation between clusters, i.e., an equivalence relation, which enables more effective and efficient merging of clusters via application of advanced data structures. Experiments based on synthetic and real datasets demonstrate that the proposed HCNN algorithms can possibly outperform the state-of-the-art clustering methods evaluated in terms of the clustering accuracy, normalized mutual information and adjust rand index. Moreover, experiments on unlabeled real datasets show that our method enables effective identification of the quantity of clusters based on Davies–Bouldin index and average silhouette coefficient.

Fluid simulations of cosmic ray-modified shocks

ABSTRACT Cosmic ray (CR)-modified shocks are a demanding test of numerical codes. We use them to test and validate the two-moment method for CR hydrodynamics, as well as characterize the realism of CR shock acceleration in two-fluid simulations which inevitably arises. Previously, numerical codes were unable to incorporate streaming in this demanding regime, and have never been compared against analytic solutions. First, we find a new analytic solution highly discrepant in acceleration efficiency from the standard solution. It arises from bi-directional streaming of CRs away from the subshock, similar to a Zeldovich spike in radiative shocks. Since fewer CRs diffuse back upstream, this favours a much lower acceleration efficiency, typically ${\lesssim}10{{\ \rm per\ cent}}$ (even for Mach number > 10) as opposed to ${\gtrsim}50{{\ \rm per\ cent}}$ found in previous analytic work. At Mach number ≳10, the new solution bifurcates into three branches, with efficient, intermediate, and inefficient CR acceleration. Our two-moment code accurately recovers these solutions across the entire parameter space probed, with no ad hoc closure relations. For generic initial conditions, the inefficient branch is robustly chosen by the code; the intermediate branch is unstable. The preferred branch is very weakly modified by CRs. At high Mach numbers (≳10), the gas jump conditions approach that of a purely hydrodynamic shock, and a sub-grid prescription for thermal injection is required for reasonable acceleration efficiencies ${\sim}10{{\ \rm per\ cent}}$. CR-modified shocks have very long equilibration times (∼1000 diffusion time) required to develop the precursor, which must be resolved by ≳10 cells for convergence. Non-equilibrium effects, poor resolution, and obliquity of the magnetic field all reduce CR acceleration efficiency. Shocks in galaxy-scale simulations will generally contribute little to CR acceleration without sub-grid modification.

On the Existence of the Plateau Emission in High-energy Gamma-Ray Burst Light Curves Observed by Fermi-LAT

The Large Area Telescope (LAT) on board the \Fermi Gamma-ray Space Telescope (\Fermi) shows long-lasting high-energy emission in many gamma-ray bursts (GRBs), similar to X-ray afterglows observed by the Neil Gehrels Swift Observatory \citep[\textit{Swift};][]{gehrels2004}. Some LAT light curves (LCs) show a late-time flattening reminiscent of X-ray plateaus. We explore the presence of plateaus in LAT temporally extended emission analyzing GRBs from the second \lat GRB Catalog \citep[2FLGC;][]{Ajello2019apj} from 2008 to May 2016 with known redshifts, and check whether they follow closure relations corresponding to 4 distinct astrophysical environments predicted by the external forward shock (ES) model. We find that three LCs can be fit by the same phenomenological model used to fit X-ray plateaus \citep{Willingale2007} and show tentative evidence for the existence of plateaus in their high-energy extended emission. The most favorable scenario is a slow cooling regime, whereas the preferred density profile for each GRBs varies from a constant density ISM to a $r^{-2}$ wind environment. We also compare the end time of the plateaus in $\gamma$-rays and X-rays using a statistical comparison with 222 \textit{Swift} GRBs with plateaus and known redshifts from January 2005 to August 2019. Within this comparison, the case of GRB 090510 shows an indication of chromaticity at the end time of the plateau. Finally, we update the 3-D fundamental plane relation among the rest frame end time of the plateau, its correspondent luminosity, and the peak prompt luminosity for 222 GRBs observed by \textit{Swift}. We find that these three LAT GRBs follow this relation.

Closure relations during the plateau emission of Swift GRBs and the fundamental plane

Abstract The Neil Gehrels Swift observatory observes gamma-ray burst (GRB) plateaus in X-rays. We test the reliability of the closure relations through the fireball model when dealing with GRB plateau emissions. We analyze 455 X-ray light curves collected by Swift from 2005 January until 2019 August for which the redshifts are either known or unknown using the phenomenological Willingale 2007 model. Using these fits, we analyze the emission mechanisms and astrophysical environments of these GRBs through the closure relations within the time interval of the plateau emission. Finally, we test the three-dimensional fundamental plane relation (Dainotti relation) which connects the prompt peak luminosity, the time at the end of the plateau (rest frame), and the luminosity at that time, for the GRBs with redshift, concerning groups determined by the closure relations. This allows us to check if the intrinsic scatter σint of any of these groups is reduced compared to previous literature. The most fulfilled environments for the electron spectral distribution, p > 2, are wind slow cooling (SC) and interstellar material (ISM) slow cooling for cases in which the parameter q, which indicates the flatness of the plateau emission and accounts for the energy injection, is 0 and 0.5, respectively, in cases with both known and unknown redshifts. We also find that for short GRBs all ISM environments with q = 0 have the smallest σint = 0.04 ± 0.15 in terms of the fundamental plane relation holding a probability of occurring by chance of p = 0.005. We have shown that the majority of GRBs presenting plateau emission fulfill the closure relations, including the energy injection, with a particular preference for the wind SC environment. The subsample of GRBs that fulfill the given relations can be used as possible standard candles and can suggest a way to reduce the intrinsic scatter of these studied relationships.

Derivation of a macroscopic mixture model for two-phase turbulent flows

This article addresses the issue of reduced models to describe turbulent two-phase flows in industrial applications. A spatially-averaged mixture or drift-flux model is derived theoretically from the local instantaneous Navier-Stokes description. Reynolds-averaging and space-averaging are applied successively. Between these two steps, a model reduction is achieved to account for the non-equilibrium between phases via algebraic relations. Applications of this work are not limited to porous media but also include macroscopic descriptions to model high-shear regions developing near the walls for internal flows. Thermal effects, heat transfer at the wall and phase-change are also considered and briefly discussed. The final model describes the evolution of mixture variables, including effects of both sub-filter spatial variations, turbulence, and local non-equilibrium in velocity, pressure and enthalpy. This analysis provides bridges between different approaches to model two-phase flows (local instantaneous description, two-fluid model, local drift-flux model and spatially-averaged drift-flux model). It clarifies the content of each model involved by defining them in terms of local instantaneous quantities. Turbulent fluctuations and phase intermittency are crucial mechanisms. Important effects to model also include void fraction dispersion and turbulent diffusion; then, it is necessary to model the relative velocity, including the drift velocity orthogonal to gravity induced by the complex interactions between turbulent velocity fluctuations and the interfacial momentum transfer.The final macroscopic (spatially-averaged) mixture formulation is open, in the sense that expressions to model the various terms representing the physics of the small scales are not provided; instead, the physical sense and the origin of these models are discussed. The paper is meant as a basis on which analyses on local imbalance assumptions or relative velocity closures can be assessed. CFD simulations can provide information to complement experiments in technically challenging physical conditions or on processes essential to the models yet difficult to access experimentally (such as interfacial transfers for instance). Different kinds of two-fluid models can be tested to analyse their consequences on the macroscopic spatially-averaged model. In addition, a new path to calibrate closure laws or propose new models is opened based on finer-scale descriptions. Guidelines to use fine simulations along with the open expressions to derive closure relations either for the local drift-flux or for the spatially-averaged models are presented. They concern the modelling of the local relative velocity, the spatial average of the diffusion of void fraction and of the pressure drop.

Smoothed particle radiation hydrodynamics: two-moment method with local Eddington tensor closure

ABSTRACTWe present a new smoothed particle hydrodynamics-radiative transfer method (sph-m1rt) that is coupled dynamically with sph. We implement it in the (task-based parallel) swift galaxy simulation code but it can be straightforwardly implemented in other sph codes. Our moment-based method simultaneously solves the radiation energy and flux equations in sph, making it adaptive in space and time. We modify the m1 closure relation to stabilize radiation fronts in the optically thin limit. We also introduce anisotropic artificial viscosity and high-order artificial diffusion schemes, which allow the code to handle radiation transport accurately in both the optically thin and optically thick regimes. Non-equilibrium thermochemistry is solved using a semi-implicit sub-cycling technique. The computational cost of our method is independent of the number of sources and can be lowered further by using the reduced speed-of-light approximation. We demonstrate the robustness of our method by applying it to a set of standard tests from the cosmological radiative transfer comparison project of Iliev et al. The sph-m1rt scheme is well-suited for modelling situations in which numerous sources emit ionizing radiation, such as cosmological simulations of galaxy formation or simulations of the interstellar medium.

A continuum mechanical framework for modeling tumor growth and treatment in two- and three-phase systems

The growth and treatment of tumors is an important problem to society that involves the manifestation of cellular phenomena at length scales on the order of centimeters. Continuum mechanical approaches are being increasingly used to model tumors at the largest length scales of concern. The issue of how to best connect such descriptions to smaller-scale descriptions remains open. We formulate a framework to derive macroscale models of tumor behavior using the thermodynamically constrained averaging theory (TCAT), which provides a firm connection with the microscale and constraints on permissible forms of closure relations. We build on developments in the porous medium mechanics literature to formulate fundamental entropy inequality expressions for a general class of three-phase, compositional models at the macroscale. We use the general framework derived to formulate two classes of models, a two-phase model and a three-phase model. The general TCAT framework derived forms the basis for a wide range of potential models of varying sophistication, which can be derived, approximated, and applied to understand not only tumor growth but also the effectiveness of various treatment modalities.

Probing the balance of Telechelic Polymers Bridging and Screened-Coulomb Interactions in Microemulsion System

The structural properties of neutral oil-in-water (O/W) microemulsions nanodroplets in the presence of telechelic polymers PEO modified with two hydrophobic end functions (PEO- 2m) were studied by Small-Angle Neutron Scattering (SANS). We show that the unknown attractive interaction introduced by the addition of PEO-2m, can could be balanced by a known repulsion interaction due to the addition of a cationic surfactant Cp +, which causes a progressive transformation of the bridges connecting the nanodroplets into loops. For this, we compute the structure factor S(q) with an effective pair potential, using the Ornstein Zernicke (OZ) integral equation approach with the Hypernetted Chain (HNC) closure relation. Generally, the agreement between the neutron scattering spectra and the numerical study is reasonable and allows a detailed description for each sample.

A self-consistent perturbative density functional theory for hard-core fluids: phase diagrams, structural and interfacial properties

The classical Density functional theory (DFT) has become a powerful tool to describe the microscopic structure of fluids as the radial distribution function. One of its particular capabilities is to express the thermodynamic properties of those fluids even under the influence of external potentials, such as fluid-solid interaction. However, good models for the Helmholtz free-energy functionals are necessary to improve the results. In this work, we present a self-consistent thermodynamic perturbation theory for the excess Helmholtz free-energy from the DFT applied to hard-core fluids. The new perturbation theory is solved self-consistently without any closure relation to solving the Ornstein-Zernike equation explicitly. We compare the performance of our self-consistent perturbation theory with the results obtained with the well-known second-order Barker-Henderson perturbation theory for the hard-core Yukawa and square-well fluids. Moreover, we propose two versions of the DFT to describe the perturbative contribution: one based on the weighted density approximation theory and another from a modified mean-field theory. The present results confirm the modified mean-field theory as a better option to calculate the thermodynamic and structural properties of hard-core fluids.

Biomolecular Simulations with the Three-Dimensional Reference Interaction Site Model with the Kovalenko-Hirata Closure Molecular Solvation Theory.

The statistical mechanics-based 3-dimensional reference interaction site model with the Kovalenko-Hirata closure (3D-RISM-KH) molecular solvation theory has proven to be an essential part of a multiscale modeling framework, covering a vast region of molecular simulation techniques. The successful application ranges from the small molecule solvation energy to the bulk phase behavior of polymers, macromolecules, etc. The 3D-RISM-KH successfully predicts and explains the molecular mechanisms of self-assembly and aggregation of proteins and peptides related to neurodegeneration, protein-ligand binding, and structure-function related solvation properties. Upon coupling the 3D-RISM-KH theory with a novel multiple time-step molecular dynamic (MD) of the solute biomolecule stabilized by the optimized isokinetic Nosé–Hoover chain thermostat driven by effective solvation forces obtained from 3D-RISM-KH and extrapolated forward by generalized solvation force extrapolation (GSFE), gigantic outer time-steps up to picoseconds to accurately calculate equilibrium properties were obtained in this new quasidynamics protocol. The multiscale OIN/GSFE/3D-RISM-KH algorithm was implemented in the Amber package and well documented for fully flexible model of alanine dipeptide, miniprotein 1L2Y, and protein G in aqueous solution, with a solvent sampling rate ~150 times faster than a standard MD simulation in explicit water. Further acceleration in computation can be achieved by modifying the extent of solvation layers considered in the calculation, as well as by modifying existing closure relations. This enhanced simulation technique has proven applications in protein-ligand binding energy calculations, ligand/solvent binding site prediction, molecular solvation energy calculations, etc. Applications of the RISM-KH theory in molecular simulation are discussed in this work.

Wall and interfacial shear stresses in laminar two-phase stratified flow in pipes

Exact solutions for separated two-phase flows are useful as a benchmark for numerical codes and for testing closure relations for simplified 1D Two-Fluid models. In particular, the correct representation of the wall and interfacial shear stresses is essential for the modeling of these flows. In this study, a complete set of exact solutions for laminar stratified two-phase flows in inclined pipes is used to explore the wall and interfacial shear stresses. We examine all possible configurations with concave and convex interfaces, which are related to the contact angle of the phases with the pipe surface. A closed-form analytical expression for the average interfacial shear stress was obtained for all cases of stratified two-phase flow. The limiting behavior of the local shear stresses at the triple-point (TP) region, where the interface meets the pipe wall, is determined by residue calculus. It is shown that the shear stresses behavior upon approaching the TP is determined by the fluid's viscosity ratio and the contact angle. The combinations of those parameters that result in convergence or divergence of the shear stresses at the TP are identified. The usefulness of the analytical expressions obtained for correctly predicting the shear stresses variation upon approaching the TP is demonstrated and discussed.

Thermodynamic consistency by a modified Perkus–Yevick theory using the Mittag-Leffler function

Closure relations that satisfies thermodynamical consistencies are very important, for it implies physical consistent Ornstein–Zernike equation solutions. The method proposed by Percus (1962), which uses the functional Taylor expansion to obtain closure relations, is presented along with the particular generating functional considered to retrieve the Percus–Yevick (PY) approximation. The main assumption for this particular case, which takes into account the low density limit, is discussed. Based on this argument, it was proposed a modified generating functional with no prior considerations about the system density. As an example, the two-parameters Mittag-Leffler function was used. Since it generalizes the exponential function, the parameters determined at low densities retrieves the PY approximation. For higher densities it was possible to obtain parameters which lead to pressure consistent results, in good agreement with those found in literature.

Comparison of Two-Fluid Model Simulations of Freely Bubbling Three-Dimensional Gas-Fluidized Beds with Magnetic Resonance Imaging Results

Two-fluid model (TFM) simulation predictions of bubble and particle dynamics in a 3D cylindrical fluidized bed are compared with magnetic resonance imaging (MRI) measurements from a prior study across a range of gas velocities, particle sizes, and bed heights. TFM simulations generally predict the number of bubbles and bubble diameters seen as a function of vertical position of the bubbles in the bed well; however, simulations generally predict particle speeds lower than those measured experimentally. The drag model and particle stress model used in the simulations were varied and found to have an insignificant effect on predictions of bubble dynamics but a significant effect on predictions of particle velocity distribution. Overall, this study demonstrates the importance of challenging model predictions of both particle and bubble dynamics and the ability of the prior MRI results to evaluate the accuracy of submodels, such as drag and particle stress closure relationships.

Dispersed-phase velocities for gas-liquid vertical slug and dispersed-bubbles flows using an ultrasonic cross-correlation technique

There is a consensus in the oil industry that multiphase flow meters (MFMs) can bring various benefits to the oil industry. MFMs would reduce the minimum space and weight requirements of the platforms as they would eliminate the separating tanks and the interconnection pipes. One of the important variables to MFMs is the dispersed-phase velocity that is a closure relationship to mechanistic models. In this context, this paper studied the use of the ultrasonic cross-correlation technique based on the energies of the reflected pulses to estimate the dispersed-phase velocities for the dispersed-bubbles and the slug flow patterns. The experiments were done in a vertical two-phase liquid-gas flow pipe, with superficial liquid and gas velocities ranging from 0.5 to 3.9m/s and 0.1 to 3.7m/s, respectively. Also, the results were compared with image analysis and mechanistic models. For dispersed-bubbles flows, the relative deviation from the mixture velocities was less than 20%. For slug flows, the relative deviations from the images were less than 15%.

Ionization of Hydrogen-Like Atoms by Proton Impact

Direct ionization processes of hydrogen-like atoms by proton impact are investigated by Born theory. The triple differential cross sections (TDCSs) and the quantities of physical interest in the process are calculated for ionization of H, He+, Li2+, and Be3+ ground-state atoms, using the first- and second-Born approximations. The closure relation is employed to come over the consideration of the intermediate states. The calculations are carried for fixed scattering angles and ejected electron energies at the impact energies 75, 200, and 500 keV. The influence of each of the incidence energy and the atomic number of the target atom on the TDCSs is studied. The contribution of the second-Born term to the cross section is illustrated.