Density Evolution Research Articles

Modeling the Evolution of the Dislocation Density and Yield Stress of Al over a Wide Range of Temperatures and Strain Rates

This paper introduces an extended model for the evolution of internal and wall dislocation densities in pure aluminum during plastic deformation. The approach takes the three internal state variables (3IVM) model as a starting point and advances it by taking into account the dynamic annihilation of immobile/locked dislocations as well as dislocations stored in the subgrain/cell walls. The strength of the material, as one of the properties affected by dislocation density, is used to validate the model. Experimental flow curves for pure Al are taken as the basis for calibration. Compression tests are performed at temperatures from − 196 °C to 500 °C with strain rates of 1, 0.1, and 0.01 s−1. The effect of temperature and strain rate on each state parameter is illustrated and discussed.

Measurement of medium-voltage AC air arc temperature and particle number density based on dual-wavelength Moiré deflection technology

AC air arcs are generated in medium-voltage (MV) power systems under the effect of harsh weather conditions, equipment aging, and high penetration of distributed generation, threatening equipment and public safety. The arc current and temperature are low due to the wide application of arc suppression devices. In this scenario, the MV AC air arc does not satisfy the local thermodynamic equilibrium (LTE) condition. In addition, the repeated arcing and extinguishing processes further complicate the arc discharge mechanism, which bring challenges in the modeling and detection of MV AC air arcs. Experimental methods are a direct and efficient approach to determine the properties of arc plasmas. In this study, a dual-wavelength Moiré deflection diagnostic system was established to determine the time evolution of the particle density and radial distribution of the temperature in an MV AC air arc without relying on the LTE assumption. The electron number density and heavy particle number density change transiently during the arc discharge process and change gradient along the radial direction. The heavy particle temperature and electron temperature were then calculated based on the measured particle number density. During the arcing stage, the temperature of the electrons exceeded that of the heavy particles significantly, and the arc deviated from LTE. Finally, the limitations of the traditional single-wavelength Moiré deflection method are analyzed. The classic single-wavelength Moiré deflection method, while capable of estimating heavy particle temperature in plasma, exhibits a significant error in electron density estimation compared to the dual-wavelength Moiré deflection method.

Estimation of regime-switching diffusions via Fourier transforms

In this article, an algorithm for maximum-likelihood estimation of regime-switching diffusions is proposed. The proposed approach uses a Fourier transform to numerically solve the system of Fokker–Planck or forward Kolmogorow equations for the temporal evolution of the state densities. Monte Carlo simulations confirm the theoretically expected consistency of this approach for moderate sample sizes and its practical feasibility for certain regime-switching diffusions used in economics and biology with moderate numbers of states and parameters. An application to animal movement data serves as an illustration of the proposed algorithm.

Hydrogen uptake kinetics of cathodic polarized metals in aqueous electrolytes

We use a unique combination of electrochemical techniques to elucidate the dependency of hydrogen evolution reaction (HER) and absorption on pH and overpotential for iron and nickel. Impedance spectroscopy shows the dominance of the Volmer–Heyrovsky reaction pathway, challenging the common consideration of Volmer–Tafel dominance. Polarization slopes agree with the Volmer or Heyrovsky rate-determining step, with limitations at high overpotential. The evolution of steady-state permeation current density with overpotential is rationalized through newly-developed theory. Surface activity and absorption trends are captured. Combined with modelling, this work provides a path for quantifying hydrogen uptake and establishing an equivalent fugacity for aqueous electrolytes.

Constraints on cosmic star formation history via a new modeling of the radio luminosity function of star-forming galaxies

Context. Radio wavelengths offer a unique possibility to trace the total star-formation rate (SFR) in galaxies, both obscured and unobscured. To probe the dust-unbiased star-formation history, an accurate measurement of the radio luminosity function (LF) for star-forming galaxies (SFGs) is crucial. Aims. We make use of an SFG sample (5900 sources) from the Very Large Array (VLA) COSMOS 3 GHz data to perform a new modeling of the radio LF. By integrating the analytical LF, we aim to calculate the history of the cosmic SFR density (SFRD) from z ∼ 5 onwards. Methods. For the first time, we use both models of the pure luminosity evolution (PLE) and joint luminosity+density evolution (LADE) to fit the LFs directly to the radio data using a full maximum-likelihood analysis, considering the sample completeness correction. We also incorporate updated observations of local radio LFs and radio source counts into the fitting process to obtain additional constraints. Results. We find that the PLE model cannot be used to describe the evolution of the radio LF at high redshift (z > 2). By construct, our LADE models can successfully fit a large amount of data on radio LFs and source counts of SFGs from recent observations. The Akaike information criterion (AIC) also demonstrates that the LADE model is superior to the PLE model. We therefore conclude that density evolution is genuinely indispensable in modeling the evolution of SFG radio LFs. Our SFRD curve shows a good fit to the SFRD points derived by previous radio estimates. In view of the fact that our radio LFs are not biased, as opposed those of previous studies performed by fitting the 1/Vmax LF points, our SFRD results should be an improvement on these previous estimates. Below z ∼ 1.5, our SFRD matches a published multiwavelength compilation, while our SFRD turns over at a slightly higher redshift (2 < z < 2.5) and falls more rapidly out to high redshift.

Exomoon Phase Curves: Toroidal Exosphere Simulations of Exo‐Ios Orbiting 8 Exoplanets in Alkali Spectroscopy

AbstractToroidal atmospheres and exospheres characterized at exoplanets may be fueled by volcanically active exomoons, often referred to as exo‐Ios. We study the neutral outgassing and volatile evolution of a close‐orbiting, evaporating satellite at eight candidate exoplanet‐exomoon systems WASP‐49,‐96,‐69,‐17 b, XO‐2N b, HAT‐P‐1 b, HD‐189733 b, and HD‐209458 b by developing a 3‐D test‐particle Monte Carlo simulation, Simulating the Evolution of Ring Particles Emergent from Natural Satellites. The module is coupled to dishoom, approximating the minimum mass‐flux needed to reproduce observations of alkali line profiles identified in dozens of transmission spectra. We focus on sputtered neutral sodium, limited by photoionization and radiative effects. By considering Earth‐, Io‐, and Enceladus‐like masses, we systematically simulate the imprint of a non‐hydrostatic medium (characteristic of volcanic exospheres) in density and velocity space using a novel Delaunay tesselation field estimator algorithm. Our results demonstrate how exomoons can considerably modulate gas density observations probed near exoplanet transit, depending on the orbital phase of the putative satellite at the time of observation. The density evolution, therefore, manifests on orbital timescales as “exomoon phase curves” from shadow to occultation. We find two regimes of density evolution, characteristic of a: (a) localized cloud and (b) an azimuthally symmetric exoring/torus, degenerate with an exoplanet atmosphere, ranging from ∼109.5±0.5 cm−2 to ∼1015±0.25 cm−2 at our leading candidate WASP‐69b I. In certain orbital architectures, the smallest evaporating satellite mass surprisingly generates the brightest sodium signal, fueling optimism for discovering photometrically indiscernible rocky exomoons. We suggest long baseline monitoring of alkali and SO2 systems in spectroscopy to search for the temporal and spatial variability predicted here.

Controlled transport of fluid particles by microrotors in a Stokes flow using linear transfer operators

The manipulation of a collection of fluid particles in a low Reynolds number environment has several important applications. As we demonstrate in this paper, this manipulation problem is related to the scientific question of how fluid flow structures direct Lagrangian transport. We investigate this problem of directing the transport by manipulating the flow, specifically in the Stokes flow context, by controlling the strengths of two rotors fixed in space. We demonstrate a novel dynamical systems approach for this problem and apply this method to several scenarios of Stokes flow in unbounded and bounded domains. Furthermore, we show that the time-varying flow field produced by the optimal control can be understood in terms of dynamical structures such as coherent sets that define Lagrangian transport. We model the time evolution of the fluid particle density using finite-dimensional approximations of the Liouville operators for the microrotor flow fields. Using these operators, the particle transport problem is framed as an optimal control problem, which we solve numerically. This framework is then applied to the problem of transporting a blob of fluid particles in domains with different boundary conditions: free space, near to a plane wall, in a circular confinement, and the transport of two distributions of particles to a common target. These examples demonstrate the effectiveness of the proposed framework and also shed light on the effects of boundaries on the ability to achieve a desired fluid transport using a rotor-driven flow.

On learning latent dynamics of the AUG plasma state

In this work, we demonstrate the utility of state representation learning applied to modeling the time evolution of electron density and temperature profiles at ASDEX-Upgrade (AUG). The proposed model is a deep neural network, which learns to map the high dimensional profile observations to a lower dimensional state. The mapped states, alongside the original profile's corresponding machine parameters, are used to learn a forward model to propagate the state in time. We show that this approach is able to predict AUG discharges using only a selected set of machine parameters. The state is then further conditioned to encode information about the confinement regime, which yields a simple baseline linear classifier, while still retaining the information needed to predict the evolution of profiles. We, then, discuss the potential use cases and limitations of state representation learning algorithms applied to fusion devices.

Interferometer study of radial and axial density evolution and temperature in N2 gas subjected to negative and positive corona discharges

In the present paper, we propose to make an optical diagnosis by laser interferometry to determine experimentally the density of neutral particles in a coronal discharge bathing in a gaseous medium N2. We are particularly interested in the measurement of variation of the optical path and therefore of the index of refraction which allows us to highlight the variation of the density of the neutrals in the discharge, and using a computer treatment (Pearce method) the refractive index of the medium is calculated to determine the spatial distribution of density and temperature (radial and axial) at the core of the corona discharge in two different situations (positive and negative). This optical diagnosis also allowed us to quantitatively determine the phenomenon of depopulation of neutral particles. These quantitative and experimental results are interesting because they allow a good correlation between the theoretical results obtained so far

Cosmic evolution of FRI and FRII sources out to z = 2.5

Context. Radio-loud active galactic nuclei (RLAGN) play an important role in the evolution of galaxies through the effects on their environment. The two major morphological classes are core-bright (FRI) and edge-bright (FRII) sources. With the LOw-Frequency ARray (LOFAR), we can now compare the FRI and FRII evolution down to lower flux densities and with larger samples than before. Aims. Our aim is to examine the cosmic space density evolution for FRIs and FRIIs by analyzing their space density evolution between L150 ∼ 1024.5 W Hz−1 and L150 ∼ 1028.5 W Hz−1 and up to z = 2.5. In particular, we look at the space density enhancements and compare the FRI and FRII evolution with the total RLAGN evolution. Methods. We constructed radio luminosity functions (RLFs) from FRI and FRII catalogues based on recent data from LOFAR at 150 MHz to study the space densities as a function of radio luminosity and redshift. These catalogues contain over 100 times the number of FRIs with associated redshifts greater than z = 0.3, compared to the most recent FRI/FRII RLF study. To derive the maximum distance according to which a source can be classified and to correct for detection limits, we conducted simulations of how sources appear across a range of redshifts. Results. Our RLFs do not show any sharp transitions between the space density evolution of FRI and FRII sources as a function of radio luminosity and redshift. We report a space density enhancement from low to high redshift for FRI and FRII sources brighter than L150 ∼ 1027 W Hz−1. Furthermore, while we observe a tentative decrease in the space densities of FRIs with luminosities below L150 ∼ 1026 W Hz−1 and at redshifts beyond z = 0.8, this may be due to residual selection biases. The FRI/FRII space density ratio does not appear to evolve strongly as a function of radio luminosity and redshift. Conclusions. We argue that the measured space density enhancements above L150 ∼ 1027 W Hz−1 are related to the higher gas availability in the earlier, denser universe. The constant FRI/FRII space density ratio evolution as a function of radio luminosity and redshift suggests that the jet-disruption of FRIs might be primarily caused by events occurring on scales within the host galaxy, rather than being driven by changes in the overall large-scale environment. The remaining selection biases in our results also highlight the need to resolve more sources at angular scales below 40″, thereby strengthening the motivation for further developing and automating the calibration and imaging pipeline of LOFAR data to produce images at a sub-arcsecond resolution.

Extending damage accumulation of commercial reactor irradiated 316 stainless steel with ion irradiation

Austenitic 316 stainless steel flux thimble tubes (FTTs) removed from a commercial pressurized water reactor (PWR) were re-irradiated using nickel ions to evolve the irradiated microstructure to higher damage levels than those achieved in reactor. The microstructures of the neutron-plus-ion irradiated samples were compared with those of neutron irradiated only samples at the same doses to evaluate the effectiveness of ion irradiation to extend the damage range and match that created in reactor. Ion irradiations effectively captured, both qualitatively and semi-quantitatively, the evolution of dislocation loops, nanocavities, nanocluster size and density, and radiation-induced segregation (RIS) with dose. Only Ni-Si clusters were observed in ion irradiated FTT samples while Ni-Si-Mn clusters that were frequently observed in reactor irradiated samples above ∼41 dpa were absent in ion irradiated samples probably due to ballistic mixing at the high ion irradiation damage rate. Two samples were ion irradiated to ∼160 dpa to predict the irradiated microstructure that may develop due to an increase of the damage accumulation by extending a PWR life from 40 years to 60 years. The extended ion irradiation results indicated that significant microstructure changes were unlikely to occur even to very high doses near common PWR relevant temperature conditions. Overall, the microstructure resulting from ion irradiation following neutron irradiation matched that of neutron to the same dose reasonably well in most cases.

Multi-hazard stochastic response analysis and reliability evaluation of a 5 MW monopile offshore wind turbine based on extended platform FAST-S

This study investigates the random dynamic response of a 5 MW monopile offshore wind turbine (MOWT) subjected to wind, wave and earthquake loads. Based on the aero-hydro-servo-elastic code FAST, a multi-field coupled simulation module considering earthquake, FAST-S, is developed for the fully coupled multi-hazard analysis of OWTs. Compared to the original FAST code, a seismic analysis module (SeismoDyn) and a soil-structure interaction (SSI) module (SoilDyn) have been supplemented in FAST-S. SeismoDyn is developed by modifying the Kane’s kinetic equation in the FAST, and the SoilDyn is developed through Winkler foundation model in order to consider the nonlinear SSI. Moreover, FAST-S has been validated by other recognized codes. Based on FAST-S, a fully coupled numerical analysis model of the NREL 5 MW MOWT is established, the random responses and reliability evaluation of the offshore wind turbine under wind, wave and earthquake loads are analyzed by probability density evolution method. The results show that the aerodynamic and hydrodynamic actions increase the damping of the wind turbine system, which has a certain effect on inhibiting the tower’s vibration caused by the earthquake. Therefore, compared with the coupled wind-wave-earthquake case, the sole earthquake case may be more unfavorable, and should be paid more attention to in high risk earthquake regions. The results show that the probability density functions of offshore wind turbine response under strong earthquakes obviously deviates from the typical regular distribution models. Based on the theory of equivalent extreme value event, the dynamic reliability of the offshore wind turbine is calculated. Seismic actions maybe the major factor in the design of the support structure. Nevertheless, owing to the aerodynamic and hydrodynamic damping effects of wind and wave loadings, the structural responses under wind-wave-earthquake are more moderate than ones subjected to solely earthquake. The strong seismic loads would cause significant complex variations of time-dependent probability density function of the OWT response.

A modified Chebyshev collocation method for the generalized probability density evolution equation

Solving the generalized density evolution equation (GDEE) is essential to obtain the structural random response by the probability density evolution method (PDEM). To improve the collocation method, this study proposes a modified Chebyshev collocation method (MCCM) scheme for the solution of the GDEE based on the pseudo-spectral method, and derives a refine time integration format for GDEE. The GDEE is numerically solved by combining the fine time integration method and MCCM, and the accuracy, efficiency and numerical stability of the algorithm are investigated. Two numerical examples indicate that the results for the proposed method are in good agreement with analytical solutions and Monte Carlo simulation results. The proposed four numerical collocation schemes could effectively suppress numerical dispersion and numerical dissipation. In addition, the modified Chebyshev collocation method based fourth-order Runge-Kutta scheme presents excellent computational accuracy, efficiency and numerical stability, especially highlighting significant advantages in the computational efficiency.

Average spectral density of multiparametric Gaussian ensembles of complex matrices

A statistical description of part of a many body system often requires a non-Hermitian random matrix ensemble with nature and strength of randomness sensitive to underlying system conditions. For the ensemble to be a good description of the system, the ensemble parameters must be determined from the system parameters. This in turn makes its necessary to analyze a wide range of multi-parametric ensembles with different kinds of matrix elements distributions. The spectral statistics of such ensembles is not only system-dependent but also non-ergodic as well as non-‘stationary’. A change in system conditions can cause a change in the ensemble parameters resulting an evolution of the ensemble density and it is not sufficient to know the statistics for a given set of system conditions. This motivates us to theoretically analyze a multiparametric evolution of the ensemble averaged spectral density of a multiparametric Gaussian ensemble on the complex plane. Our analysis reveals the existence of an evolutionary route common to the ensembles belonging to same global constraint class and thereby derives a complexity parameter dependent formulation of the spectral density for the non-equilibrium regime of the spectral statistics, away from Ginibre equilibrium limit.

Introducing electromagnetic effects in Soledge3X

AbstractIn the pedestal region, electromagnetic effects affect the evolution of micro‐instabilities and plasma turbulence. The transport code Soledge3X developed by the CEA offers an efficient framework for turbulent 3D simulation on an electrostatic model with a fixed magnetic field. The physical accuracy of the model is improved with electromagnetic induction, driven by the local value of the parallel component of the electromagnetic vector potential , known from Ampère's law. It is solved implicitly in a coupled system with the vorticity equation on the electric potential . The consequence is a basic electromagnetic behavior in the form of shear Alfvén waves. A finite electron mass prevents unphysical speeds but requires solving for the time evolution of the parallel current density in the generalized Ohm's law. This term can be analytically included with little computational overhead in the system on and and improves its numerical condition, facilitating the iterative solving procedure. Simulations on a periodic slab case let us observe the predicted bifurcation of the wave propagation speed between the Alfvén wave and the electron thermal wave speeds for varying perpendicular wavenumbers. The first results on a circular geometry with a limiter attest to the feasibility of turbulent electromagnetic scenarios.

A physics-based plasticity study of the mechanism of inhomogeneous strain evolution in dual phase 600 steel

In this study, we employ a physics-based crystal plasticity finite element method (CPFEM) approach to study the microstructural response of DP600 steel to various loading types. With the pursued approach, it is possible link microstructural changes in the dual phase (DP) steel to its macroscopic response in a convenient and efficient manner. Microstructural data were acquired first from experimental work via the electron backscattered diffraction (EBSD) method. DP crystallographic features were then fed to a numerical hybrid calculation subroutine combining a comprehensive physics-based crystal plasticity scheme for ferrite along with an isotropic plasticity scheme for martensite defined on a single-layer representative volume element (RVE). The developed single-layer 3D mode, while allowing for high computational efficiency, can capture the stress/strain distribution, revealing the microscopic features of the deformation process including the physical evolution of the dislocation density, retexturing, and morphology. The single-layer RVE physics-based method employed herein establishes the strong correlation between morphology, dislocation density and deformation response of the material under various loading types. Specifically, findings reveal that the tendency of the ferrite phase neighboring the martensite regions in the DP steel to retexture is much more profound than pure ferrite when subjected to the same loading conditions, pointing out to the steep strain gradients in the former component. Martensite phase facilitates considerable heterogeneity in the stress and strain distribution, increases the misorientation polarization as well as leading to local strain hardening in the neighboring ferrite volume that stem from profound increase in spatial dislocation density. Stress-strain curves under various loading schemes are simulated and discussed in the light of the experimental findings in literature, revealing the dependence of the microstructural evolution on the loading type.

Simulating the replication and entanglement of semi-rigid polymers in nano-injection moulding

Many polymers have been used to design polymer/metal composite structures with high bond strength through nano-moulding technology. However, whether high-molecular-weight polymers flow deeply into nanostructures and whether polymer entanglement hinders complete infiltration remain contentious issues in theoretical studies. In this study, the effects of the injection pressure, molecular weight of the semi-rigid polymer [polyphenylene sulphide (PPS)], and nanostructure size of the metal surface on the replication quality were investigated by molecular dynamics simulations. Increasing the injection pressure and polymer molecular weight increased the replication quality at practical temperatures. PPS with various chain lengths could completely infiltrate the nanopores. The nanostructure size of the metal surface was weakly negatively correlated with the filling rate, but it was substantially negatively correlated with the infiltration behaviour of the entire PPS chain. The reasons for infiltration of long-chain PPS and the steady evolution of the entanglement density were investigated. The steady entanglement density of PPS indicates that entanglement is not the main reason for the low filling rate. From the mobility of a single chain, the PPS chain flows into nanopores in a snake-like fashion. These results provide new insights to improve the adhesion strength between polymers and metals in nano-injection moulding.

A unified physically-based model describing complex unloading behavior in cold and hot deformation process of aluminum alloys

During the cold and hot deformation of metals, the nonlinearity in unloading is mainly caused by dislocation-associated behaviors and creates challenges for the prediction of springback law and residual stress distribution after springback. In this paper, a unified physically-based model for describing the dislocation density evolution in unloading after cold and hot deformation was proposed firstly by considering the combined effects of reverse slip of reversible mobile dislocations (under back stress and the static recovery) and recrystallization (after deformation) on the dislocation density evolution in unloading. Furthermore, an anelastic strain model associated with dislocation behaviors was established for predicting the microplastic strain in unloading after cold and hot deformation by considering the effects of strain-induced dislocation multiplication and the dislocation annihilation/consumption caused by dynamic recovery and dynamic recrystallization on the reversible mobile dislocation density evolution during deformation. The established dislocation density evolution model and anelastic strain model were evaluated using the experimental results obtained from the hot compression of homogenous aluminum alloys and the cold compression of extruded aluminum alloys. The results indicated that the established models are able to accurately predict nonlinear unloading behavior and nonlinearly varying microplastic strain. Meanwhile, it was also found that, for cold deformation, the reverse slip of reversible mobile dislocations driven by back stress is the leading reason for the nonlinearity of unloading, and the strain hardening and recovery during deformation are main factors affecting the anelasticity of unloading; for hot deformation, the reverse dislocation slip, static recovery and recrystallization collectively affect the nonlinearity in unloading, and the anelasticity is affected not only by strain hardening and dynamic recovery but also by dynamic recrystallization during loading process.

A full‐probabilistic cloud analysis for structural seismic fragility via decoupled M‐PDEM

AbstractPerformance‐based earthquake engineering (PBEE) is essential for ensuring engineering safety. Conducting seismic fragility analysis within this framework is imperative. Existing methods for seismic fragility analysis often rely heavily on double loop reanalysis and empirical data fitting, leading to challenges in obtaining high‐precision results with a limited number of representative structural analysis instances. In this context, a new methodology for seismic fragility based on a full‐probabilistic cloud analysis is proposed via the decoupled multi‐probability density evolution method (M‐PDEM). In the proposed method, the assumption of a log‐normal distribution is not required. According to the random event description of the principle of preservation of probability, the transient probability density functions (PDFs) of intensity measure (IM) and engineering demand parameter (EDP), as key response quantities of the seismic‐structural system, are governed by one‐dimensional Li‐Chen equations, where the physics‐driven forces are determined by representative analysis data of the stochastic dynamic system. By generating ground motions based on representative points of basic random variables and performing structural dynamic analysis, the decoupled M‐PDEM is employed to solve the one‐dimensional Li‐Chen equations. This yields the joint PDF of IM and EDP, as well as the conditional PDF of EDP given IM, resulting in seismic fragility analysis outcomes. The numerical implementation procedure is elaborated in detail, and validation is performed using a six‐story nonlinear reinforced concrete (RC) frame subjected to non‐stationary stochastic ground motions. Comparative analysis against Monte Carlo simulation (MCS) and traditional cloud analysis based on least squares regression (LSR) reveals that the proposed method achieves higher computational precision at comparable structural analysis costs. By directly solving the physics‐driven Li‐Chen equations, the method provides the full‐probabilistic joint information of IM and EDP required for cloud analysis, surpassing the accuracy achieved by traditional methods based on statistical moment fitting and empirical distribution assumptions.

A Model for Membrane Degradation Using a Gelatin Invadopodia Assay.

One of the most crucial and lethal characteristics of solid tumors is represented by the increased ability of cancer cells to migrate and invade other organs during the so-called metastatic spread. This is allowed thanks to the production of matrix metalloproteinases (MMPs), enzymes capable of degrading a type of collagen abundant in the basal membrane separating the epithelial tissue from the connective one. In this work, we employ a synergistic experimental and mathematical modelling approach to explore the invasion process of tumor cells. A mathematical model composed of reaction-diffusion equations describing the evolution of the tumor cells density on a gelatin substrate, MMPs enzymes concentration and the degradation of the gelatin is proposed. This is completed with a calibration strategy. We perform a sensitivity analysis and explore a parameter estimation technique both on synthetic and experimental data in order to find the optimal parameters that describe the in vitro experiments. A comparison between numerical and experimental solutions ends the work.