Potential Equation Research Articles

Numerical modeling of the wind regime on the beaches of the wash of the artificial storage facilities for mineral processing waste

A 2D numerical model has been developed to estimate the airflow velocity field when flowing around the dam of an artificial storage facility for mineral processing waste. To solve the aerodynamic problem of determining the air flow velocity field when flowing around such hydraulic structures with a complex geometric shape, a potential motion model was applied. The numerical integration of the equation for the velocity potential is carried out using the Liebman method. The geometric shape of the tailings storage facility is formed in a discrete model using the marking method. A computer program was created to implement the developed numerical aerodynamics model. Based on the processing of the results of computational experiments, coefficients were obtained that allow us to quickly determine the value of the air flow velocity at the beginning and end of the tailing pond beach, i.e. in the area of the most intense dust emission. This allows for a quick prediction of the risk of dust air pollution at different tailing pile heights.

Eigenmodes and eigenfunctions of high-power electromagnetic waves in inhomogeneous magnetized plasmas and their stability to stimulated Raman backscattering

Eigenmodes and eigenfunctions of high-power electromagnetic waves in inhomogeneous magnetized plasmas are investigated. Considering the dielectric permittivity in the presence of an external magnetic field and relativistic nonlinearity and considering the inhomogeneity of the electron density in the linear and rippled structures, two nonlinear wave equations are obtained. In the linear regime, eigenmodes and eigenfunctions of the electric field for two-electron densities are obtained. In the nonlinear regime, results indicate that by increasing the external magnetic field, the amplitude of the electric field increases, which affects the dielectric permittivity. The conditions of the stimulated Raman backscattering and the equation of the space-charge potential are obtained. Results also indicate that in the presence of the external magnetic field, plasma inhomogeneity, and relativistic nonlinearity, the stimulated Raman backscattering becomes unstable.

MODELLING OF MASS TRANSFER IN WASTEWATER FACILITIES

Problem statement. The design of wastewater treatment systems is a complex process and requires the use of special mathematical models. As a rule, empirical models are used at the stage of designing structures of water drainage systems, which allow obtaining only an “integral” characteristic of the efficiency of wastewater treatment. But in a number of cases, it is important to have information about the spatial distribution of the impurity concentration in the structure. To solve this problem, you need to have three-dimensional mathematical models. In the future, there is a shortage of such models, so the creation of three-dimensional multifactorial models for the analysis of the efficiency of drainage system structures is an urgent task. The purpose of the article. Development of a three-dimensional numerical model for the analysis of the mass transfer process to determine the impurity concentration in the clarifier. Methodology. The analysis of impurity concentration fields in the clarifier is carried out by numerical integration of the three-dimensional equation for the velocity potential and the three-dimensional equation of the convective-diffusion transport of the impurity. For the numerical integration of the Laplace equation for the velocity potential, the variable-triangular method and the Liebmann method are used. Finite-difference splitting schemes are used for numerical integration of the three-dimensional equation of convective-diffusion transport of impurities. Scientific novelty. A dynamic multifactorial numerical model was created for the analysis of the process of mass transfer of impurities in a settling tank by conducting a computational experiment. Practical value. The built multifactorial numerical model makes it possible to analyze the efficiency of wastewater treatment in clarifiers that have a complex geometric shape and cannot be calculated on the basis of existing engineering methods. Conclusions. On the basis of the developed three-dimensional numerical model, a computer code was created, which allows you to quickly obtain information about the distribution of the impurity concentration in the settling tank.

Numerical Solutions of Potential Flow Equations using Finite Differences

This article delves into the numerical solutions of potential flow equations using finite differences, aiming to enhance our understanding of fluid dynamics. The general objective is to obtain numerical solutions to potential flow equations using finite differences, with specific objectives including the investigation of potential flow equations, the solutions of associated PDE and the analysis of the stability of employed numerical schemes. The study employs a combination of numerical methods to achieve its objectives; MATLAB is utilized as a computational tool, while the Gauss-Seidel and Jacobi’s iterative methods are implemented for solving PDEs. Central differences are employed for discretization. The study yields valuable insights into the behaviour of potential flow systems. The significance of this research lies in its contribution to advancing our comprehension of fluid dynamics with potential applications. Generally, this work provides a foundation for further exploration and application of numerical methods in the study of potential flow.

Phase-field modeling of fracture for ferromagnetic materials through Maxwell’s equation

Electro-active materials are classified as electrostrictive and piezoelectric materials. They deform under the action of an external electric field. Piezoelectric material, as a special class of active materials, can produce an internal electric field when subjected to mechanical stress or strain. In return, there is the converse piezoelectric response, which expresses the induction of the mechanical deformation in the material when it is subjected to the application of the electric field. This work presents a variational-based computational modeling approach for failure prediction of ferromagnetic materials. In order to solve this problem, a coupling between magnetostriction and mechanics is modeled, then the fracture mechanism in ferromagnetic materials is investigated. Furthermore, the failure mechanics of ferromagnetic materials under the magnetostrictive effects is studied based on a variational phase-field model of fracture. Phase-field fracture is numerically challenging since the energy functional may admit several local minima, imposing the global irreversibility of the fracture field and dependency of regularization parameters related discretization size. Here, the failure behavior of a magnetoelastic solid body is formulated based on the Helmholtz free energy function, in which the strain tensor, the magnetic induction vector, and the crack phase-field are introduced as state variables. This coupled formulation leads to a continuity equation for the magnetic vector potential through well-known Maxwell’s equations. Hence, the energetic crack driving force is governed by the coupled magneto-mechanical effects under the magneto-static state. Several numerical results substantiate our developments.

Approaching certain central-field Schrödinger problems through the harmonic oscillator green's function

We propose a method to analytically solve the Schrödinger equation with central potentials. By means of Sturm–Liouville expansion, the Green's function (GF) of the Schrödinger differential operator can be represented in the GF of the isotropic harmonic oscillator. Applying this straightforward approach, we solve the Schrödinger problem for the Coulomb potential, for the Morse potential, and for the respective inverted potentials. Using the obtained results, we formulate a method to analytically solve Schrodinger equations for multi-well potentials, demonstrating that classically forbidden regions can be treated as inverted harmonic oscillators, whereas harmonic oscillators or Morse potentials can approximate the wells.

Involvement of ANO1 currents in pacemaking of PDGFRα-positive specialised smooth muscle cells in rat caudal epididymis.

The epididymal duct exhibits spontaneous phasic contractions (SPCs) to store and transport sperm. Here, we explored molecular identification of pacemaker cells driving SPCs in the caudal epididymal duct and also investigated properties of pacemaker currents underlying SPCs focusing on ANO1 Ca2+-activated Cl- channels (CaCCs). Immunohistochemistry was performed to visualise the distribution of platelet-derived growth factor receptor α (PDGFRα)- or ANO1-positive cells in the rat caudal epididymal duct. Perforated whole-cell patch clamp technique was applied to enzymatically isolated epididymal cells, while SPCs were recorded with video edge-tracking technique. Immunohistochemistry revealed the distribution of α-smooth muscle actin (α-SMA)-positive cells co-expressing both PDGFRα and ANO1 in the innermost smooth muscle layer. Approximately one-third of isolated epididymis cells exhibited spontaneous transient inward currents (STICs) at the holding potential -60 mV. The reversal potential for STICs was close to the calculated chloride equivalent potential depending on intracellular Cl- concentrations. Ani9 (3 µM), the ANO1 specific inhibitor, decreased both amplitude and frequency of STICs, while cyclopiazonic acid (CPA, 30 µM), a sarco-/endoplasmic reticulum Ca2+-ATPase (SERCA) inhibitor, abolished STICs. Ani9 (3 or 10 µM) reduced the frequency of SPCs without changing their amplitude. Thus, PDGFRα+, ANO1+ specialised smooth muscle cells (SMCs) appear to function as pacemaker cells to electrically drive epididymal SPCs by generating ANO1-dependnet STICs. STICs arising from spontaneous Ca2+ release from intracellular Ca2+ store and subsequent opening of ANO1 result in depolarisations that spread into adjacent SMCs where L-type voltage-dependent Ca2+ channels are activated to develop SPCs.

ANALYSIS OF THE EFFICIENCY OF THE USE OF PROTECTIVE SCREENS TO REDUCE THE LEVEL OF AIR POLLUTION

The task of assessing the areas of chemical pollution near the highway, where protective screens of different geometric shapes are located, is considered. The purpose of the work is to develop numerical models for calculating pollution zones formed near protective screens, as well as conducting a laboratory experiment to analyze the patterns of formation of pollution zones near screens of a complex geometric shape. For mathematical modeling of the process of formation of pollution zones near the protective screen, the equation of convective-diffusion transfer of impurities is used. This equation takes into account atmospheric diffusion, wind speed, emission intensity of a chemically hazardous substance, the location of the emission source, and the shape of the protective screen. The Navier-Stokes equation and the Laplace equation for the velocity potential are used to solve the problem of aerodynamics. Finite-difference methods are used for numerical integration of modeling equations. A package of application programs was created on the basis of the developed numerical models. Numerical models and a package of programs have been built, allowing to study the process of the formation of areas of pollution near the highway in almost real time. The results of the computational experiment are presented.

Lump–kink and hybrid solutions of the extended (3+1)-dimensional potential KP equation in fluid mechanics

In this paper, the extended (3+1)-dimensional potential KP equation in fluid mechanics is studied through Hirota bilinear method. Many types of hybrid solutions, such as the lump–kink solution, lump-two kink solution and periodic lump solution are obtained by assuming different functions in the bilinear equation. The interaction solution between lump and triangular periodic wave is also derived by combining sine and cosine functions with quadratic functions. Dynamical structures of these exact solutions are depicted by presenting the corresponding three-dimensional, two-dimensional structures and density graphs. These diverse interaction solutions could be helpful for understanding physical phenomena in fluid mechanics.

A (2+1) -dimensional evolution model of Rossby waves and its resonance Y-type soliton and interaction solutions

In this paper, we derive a Kadomtsev-Petviashvili equation by using the multi-scale expansion and perturbation method, which is a model from the potential vorticity equation in the traditional approximation and describes the Rossby wave propagation properties. The N-soliton solutions, resonance Y-type soliton solutions, resonance X-type soliton solutions and interaction solutions of the equation are obtained with the help of dependent-variable transformation. In addition, the composite graphs are given to view the resonance phenomenon of Rossby waves. The results better enrich the research of Rossby waves in ocean dynamics and atmospheric dynamics.

Numerical treatment of entropy generation and Bejan number into an electroosmotically-driven flow of Sutterby nanofluid in an asymmetric microchannel

This article aims to investigate the electrokinetic peristaltic flow of the Sutterby nanofluid through an asymmetric microchannel with a porous medium and the Joule heating parameter. The magnetic field is applied in the transverse direction and the electric field on the flow direction. The flow model consists of the continuity, momentum, heat, and electric potential equations, with appropriate boundary conditions. The Debye-Hückel linearization approximation is considered. Under the long wavelength and small Reynolds number approximation, the lengthy equations were reduced. The resultant coupled equations are numerically solved by the renowned NDSolve coding using Mathematica software. Entropy and Bejan number are also incorporated in this study. The velocity, temperature, and the phenomenon of entrapment are analyzed in depth with the aid of graphical representations. Streamlines of various waveforms are also discussed. The analysis discovered that the velocity of the fluid enhanced for porosity parameter and reverse effects is observed on the quantity of the magnetic parameter.

The role of above-code labeling programs in reducing CO2e emissions in residential buildings

Residential buildings account for 15 % of total U.S. carbon dioxide (CO2) emissions. Voluntary, above-code labeling programs, such as the ENERGY STAR® and Zero Energy Ready Home™ programs, can play a crucial role in reducing the carbon footprint of the new construction residential building sector by encouraging adoption of more energy-efficient and lower emissions building practices and technologies. It’s important for the industry to understand the carbon dioxide equivalent (CO2e) and energy cost savings potential of these programs. To estimate the potential CO2e emissions savings for homes built to above-code labeling program levels, it is imperative to have reliable and accurate tools for estimating emissions from buildings. The open-source OpenStudio-ERI workflow employs detailed physics-based building energy modeling (BEM) to estimate energy usage and operational CO2e emissions. In this study we use the OpenStudio-ERI workflow and regional, time-varying emissions factors from the Cambium database to compare the CO2e emissions and energy costs of homes built to performance levels required by these above-code labeling programs relative to homes built to model energy codes. We present the results of our analysis and provide a range of emissions and annual energy cost savings for the above-code labeling programs.

Analysis of Self-Gravitating Fluid Instabilities from the Post-Newtonian Boltzmann Equation.

Self-gravitating fluid instabilities are analysed within the framework of a post-Newtonian Boltzmann equation coupled with the Poisson equations for the gravitational potentials of the post-Newtonian theory. The Poisson equations are determined from the knowledge of the energy-momentum tensor calculated from a post-Newtonian Maxwell-Jüttner distribution function. The one-particle distribution function and the gravitational potentials are perturbed from their background states, and the perturbations are represented by plane waves characterised by a wave number vector and time-dependent small amplitudes. The time-dependent amplitude of the one-particle distribution function is supposed to be a linear combination of the summational invariants of the post-Newtonian kinetic theory. From the coupled system of differential equations for the time-dependent amplitudes of the one-particle distribution function and gravitational potentials, an evolution equation for the mass density contrast is obtained. It is shown that for perturbation wavelengths smaller than the Jeans wavelength, the mass density contrast propagates as harmonic waves in time. For perturbation wavelengths greater than the Jeans wavelength, the mass density contrast grows in time, and the instability growth in the post-Newtonian theory is more accentuated than the one of the Newtonian theory.

Deciphering the linear relationship in the activity of the oxygen reduction reaction on Pt electrodes: A decisive role of adsorbates

Despite substantial efforts in developing high-performance catalysts for the oxygen reduction reaction (ORR), the persistent challenge lies in the high onset overpotential of the ORR, and the effect of the electrolyte solution cannot be ignored. Consequently, we have systematically investigated the impact of adsorbate species and concentration, as well as solution pH, on the ORR activity on Pt(111) and Pt(poly) electrodes. The results all tend to establish a linear quantitative relationship between the onset potential for ORR and the adsorption equilibrium potential of the adsorbate. This finding indicates the decisive role of adsorbates in the onset potential for ORR, suggesting that the adsorption potential of adsorbates can serve as an intuitive criterion for ORR activity. Additional support for this conclusion is derived from experimental results obtained from the oxygen evolution reaction on Pt(poly) with different adsorbate species and from the hydrogen evolution reaction on Pt(111) with iodine adsorption. We further propose both an empirical equation for the onset potential for ORR and the concept of a potential-regulated adsorbate shielding effect to elucidate the influence of adsorbates on ORR activity. This study provides new insights into the high onset overpotential of the ORR and offers potential strategies for predicting and enhancing ORR activity in the future.

Modeling of inner-outer gates and temperature dependent gate-induced drain leakage current of junctionless double-gate-all-around FET

In this paper, the temperature-dependent gate-induced drain leakage (GIDL) current model is proposed with the help of a lateral electric field (EL) across the inner and outer gate interfaces of the junctionless double-gate-all-around (JL-DGAA) field-effect transistor (FET). The EL at the interface is obtained from the surface potential equation after solving the 3D Poisson equation with appropriate boundary conditions. The derived model is validated using the well-calibrated TCAD simulator platform with experimental data. A higher GIDL current at the outer gate-channel interface is observed compared to the inner gate-channel interface in the OFF-state conditions due to the high EL between the channel and drain, which is reported for the first time here. The effect of temperature (from 200 K to 500 K) on the GIDL current is also observed, which increases linearly with increasing temperature. Further, the impact of induced mechanical stress (MS) is investigated on the GIDL current, where the GIDL current increases exponentially with increasing induced MS due to effective mass and band gap reduction. Finally, the dynamic performance is investigated in terms of the gate delay time as a function of channel length and temperature. The gate delay diminishes with decreasing channel length and increasing temperature, which shows the switching speed and ability of the device.

Modeling of Self Potential (SP) Anomalies over a Polarized Rod with Finite Depth Extents

Modeling is a powerful tool used by Geoscientists to provide pre-knowledge about the expectations of any geophysical field measurements. This study generates Self Potential (SP) anomalies over a typical dike-like structure to observe the influence of depth of burial and dip on SP anomalies. A computer program was developed from the potential distribution equation of an inclined polarized rod with a limited depth extent using Visual Basic (VB) programming language to produce synthetic data for potential distribution. The potential distribution data were used to generate theoretical SP anomaly curves for a polarized rod for varying depth of burial and dip. Twenty SP anomaly curves were generated with different dip values and depth of burial and from these curves, superimposed curves were also generated. The anomalies were analyzed for the effect of depth of burial and attitude or dip. The SP anomaly curves generated show that an increase in depth of burial causes a reduction in the peak negative amplitude of SP anomaly curves. For inclined polarized rod at relatively shallow depth (<2.0 m), the peak negative amplitude remains virtually the same with a positive shoulder over the down dip side of the target. Also as the dip angle decreases from 90o for a fixed depth of burial, the anomaly curves become asymmetrical. At 0o, the distance between the peak negative and peak positive amplitude of the anomaly curve is equal to the linear extent of the rod. Therefore, this study shows that the depth of burial inversely influences the amplitude of self-potential (SP) anomalies while the dip angle affects the form or symmetry of anomaly curves.

A quadratic conservation algorithm for non‐hydrostatic models

AbstractWe propose a quadratic conservation algorithm for non‐hydrostatic models, focusing on energy conservation and consistency. To ensure that the discrete potential and continuity equations are equivalent and to achieve conservation of mass and momentum transport, we introduce a potential reference height into the discrete potential equation. The time integration scheme is based on an explicit fourth‐order Runge–Kutta method with varying time steps, which is specially designed to maintain the quadratic conservation property. To suppress numerical noise, we suggest an adaptive diffusion method that does not lose energy. Numerical noise is estimated by higher order terms of polynomial equations, and diffusion coefficients are determined using a least‐squares method. Numerical tests demonstrate that the quadratic conservation algorithm accurately maintains total energy conservation and produces solutions of comparable quality to those reported in existing literature. Furthermore, it can resolve problems with small‐scale features.

Reconstructing the baryonic acoustic oscillations in the presence of photo-z uncertainties

ABSTRACT The reconstruction method has been widely employed to improve the baryon acoustic oscillations (BAO) measurement in spectroscopic survey data analysis. In this study, we explore the reconstruction of the BAO signals in the realm of photometric data. By adapting the Zel’dovich reconstruction technique, we develop a formalism to reconstruct the transverse BAO in the presence of photo-z uncertainties under the plane-parallel approximation. We access the performance of the BAO reconstruction through comoving N-body simulations. The transverse reconstruction potential can be derived by solving a 2D potential equation, with the surface density and the radial potential contribution acting as the source terms. The solution is predominantly determined by the surface density. As is evident in dense samples, such as the matter field, the transverse BAO reconstruction can enhance both the strength of the BAO signals and their cross correlation with the initial conditions. At z = 0, the cross-correlation is increased by a factor of 1.2 at $k_\perp = 0.2 \, \mathrm{Mpc}^{-1}h$ and 1.4 at $k_\perp = 0.3 \, \mathrm{Mpc}^{-1}h$, respectively. We contrast the 2D potential results with the 3D Poisson equation solution, wherein we directly solve the potential equation using the position in photo-z space, and find good agreement. Additionally, we examine the impact of various conditions, such as the smoothing scales and the level of photo-z uncertainties, on the reconstruction results. We envision the straightforward application of this method to survey data.

Three-dimensional modeling of Marine Controlled Source Electromagnetic field using a space-wavenumber domain method

Under complex conditions, the three-dimensional modeling of Marine Controlled Source Electromagnetic field requires a significant amount of computation, resulting in slow calculation speed and high storage requirements. To solve these problems, we propose a 3D numerical simulation method of electromagnetic field in the space-wavenumber domain under the Lorenz gauge. Firstly, the new method utilizes the two-dimensional Fourier transform in the horizontal direction to transform the 3D partial differential equations of the Lorenz vector potentials into multiple independent ordinary differential equations. Secondly, the finite element method is used to solve the ordinary differential equations. The fields in the spatial domain are then obtained by using the inverse Fourier transform, and finally, an iterative method with a compact operator is applied to approximate the true solution. The approach requires less computation and storage, and the algorithm is highly parallel, which effectively improves the efficiency of 3D forward modeling of electromagnetic fields. The correctness, effectiveness, and computational efficiency of the method are verified by the design model.

Three-dimension holographic numerical simulation of gravity anomalies

In gravity-anomaly-based prospecting, it is difficult to achieve large-scale and high-precision inversion imaging of complex geological models. To address this issue, this paper proposes a three-dimensional holographic numerical simulation method for gravity anomalies. This method transforms the 3D partial differential equation of gravity potential into many independent 1D differential equations with different wavenumbers by performing a 2D Fourier transform along the horizontal direction. It decomposes a large-scale 3D numerical modeling problem into many 1D numerical modeling problems, which greatly reduces the computation and memory requirements of modeling, and each 1D differential equation is independent, so it has high parallelism. The vertical direction is reserved as the spatial domain, which has strict upper and lower boundary conditions; The horizontal 2D Fourier transform uses a holographic Fourier transform (Holo-FT) with a full interval of integration, consistent with the physical boundary in the horizontal direction, thus the physical information of the gravity potential described by the three-dimensional partial differential equation is fully and accurately simulated. Using the high parallelism of the algorithm, the CPU is used for parallel solving ordinary differential equations, while the GPU is used for parallel computing of Holo-FT, which implements the CPU-GPU parallel acceleration scheme and further improves the efficiency of this algorithm.