Equation For Electric Potential Research Articles

Lattice Boltzmann modelling of electro-thermo-convection in a planar layer of dielectric liquid subjected to unipolar injection and thermal gradient

In this paper, the electro-thermo-convective phenomena induced by the simultaneous action of a unipolar injection of ions and a thermal gradient in a dielectric liquid lying between two parallel plates are studied. We develop a lattice Boltzmann model (LBM) to solve the whole set of coupled governing equations, including the Navier–Stokes equations, the conservation equation of charge density, the Poisson’s equation for electric potential and the energy equation. In this method, four different particle distribution functions are used to calculate the flow field, electric potential, charge density distribution and temperature field, respectively. A multi-scale analysis is also performed to recover the macroscopic equations from the discrete lattice Boltzmann equations (LBEs). The present method is validated with several carefully chosen test cases, and all LBM results are found to be highly consistent with available analytical solutions or other numerical works. Besides, the typical subcritical bifurcations and the hysteresis loops in electro-thermo-convective are clearly presented and analyzed.

Operation of a semiconductor microcavity under electric excitation

We present a microscopic theory for the description of the bias-controlled operation of an exciton-polariton-based heterostructure, in particular, the polariton laser. Combining together the Poisson equations for the scalar electric potential and Fermi quasi-energies of electrons and holes in a semiconductor heterostructure, the Boltzmann equation for the incoherent excitonic reservoir and the Gross-Pitaevskii equation for the exciton-polariton mean field, we simulate the dynamics of the system minimising the number of free parameters and build a theoretical threshold characteristic: number of particles vs applied bias. This approach, which also accounts for the nonlinear (exciton-exciton) interaction, particle lifetime, and which can, in principle, account for any relaxation mechanisms for the carriers of charge inside the heterostructure or polariton loss, allows to completely describe modern experiments on polariton transport and model devices.

Continuous Modeling of Calcium Transport Through Biological Membranes

In this work an approach to the modeling of the biological membranes where a membrane is treated as a continuous medium is presented. The Nernst-Planck-Poisson model including Poisson equation for electric potential is used to describe transport of ions in the mitochondrial membrane—the interface which joins mitochondrial matrix with cellular cytosis. The transport of calcium ions is considered. Concentration of calcium inside the mitochondrion is not known accurately because different analytical methods give dramatically different results. We explain mathematically these differences assuming the complexing reaction inside mitochondrion and the existence of the calcium set-point (concentration of calcium in cytosis below which calcium stops entering the mitochondrion).

Relevance of Hydrodynamic Effects for the Calculation of Outer Surface Potential of Biological Membrane Using Electrophoretic Data

In this paper, we present the results of a study on the influence of hydrodynamic effects on the surface potentials of the erythrocyte membrane, comparing two different models formulated to simulate the electrophoretic movement of a biological cell: the classical Helmholtz-Smoluchowski model and a model presented by Hsu et al. (1996). This model considers hydrodynamic effects to describe the distribution of the fluid velocity. The electric potential equation was obtained from the non-linear Poisson-Boltzmann equation, considering the spatial distribution of electrical charges fixed in glycocalyx and cytoplasmic proteins, as well as electrolyte charges and ones fixed on the surfaces of lipidic bilayer. Our results show that the Helmholtz-Smoluchowski model is not able to reflect the real forces responsible to the electrophoretic behavior of cell, because it does not take account the hydrodynamic effects of glycocalyx. This charged network that covers cellular surface constitutes a complex physical system whose electromechanical characteristics cannot be neglected. Then, supporting the hypothesis of other authors, we suggest that, in electrophoretic motion analyses of cells, the classical model represents a limiting case of models that take into account hydrodynamic effects to describe the velocity distribution of fluid.

Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids.

In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work.

Numerical Modeling of Dielectric Barrier Discharge Plasma Actuation

A computational method has been developed to couple the electrohydrodynamic (EHD) body forces induced by dielectric barrier discharge (DBD) actuation with unsteady Reynolds-Averaged Navier–Stokes (URANS) model or large eddy simulation (LES) for incompressible flows. The EHD body force model is based on solving the electrostatic equations for electric potential and net charge density. The boundary condition for net charge density on the dielectric surface is obtained from a space–time lumped-element (STLE) circuit model or an empirical model. The DBD–URANS/LES coupled solver has been implemented using a multiple-domain approach and a multiple subcycle technique. The DBD plasma-induced flow in a quiescent environment is used to validate the coupled solver, evaluate different EHD body force models, and compare the performance of the actuator driven by voltage with various waveforms and amplitudes.

Radio-Frequency Discharge at Low Pressure: A Non-local Problem Statement Approach

Self-consistent nonlocal nonlinear mathematical CCRF discharge model is constructed. The model includes the convection – diffusion equations for electronic and ionic gas, Poisson's equation for electric field potential, the equation of balance of metastable as well as ground states neutral atoms, and stationary equation of heat conductivity for electronic and ionic gases. An algorithm for the numerical solutions of the model has been described. The results of test calculations of CCRF discharge characteristics at the interelectrode distance of 2.2cm, the gas pressure of 13.3Pa, the voltage amplitude of 65V were obtained. Comparison of the results showed the accuracy of the results and proved the adequacy of the mathematical model and the method of calculation.

キャパシタンスCTを用いた土中水分評価の試み

This paper proposes a capacitance CT technique with two sets of in-line electrodes arranged in parallel to evaluate soil-water distribution under the ground and numerical simulation is carried out to verify the possibility of measuring the soil-water distribution. Volumetric water content is used as an index to estimate the wet state in soil to solve mechanism of landslides in research. The proposed CT technique can be applied to the measurement of volumetric water content without interfering the water penetration into the measuring region. The Laplace equation for electric potential is solved by a finite difference method to evaluate the capacitance under the Gauss’ law in numerical simulation. From the simulation results, it is found that the soil-water distribution can be successfully estimated for various sizes and numbers of water elements in soil. Furthermore, it is confirmed that the permittivity estimated by the present capacitance CT corresponds to the given volumetric water content.

An Unconditional Stable Meshless ADI-RPIM for Simulation of Coupled Transient Electrothermal Problems

Joule heat losses of electronics components may change behaviors of electrical and electronic devices, especially for semiconductor components whose electrical parameters are temperature-dependent. In this paper, a meshless approach based on radial point interpolation method (RPIM) is developed to numerically simulate the coupled electrical–thermal process that consists of electric potential equations and time-domain heat transfer equations in two dimensions. The RPIM can handle multiscale structures well with their nonuniform node placements. Furthermore, an alternating-direction-implicit technique is introduced to resolve the numerical stability issue of time-domain simulation of the multiscale electrothermal coupled problems. Detailed formulations are derived, and a fictitious point method is developed to handle the Neumann boundary condition pertaining to electric potential problems and the Robin condition pertaining to multiphysics problems. The efficiency and effectiveness of the proposed RPIM are verified through simulation of thin-film resistors. In comparisons with the existing methods, computational speed up of at least ten times is achieved.

Numerical Investigation on the Impact of Anode Change on Heat Transfer and Fluid Flow in Aluminum Smelting Cells

In order to understand the impact of anode change on heat transfer and magnetohydrodynamic (MHD) flow in aluminum smelting cells, a transient three-dimensional (3D) coupled mathematical model has been developed. The solutions of the mass, momentum and energy conservation equations were simultaneously implemented by the finite volume method with full coupling of the Joule heating and Lorentz force through solving the electrical potential equation. The volume of fluid (VOF) approach was employed to describe the two-phase flow. The phase change of molten electrolyte (bath) as well as molten aluminum (metal) was modeled by an enthalpy — based technique, where the mushy zone is treated as a porous medium with porosity equal to the liquid fraction. The effect of the new anode temperature on recovery time was also analyzed. A reasonable agreement between the test data and simulated results is obtained. The results indicate that the temperature of the bath under cold anodes first decreases reaching the minimal value and rises under the effect of increasing Joule heating, and finally returns to steady state. The colder bath decays the velocity, and the around ledge becomes thicker.

Electroviscous effect on fluid drag in a microchannel with large zeta potential.

The electroviscous effect has been widely studied to investigate the effect of surface charge-induced electric double layers (EDL) on the pressure-driven flow in a micro/nano channel. EDL has been reported to reduce the velocity of fluid flow and increase the fluid drag. Nevertheless, the study on the combined effect of EDL with large zeta potential up to several hundred millivolts and surface charge depenedent-slip on the micro/nano flow is still needed. In this paper, the nonlinear Poisson–Boltzmann equation for electrical potential and ion distribution in non-overlapping EDL is first analytically solved. Then, the modified Navier–Stokes equation for the flow considering the effect of surface charge on the electrical conductivity of the electrolyte and slip length is analytically solved. This analysis is used to study the effect of non-overlapping EDL with large zeta potential on the pressure-driven flow in a microchannel with no-slip and charge-dependent slip conditions. The results show that the EDL leads to an increase in the fluid drag, but that slip can reduce the fluid drag. When the zeta potential is large enough, the electroviscous effect disappears for flow in the microchannel under a no-slip condition. However, the retardation of EDL on the flow and the enhancement of slip on the flow counteract each other under a slip condition. The underlying mechanisms of the effect of EDL with large zeta potential on fluid drag are the high net ionic concentration near the channel wall and the fast decay of electrical potential in the EDL when the zeta potential is large enough.

Numerical simulation of electrohydrodynamic spray with stable Taylor cone–jet

A numerical model has been developed to estimate the shape of the liquid cone and the resulting jet in presence of external electric field. The model estimates the liquid velocity at the cone surface to calculate the electrical current. The equations of electric potential are solved numerically with an iterative procedure and then continuity and momentum equations solved in a commercial CFD code with electrical body forces. The model is validated by comparing with experimental results.

Domain Decomposition Modified with Characteristic Finite Element Method for Numerical Simulation of Semiconductor Transient Problem of Heat Conduction

A characteristic finite element algorithm based on domain decomposition is structured in this paper to approximate numerically multi-dimensional semiconductor transient problems of heat conduction. Finite element approximation is presented for the electric field potential equation, and a domain decomposition discretization with characteristic finite element is put forward for the electron concentration equation, hole concentration equation and heat conductor equation. An optimal order error estimate in L2 norm is derived for the coupled system by using some techniques such as variation, domain decomposition, the method of characteristics, the principle of energy, negative norm estimates, induction hypothesis, prior estimates theory and other techniques of partial differential equations. Finally, experimental data consistent with theoretical convergence rate are shown. This type of numerical method is of high computational efficiency and can successfully solve this international problem.

Effect of induced electric field on magneto-natural convection in a vertical cylindrical annulus filled with liquid potassium

Laminar steady magneto-natural convection in a vertical cylindrical annulus formed by two coaxial cylinders filled with liquid potassium is studied numerically. The cylindrical walls are isothermal, and the other walls are assumed to be adiabatic. A constant horizontal magnetic field is also applied on the enclosure. The results show that flow is axisymmetric in the absence of the magnetic field; but by applying the horizontal magnetic field, it becomes asymmetric. This is due to the growth of Roberts and Hartmann layers near the walls parallel and normal to the magnetic field, respectively. The applied magnetic field results in a reduction in the Nusselt number in most of the regions of the annulus. This reduction is high in the Hartmann layers but low in the Roberts layers. Moreover, it was found that for a given value of Hartman number, the average Nusselt number is greater in the case of solving the electric potential equation. The results show that there is a large difference in the Nusselt number obtained by solving the electric potential equation, compared with by neglecting the electric potential.

Fractal electrodynamics via non-integer dimensional space approach

Using the recently suggested vector calculus for non-integer dimensional space, we consider electrodynamics problems in isotropic case. This calculus allows us to describe fractal media in the framework of continuum models with non-integer dimensional space. We consider electric and magnetic fields of fractal media with charges and currents in the framework of continuum models with non-integer dimensional spaces. An application of the fractal Gauss's law, the fractal Ampere's circuital law, the fractal Poisson equation for electric potential, and equation for fractal stream of charges are suggested. Lorentz invariance and speed of light in fractal electrodynamics are discussed. An expression for effective refractive index of non-integer dimensional space is suggested.

Entropy of a self-gravitating electrically charged thin shell and the black hole limit

A static self-gravitating electrically charged spherical thin shell embedded in a ($3+1$)-dimensional spacetime is used to study the thermodynamic and entropic properties of the corresponding spacetime. Inside the shell, the spacetime is flat, whereas outside it is a Reissner-Nordstr\"om spacetime, and this is enough to establish the energy density, the pressure, and the electric charge in the shell. Imposing that the shell is at a given local temperature and that the first law of thermodynamics holds on the shell one can find the integrability conditions for the temperature and for the thermodynamic electric potential, the thermodynamic equilibrium states, and the thermodynamic stability conditions. Through the integrability conditions and the first law of thermodynamics an expression for the shell's entropy can be calculated. It is found that the shell's entropy is generically a function of the shell's gravitational and Cauchy radii alone. A plethora of sets of temperature and electric potential equations of state can be given. One set of equations of state is related to the Hawking temperature and a precisely given electric potential. Then, as one pushes the shell to its own gravitational radius and the temperature is set precisely equal to the Hawking temperature, so that there is a finite quantum backreaction that does not destroy the shell, one finds that the entropy of the shell equals the Bekenstein-Hawking entropy for a black hole. The other set of equations of state is such that the temperature is essentially a power law in the inverse Arnowitt-Deser-Misner (ADM) mass and the electric potential is a power law in the electric charge and in the inverse ADM mass. In this case, the equations of thermodynamic stability are analyzed, resulting in certain allowed regions for the parameters entering the problem. Other sets of equations of state can be proposed. Whatever the initial equation of state for the temperature, as the shell radius approaches its own gravitational radius, the quantum backreaction imposes the Hawking temperature for the shell in this limit. Thus, when the shell's radius is sent to the shell's own gravitational radius the formalism developed allows one to find the precise form of the Bekenstein-Hawking entropy of the correlated black hole.

Analysis of electro-osmotic flow in a microchannel with undulated surfaces

The electro-osmotic flow through a channel between two undulated surfaces induced by an external electric field is investigated. The gap of the channel is very small and comparable to the thickness of the electrical double layers. A lattice Boltzmann simulation is carried out on the model consisting of the Poisson equation for electrical potential, the Nernst–Planck equation for ion concentration, and the Navier–Stokes equations for flows of the electrolyte solution. An analytical model that predicts the flow rate is also derived under the assumption that the channel width is very small compared with the characteristic length of the variation along the channel. The analytical results are compared with the numerical results obtained by using the lattice Boltzmann method. In the case of a constant surface charge density along the channel, the variation of the channel width reduces the electro-osmotic flow, and the flow rate is smaller than that of a straight channel. In the case of a surface charge density distributed inhomogeneously, one-way flow occurs even under the restriction of a zero net surface charge along the channel.

Numerical calculation of particle collection efficiency in an electrostatic precipitator

The present numerical study involves the finding of the collection efficiency of an electrostatic precipitator (ESP) using a finite volume (ANUPRAVAHA) solver for the Navier–Stokes and continuity equations, along with the Poisson’s equation for electric potential and current continuity. The particle movement is simulated using a Lagrangian approach to predict the trajectory of single particles in a fluid as the result of various forces acting on the particle. The ESP model consists of three wires and three collecting plates of combined length of L placed one after another. The calculations are carried out for a wire-to-plate spacing H= 0.175 m, length of ESP L= 2.210 m and wire-to-wire spacing of 0.725 m with radius of wire R wire= 10 mm and inlet air-particle velocity of 1.2 m/s. Different electrical potentials (φ= 15–30 kV) are applied to the three discharge electrodes wires. It is seen that the particle collection efficiency of the ESP increases with increasing particle diameter, electrical potential and plate length for a given inlet velocity.

Electric field penetration into the ionosphere in the presence of anomalous radon emanation

The scientists came to consensus that electric field driven mechanism is more probable to explain ionospheric anomalies before earthquakes than the acoustic-driven mechanism (Pulients and Davidenko, 2014), and it is essential to understand how a vertical electric field from the ground penetrates into the ionosphere. Anomalous radon emanation in the epicentral area is believed to change the atmospheric electrical conditions. Considering the effect of radon emanation on atmospheric conductivity, the electric potential equation is established and solved numerically in the presence of a vertical electric field of 1kV/m on the ground. The results show that radon emanation can strengthen atmospheric conductivity, as a consequence, the resulting electric field is increased by about 60% in the daytime ionosphere. However, the resulting electric field in the ionosphere is very weak (only about 0.3μV/m), which implies that the penetration of vertical electric field of 1kV/m in the seismic area is unlikely to produce daytime ionospheric anomalies before earthquakes.

AETHER: A simulation platform for inductively coupled plasma

An in-house code is developed to simulate the inductively coupled plasma (ICP). The model comprises the fluid, electromagnetic and transformer submodels. Fluid equations are solved to evaluate the plasma flow parameters, including the plasma and neutral densities, ion and neutral velocities, electron flux, electron temperature, and electric potential. The model relies on the ambipolar approximation and offers the evaluation of plasma parameters without solving the sheath region. The electromagnetic model handles the calculation of the electric and magnetic fields using the magnetic vector potential. The transformer model captures the effect of the matching circuit utilized in laboratory experiments for RF power deposition. The continuity and momentum equations are solved using finite volume method. The energy, electric potential, and magnetic vector potential equations are solved using finite difference method. The resulting linear systems of equations are solved with iterative solvers including Jacobi and GMRES. The code is written using the C++ programming language, it works in parallel and has graphical user interface. The model is applied to study ICP characteristics of a plasma confined within a cylindrical chamber with dielectric walls for two different power deposition cases. The results obtained from the developed model are verified using the plasma module of COMSOL Multiphysics. The model is also applied to a plasma source configuration, and it is demonstrated that there is an overall increase in the plasma potential when current is extracted from ICP with a biased wall electrode.