Equation For Electric Potential Research Articles

Mathematical modeling of inhomogeneous electric field impact on a liquid crystal layer

AbstractThe authors analyze the Fréedericksz transition phenomenon of reorientation of molecules in the plane liquid crystal layer in inhomogeneous electric field generated by parallel plate capacitor using the Oseen–Frank mathematical model. The capacitor is a set of short metal plates periodically located on both sides of the extended liquid crystal layer, connected in series in two circuits. When electric charges of different signs appear on opposite plates of the capacitor, the electric field in the periodicity cell turns out to be significantly inhomogeneous both in length and in thickness of the layer. Governing equations of the model represent Euler's variational equations in the problem on minimization of the potential energy of liquid crystal. Numerical solution of the equations uses the authorial variational difference scheme. The Laplace equation for electric potential in the exterior of the liquid crystal layer is solved using the method of lines. The electric field and the patterns of orientation angles of molecules inside the layer are determined by iterating with numerical solution of Poisson's equation using the fast Fourier transform at each iteration step. The algorithm is implemented by means of CUDA technology for computing systems with graphics accelerators. The scale of the mesh to ensure the computation accuracy is selected from calculation of Tsvetkov's order parameter in a rectangular cell pattern. Verification of the algorithm and program is implemented by way of comparing the computational results and the exact solution of the problem on the homogenous electric field with the constant initial orientation angle of liquid crystal molecules. The computations accomplished yield the patterns of the electric potentials and orientation angles of molecules depending on the position of the capacitor plates at uniform and s‐shaped initial distribution of the orientation angles through the thickness of the liquid crystal layer.

Numerical Calculations of Liquid-Metal Magnetohydrodynamic Flows Through Circular Pipe in Magnetic Field Inlet Region and Outlet Region

Numerical calculations are conducted for liquid-metal magnetohydrodynamic flows through a circular pipe with an electrically conducting wall in both the magnetic field inlet region and the outlet region. Conservation equations of fluid mass and of fluid momentum and the Poisson equation for electrical potential are solved numerically. The calculations are performed by a cylindrical coordinate system using a staggered grid in order to obtain numerically stable solutions, covering Hartmann numbers up to the order of 10 000. As to the loss coefficient ζ for the pressure drop, the value of ζ/(Ha2/Re) does not depend on the Ha number, the Re number, and the wall conductance ratio very much for both the magnetic field inlet section and the outlet section. The value of ζ/(Ha2/Re) changes mainly with the gradient of the applied magnetic field for both the magnetic field inlet section and the outlet section.

Electrostatic derivation for the van der Pauw formula and simulation using arbitrarily shaped resistive materials

The van der Pauw method is an important technique in measuring the resistivity of flat materials by means of the van der Pauw formula. The formula is derived analytically from a rectangular material by solving the two-dimensional Laplace equation of electric potential. The input and output current densities are two Neumann boundary conditions. The use of the δ function as the current boundary condition, in comparison with the rect function in the literature, makes the deduction simple. The van der Pauw measurement is simulated with the finite element method by using two arbitrarily shaped materials of different resistivities and four point contacts around the periphery. The input current is used as the boundary condition at one contact and electric potential as the variable to be solved. The potential differences between two voltage contacts are employed to calculate the two resistances for the formula. With the reduction of mesh size, the simulated resistivity approaches the actual value progressively. The minimum relative error is on the order of ppm. In addition, one circular hole is produced in the irregular material, and the relative error is investigated in response to the hole position and radius. The equipotential curves and current pattern are displayed for demonstration. This confirms that the van der Pauw method is valid for measurement of arbitrarily shaped materials without an inner hole.

Solid-liquid phase change subjected to unipolar charge injection from a circular wire electrode

A numerical investigation of isothermal melting of a dielectric phase change material in a square cavity subjected to unipolar charge injection from a circular wire electrode is reported. Governing equations for the solid-liquid phase change under the influence of an electric field include the Navier–Stokes equations for fluid flow, energy equation for thermal transport, Poisson equation for electric potential, and the Nernst-Planck equation for charge conservation. An enthalpy source-term based fixed grid approach is adopted to model the melting process. The governing equations that consider the coupled interactions between flow, thermal and electric fields are solved using the finite volume framework of OpenFOAM®. Time evolution of the melting rate, maximum flow velocity, mean Nusselt number, and mean Coulomb force in the electrohydrodynamic flow assisted melting process is mapped. The mechanism and role of electrohydrodynamic forces on influencing the net flow and melt interface morphology is studied. The competing viscous, buoyancy, and electric forces result in an altered flow pattern that aids the melting process. The enhancement in the rate of melting at different strengths of buoyancy and electrostatic forces is quantified. Furthermore, the effects of the radius of the circular wire electrode in relation to the cavity size are highlighted. A maximum of 63% enhancement in melting rate is observed within the considered parameter space.

Numerical analysis of deflection control of a gas plasma jet based on magnetohydrodynamic staggered electrode configuration

To control the deflection of the gas plasma jet, a new analytical method is proposed based on the Magnetohydrodynamic (MHD) technique. Based on the typical MHD power generation model, the applied voltage is applied to the staggered electrodes, that is, a pair of electrodes on the same side wall are connected to generate an axial current in the channel. Under the action of the magnetic field perpendicular to the direction of the flow, the plasma is subjected to electromagnetic forces perpendicular to these two directions, and the jet is deflected. The computational model including the Navier-Stokes equations coupled with electromagnetic source terms, the electric potential equation and Ohm’s law is solved. The deflection of the gas jet under the action of an electromagnetic field is observed, and the maximum deflection angle is about 14.8°. The influences of the electric field, magnetic field, and conductivity on the jet deflection are studied. Results show that although the influences of these three factors on the deflection are similar, and the effect of increasing the electric field strength is slightly greater, priority should be given to increasing the magnetic field strength from the perspective of reducing energy consumption. The Stuart number is introduced to assess the ability of electromagnetic force to control jet deflection. When the electromagnetic parameters are constant, this solution provides better control of low-density and low-speed fluid flows. The calculation results show that using the staggered electrode method configuration is feasible in terms of controlling the deflection of a plasma jet deflection.

Influence of fluid flows on electric double layers in evaporating colloidal sessile droplets.

A model is developed for describing the transport of charged colloidal particles in an evaporating sessile droplet on the electrified metal substrate in the presence of a solvent flow. The model takes into account the electric charge of colloidal particles and small ions produced by electrolytic dissociation of the active groups on the colloidal particles and solvent molecules. We employ a system of self-consistent Poisson and Nernst-Planck equations for electric potential and average concentrations of colloidal particles and ions with the appropriate boundary conditions. The fluid dynamics, temperature distribution and evaporation process are described with the Navier-Stokes equations, equations of heat conduction and vapor diffusion in air, respectively. The developed model is used to carry out a first-principles numerical simulation of charged silica colloidal particle transport in an evaporating aqueous droplet. We find that electric double layers can be destroyed by a sufficiently strong fluid flow.

Combined effects of buoyancy and electric forces on non-isothermal melting of a dielectric phase change material

Numerical investigations of non-isothermal melting of a dielectric PCM under the combined influence of buoyancy and electric forces have been performed. The coupled set of governing equations which include the Poisson equation for electric potential, Nernst–Planck equation for charge transport, Navier–Stokes equations and the energy equation are implemented in the finite-volume framework of OpenFOAM®. The solid–liquid phase change is modeled by an enthalpy source based approach and temperature dependent dielectric properties have been considered. The dynamic evolution of the electrohydrodynamics (EHD) assisted melting process is analyzed. In the presence of electric field, melting rate curve undergoes a deflection at a point that indicates the shift from natural convection dominated melting to electroconvection dominated melting. The EHD flow compliments the buoyancy driven flow, and the multiple smaller convective rolls of EHD flow enhances the mixing in liquid region. The melting time is a decreasing function of electric Rayleigh number T, whereas, the maximum melt fraction is an increasing function of T. Although the role of Rayleigh number Ra is minimal at higher values of T, Ra determines the occurrence of deflection point. The dielectric force induced by the temperature dependent dielectric properties does not have a significant influence on the melting process, for the parameters considered in this study.

DC IR-Drop Analysis of Power Distribution Networks by a Robin Transmission Condition-Enhanced Discontinuous Galerkin Method

In this work, a novel Robin transmission condition (RTC)-enhanced discontinuous Galerkin (DG) method is proposed for the dc IR-drop analysis of power distribution networks with Joule heating effects included. Unlike the conventional DG method, the proposed DG method straightforwardly applied to discretize the second-order spatial partial differential governing equations for the electrostatic potential <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi $ </tex-math></inline-formula> and the steady-state temperature <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> . The numerical flux in DG used to facilitate the information exchange among neighboring subdomains introduces two additional variables: the current density <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$J$ </tex-math></inline-formula> for the electrical potential equation and the thermal flux <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> for the thermal equation. To solve them, at the interface of neighboring subdomains, an RTC is presented as the second equation to establish another connection for solutions in neighboring subdomains. With this strategy, the number of degrees of freedom (DoFs) involved in the proposed RTC-DG method is dramatically reduced compared with the traditional DG method. The finalized matrix system is solved in an finite-element tearing and interconnecting (FETI)-like procedure, namely the unknowns are obtained in a subdomain-by-subdomain scheme. Finally, the accuracy and the efficiency of the proposed RTC-DG method are validated by several representative examples.

Interplay of dual streaming potentials on electroosmotic mixed convection flow in a vertical microannulus with Joule heating effect

A comparative analysis of dual streaming potentials on electroosmotic mixed convection flow formation in a vertical microannulus with Joule heating effect is carried out in this article. The electric potential, momentum and energy equations governing flow formations and heat transfer are derived and solved exactly using the method of undetermined coefficients. Relative to what is known in the literature, dual distinct streaming potentials are found to be induced due to the Joule heating term entering the energy equation, which led to a quadratic function of streaming potential in the momentum equation. Results obtained showed that these streaming potentials are real, distinct and behave differently on flow formations and heat transfer. It is found that heat transfer and skin-friction for normal streaming potential are increasing function of Joule heating term and magnitude of electroosmotic parameter respectively. The findings in this article has significant applications in liquid separations at microlevel, enhancements of flow rate in micropumps and soil conditioning.

Electrodeposition mechanism of chromium nanoparticle coatings: Modeling and experimental validation

In this paper the deposition and surface growth of chromium nanoparticles (CrNPs) coatings on a commercial steel substrate by electrodeposition at a constant current, is presented; both theoretically through a theoretical-computational model applying a modified Schrödinger equation, energy potentials and electrochemistry and phitosynthesis of Cr Nanoparticles . For the simulation process, the finite element method (FEM) was used considering the equation of electric potential and electroneutrality with and without the inclusion of a quantum leap. The theoretical results are in accordance with the experimental results, considering a three-level multifactorial experiment design with current, voltage and ion concentration as factors and the formation and growth of the coatings as a response function. The results obtained indicate the presence of electro-diffusion at the substrate-coating interface. The developed model has the predictive ability of 97 ± 2 % for the electrochemical deposition of CrNPs on a steel substrate.

A mathematical modelling of multiphysics-based propagation characteristics of surface wave in piezoelectric - hydrogel layer on an elastic substrate

This paper presents a mathematical modelling of a novel multiphysics (electro-mechanic coupling) problem of the shear wave propagation in laminated structures (piezoelectric - hydrogel - elastic substrate) using the wave mode method while finding the general solution for each medium. A dynamic mechanical equilibrium equation (for transverse deflection) and Maxwell equation (for electric potential) are solved in a coupled manner over each domain resulting a general closed-form solution. A specific analytical solution is then obtained enforcing the boundary conditions at the top of piezoelectric layer and the interface continuity requirements between each layer and the elastic half-space. A novel approach, to uncouple the coupled equations, is presented resulting in a final systematic solution. The effect of x coordinate (thickness) on the field variables is carefully examined for the electrically open and short cases. The jump in the stress and electric displacement components (causing delamination due to shearing mode of fracture) is present across all the interfaces due to the different bulk material constants. The key contribution of the current work is demonstrating the influence of fixed-charge concentration inside the hydrogel layer on the shear wave propagation. The present study thus provides a new concept for adjusting and controlling the elastic wave propagation in the composite structures, and provides a proper theoretical understanding of wave propagation in the damage tolerance-based design of piezotronics devices.

The pre -breakdown characteristics of weakly ionized media in the high non-uniform electric field using calculations with refined grid cells

The theoretical model of a pre-breakdown ions formation in a liquid dielectric and their flows, caused by high non-uniform electric field is represented. The 3D system of the macroscopic pre-breakdown electrohydrodynamic equations is written. Influence of the electric field on the molecule dissociation rate is taken into account. The system includes the Poison equation for electric field potential, the equation of ions formation and the Navier-Stokes equations with electric force. In addition, results of modelling using refined mesh are presented.

Finite element simulation of disk‐shaped HfB2 ceramics during spark plasma sintering process

AbstractSpark plasma sintering is an interesting manufacturing method, which benefits electric current heating. In the present study, the sintering process of hafnium diboride (HfB2) ceramics was simulated using the finite element method. The governing equations of electric potential, current conservation, and energy conservation were solved using COMSOL Multiphysics software. The electric potential of the whole system versus time and current distribution at different time steps were obtained. The heat generation was calculated using the Joule heating effect, and the needed temperature for sintering was obtained. The required temperature for process progression was 2100°C that achieved after 1380 s. Moreover, the maximum electric current was 5942.863 A, which remained constant until the end of the operation. Since HfB2 is an electrical conductor, the current easily passed through the sample with an approximately uniform distribution. To provide a better understanding of the sintering mechanisms, an electrically resistive material (Al2O3) was also simulated. The results showed a more uniform current distribution in HfB2 compared to Al2O3. It was observed that the needed temperature for sintering of alumina was obtained only by heat conduction from graphite die toward the sample, whereas HfB2 benefited both mechanisms of heat conduction and Joule heating inside the sample.

A study on the impacts that altered geometries containing a Ferro-Nano fluid affected by a magnetic field could have on a device natural convection

An investigative approach on study of a laminar natural heat convection on different geometric annuluses under impact of a constant magnetic field, using Ferro-Nano has been carried out. The three different cases of vertical cylindrical, cubic, and twisted cubic geometries, each are presented by a 3D model. These geometries are affected by a magnetic field which affects the variables such as heat transfer, natural heat convection, and entropy generation will be investigated. The chosen approach for this article is a numerical calculation, including Energy equations, Electrical potential equation, Hartmann number, Rayleigh, and Nusselt number. Results gathered by current study indicate that magnetic field is the primary reason of natural convection reduction phenomenon on each of the presented geometries. In the end it was illustrated that the average heat transfer has been reduced to 0.6% when the magnetic field is applied. Moreover, on the subject of the geometry amends, the twisted cubic had the highest amount or reduction of all the considered cases.

Numerical Calculations on Liquid-Metal Magnetohydrodynamic Flow in Magnetic Field Inlet Region and Outlet Region

Numerical calculations have been performed on liquid-metal magnetohydrodynamic flows through a rectangular channel in the magnetic field inlet region and magnetic field outlet region. The conservation equations of fluid mass and fluid momentum and the Poisson equation for electrical potential have been solved numerically. The numerical calculations have been carried out for Hartmann (Ha) numbers up to the order of 10 000 and a rectangular channel with electrically conducting channel walls. Attention is focused on pressure drops along the flow channel in the magnetic field inlet region and outlet region. The loss coefficients ζ can be represented by for both the magnetic field inlet region and outlet region, where k is a coefficient, and Ha, Re, and β are the Hartmann number, the Reynolds number, and the channel aspect ratio, respectively. The coefficient k depends on the gradient of applied magnetic field in the magnetic field inlet region and outlet region. However, the coefficient k does not change with the Ha number, the Re number, the wall conductivity number, and the aspect ratio very much.

Electro-thermo-convection in a differentially heated square cavity under arbitrary unipolar injection of ions

A numerical investigation of electro-thermo-convection in a 2-D differentially heated square cavity filled with a dielectric liquid is presented. Fully coupled governing equations of electric potential, charge transport, Navier–Stokes equations, and the energy equation are implemented in the finite-volume framework of OpenFOAM®. For this kind of electro-thermo-convection, previous studies mainly focused on the strong injection regime. This study extends the analysis to arbitrary injection strengths in weak, medium and strong regimes. Moreover, the flow configuration considered in this study investigates the simultaneous action of buoyancy and Coulomb forces acting in orthogonal direction to each other. For strong and medium injection, the flow transforms from a steady two-cell flow to a periodic two-cell flow and finally evolves into a chaotic flow with multiple cells, as the value of T is increased. In the weak injection regime, chaotic flow with multiple flow cells is observed right from the onset of instability. Heat transfer rates and the maximum velocity are directly proportional to the electric Rayleigh number T. Present study gives an insight into different flow structures and the related heat transfer phenomenon at arbitrary injection strengths.

Heat stream in electroosmotic bio-fluid flow in straight microchannel via peristalsis

Current study investigates the heat characteristics of electroosmotic phenomenon in bio fluid through a peristaltic wavy microchannel. To analyze the heat transfer, viscous dissipation is well-thought-out. The transport model includes of continuity, momentum, heat and electric potential equations with suitable boundary conditions. Debye Hückel linearization approximation is considered. Analytical solution of subsequent non-linear problem is presented to evaluate the variation in velocity, pressure gradient, pressure rise and temperature through the perturbation technique. We validate the approximate analytical solutions by comparing those with results from numerical analysis. Shear stresses and Nusselt number are calculated indeed. Presented graphical results explore the impact of the factors affecting the physical problem. Eyring fluid parameter expressively affects all the transport characteristics.Pressure gradient is more for Eyring fluid parameter and Helmholtz-Smoluchowski velocity and declines for averaged time flow rate and electroosmotic parameter. Shear stress strongly depends on Eyring-Powell fluid parameter A, electroosmotic parameter and Helmholtz-Smoluchowski velocity. The results reported provides a good insight into biological systems in biomedical engineering.

Electric-field-based Poisson-Boltzmann theory: Treating mobile charge as polarization.

Mobile charge in an electrolytic solution can in principle be represented as the divergence of ionic polarization. After adding explicit solvent polarization a finite volume of an electrolyte can then be treated as a composite nonuniform dielectric body. Writing the electrostatic interactions as an integral over electric-field energy density we show that the Poisson-Boltzmann functional in this formulation is convex and can be used to derive the equilibrium equations for electric potential and ion concentration by a variational procedure developed by Ericksen for dielectric continua [J.L. Ericksen, Arch. Rational Mech. Anal. 183, 299 (2007)AVRMAW0003-952710.1007/s00205-006-0042-4]. The Maxwell field equations are enforced by extending the set of variational parameters by a vector potential representing the dielectric displacement which is fully transverse in a dielectric system without embedded external charge. The electric-field energy density in this representation is a function of the vector potential and the sum of ionic and solvent polarization making the mutual screening explicit. Transverse polarization is accounted for by construction, lifting the restriction to longitudinal polarization inherent in the electrostatic potential based formulation of Poisson-Boltzmann mean field theory.

Determination of the band structure diagram of semiconductor heterostructures applied in photovoltaics

Recently it has been found that the heterostructures of n-ZnO/p-Si are promising photovoltaic alternatives to silicon homojunctions. It is well known that the energy band diagram of a heterostructure is crucial for the understanding of its operation. This paper analyzes the ZnO/p-Si heterostructure band by using free AMPS-1D computer program simulations. The obtained numerical results are compared with theoretical calculations based on the depletion region approximation model and the Poisson’s equation for electric potential. The results of the simulation are also compared with the experimental C-V characteristics of the test n-ZnO/p-Si heterostructure. The simulated C-V characteristics is qualitatively consistent with the experimental C-V curve, which confirms the correctness of the determined band diagram of the n-ZnO/p-Si heterostructure.

Numerical analysis of electrohydrodynamic flows of a dielectric liquid in a wire–plate configuration with cross Poiseuille flow

A numerical investigation of electrohydrodynamic flows of a dielectric liquid in a single wire–plate configuration with a cross Poiseuille flow has been presented. Unipolar charge injection takes place from a metallic wire electrode immersed in a dielectric liquid at the center of the channel. Although this configuration is frequently studied with gas as a working fluid in electrostatic precipitators, the flow of a dielectric liquid remains unexplored. Two-way coupled governing equations that include the Navier–Stokes equations for fluid flow, the charge transport equation, and the Poisson equation for electric potential are solved using a finite-volume method. A systematic analysis of flow characteristics with respect to the hydrodynamic Reynolds number (Re) and electric Reynolds number (RE) has been carried out. The transition process with four distinct flow patterns and two different flow mechanisms are discussed in detail. A comprehensive map of flow patterns with respect to various dimensionless parameters has been proposed. The results show that a higher Re can reduce the effect of electric field, and vice versa. The main flow pattern is found to be a strong function of the dimensionless external velocity.