Biot-Savart Research Articles

Research on the mechanism of the effect of vortex on the hydraulic loss of pump as turbine units based on entropy production theory

Pump as turbine (PAT) is a common method of energy recovery, however, vortices are a negative phenom for these units. The objective of this research is to study the effect of vortex motion on the hydraulic loss of pump as turbine, and establishing the correlation mechanism between vortex intensity and turbulence loss. The research method adopts theoretical analysis and model test and numerical simulation. Based on the entropy production theory, the hydraulic loss and the turbulent dissipation in boundary layer induced by vortex motion are studied, revealing the influence of vortices on the energy loss. Results show that the vortex motion can be decomposed into a synchronous component v˜sy and a rotational component v˜ro, among them, the rotational component v˜ro meets to Biot-Savart Law. The turbulent dissipation rate in the boundary layer is closely to the vortex motion, which can characterize the boundary turbulence height. Turbulent flow induced by vortex can propagate in the flow channel of the unit, causing lot of additional hydraulic loss. In the end, a mathematical model between entropy production (Sk) induced by vortex and vortex strength (Гk) was established, indicating that Sk changes with Гk in the form of a quadratic function.

Implementation of a non-axisymmetric magnetic configuration in SOLEDGE3X to simulate 3D toroidal magnetic ripple effects: Application to WEST

The fluid-drift code SOLEDGE3X, developed by CEA/IRFM in collaboration with Aix-Marseille University, is a powerful tool for simulating transport and turbulence in tokamak edge plasmas with axisymmetric magnetic configurations. In tokamaks such as WEST, the pronounced toroidal magnetic ripple significantly affects plasma confinement and power exhaust, modulating both the poloidal and toroidal components of the equilibrium field. Using a discrete Biot–Savart law, the ripple field is calculated as a magnetic perturbation on the SOLEDGE3X mesh. The transport model and parallel gradient solvers have been enhanced to incorporate the new radial magnetic field component. Preliminary simulations of a WEST scenario reveal a heat deposition pattern in the divertor region consistent with observations from infrared camera experiments.

Beamline analysis for a laser-driven proton therapy accelerator using superconducting bends

Peking University is implementing new superconducting magnets in a laser-driven proton accelerator, resulting in a lighter and more compact proton therapy facility. To efficiently transport protons with a wide energy spread, we propose a double-bend achromat design that rotates particles by 90 degrees and suppresses dispersion. Two focusing options are considered: integrating quadrupole magnets within the dipole magnets to create a hybrid field, or separating the quadrupole for independent focusing and bending. We first delve into the related lattice design, followed by an in-depth analysis of coil selection for each option. The magnetic field of the coil is indeed calculated using a combination of the Biot–Savart law and the Multi-Level Fast Multipole Method (MLFMM). We assess field quality and feasibility using TraceWin and zgoubi simulations in fieldmap, accounting for dipole-induced dispersion and quadrupole-induced chromatic effect. Through the analysis of superconducting magnets and beam dynamics, and by considering the transmission requirements of laser-plasma-accelerated protons, we have indeed developed a beam transport scheme. This scheme is anticipated to integrate laser-accelerated protons into a compact proton therapy facility, and it will eventually be implemented.

Quantum turbulence in superfluid helium: Decay and energy spectrum

The paper presents a comprehensive numerical study of the free decay of vortex tangle in superfluid helium. The initial vortex tangle represents one of the stationary configurations of loops in a counterflow with a laminar normal fluid component. The calculations are carried out in the framework of the vortex line method using the full Biot–Savart law over a wide range of temperatures. The aim of the study is to identify the role of various factors introduced into the numerical procedures (removal of small loops and segments during reconnections, the addition of and exclusion of vortex points on loops) and to determine the evolution of energy spectrum during the decay of quantum turbulence. A statistical approach is used to calculate the kinetic energy distribution on length scales. The calculations are carried out using periodic boundary conditions in a cube. The results show that, in agreement with the Feynman–Vinen theory, initially the rates of reduction in vortex line density at different temperatures are the same. However, when the vortex structure becomes rarefied, the influence of the mutual friction force becomes apparent, in agreement with Schwarz's theory. Statistical method for determining the energy spectrum is used. The Kolmogorov spectrum is not observed during decays at any temperature.

An experimental and numerical study on the behavior of finite-length column vortex

This paper explores experimental and numerical investigation of the spatiotemporal dynamics of a finite-length vortex column in three dimensions using particle image velocimetry and large eddy simulation. The research examines the combined impact of bending, buckling, and core-splitting on a finite-length vortex column. More precisely, the work centers on the fundamental motions, evolutions in flow along the axis, how the shape of the vortex core changes over time, the instability caused by the long waves on the vortex column, and the division of the vortex core. A novel prototype is developed that utilizes piston-driven inflow and produces a vortex column through the principles of flow separation and Biot–Savart induction, which is studied for predicting an ex situ cyclonic line vortex. The present study discusses the complete three-dimensional overview of the same columnar vortex. The meridional swirl and the axial transport of vorticity deform the shape of the core cross sections in different lengths of the column. The jump in the axial velocity at the core boundary allows for the occurrence of bending instabilities with a left-handed helical structure. These instabilities have a long wavelength, similar to the length of the vortex column, and belong to the m = +1 mode. The vortex column experiences a non-uniform curvature and torsion caused by the intricate shape of the laboratory model and varying flow speeds at different heights of the vortex column. The results offer valuable insights into the role of inertia, Coriolis, and viscous forces on the dynamics of the vortex column.

A relativistic approach to teaching electrodynamics: Deriving Maxwell's equations from first principles

This paper presents a novel methodology for teaching classical electrodynamics based on special relativity theory. Traditional approaches often rely heavily on empirical laws, potentially obscuring the fundamental unity of electromagnetic phenomena. We demonstrate how the core principles of electrodynamics, including Maxwell's equations, can be derived from just two fundamental postulates: the principle of relativity and Coulomb's law. By consistently applying relativistic reasoning, we show how key concepts such as the Lorentz force, Biot-Savart law, and electromagnetic induction emerge naturally. This approach provides a logically coherent framework for understanding electrodynamics, emphasizes its inherently relativistic nature, and resolves apparent paradoxes that arise in non-relativistic treatments. We discuss the pedagogical advantages of this method, including its potential to foster deeper conceptual understanding and develop students' theoretical reasoning skills. The paper also addresses practical aspects of implementing this approach in higher education physics curricula and its implications for physics education research.

A generalized Biot–Savart law and its application to the active scalar equations

A generalized Biot–Savart law and its application to the active scalar equations

Effect of salinity-clay variation on the transient magnetic field in the Quaternary aquifer, theoretically and practically

This study focuses on understanding the physical foundations of the transient electromagnetic method, such as the transition from a simple basic principle to a more complex principle, starting from the Biot-Savart law until the current concept. It calculates the steady and transient magnetic fields around any electrical wire system. Accordingly, this law will be used to determine the magnetic field around the square loop configuration, which matches the magnetic field arising from the circular loop configuration. The square loop equation was derived and extended to include changes in the half-space, which allows us to understand how the electromagnetic waves travel in the subsurface and the farthest point, that can record the TEM response from the TEM loop, according to the loop size and loop current. Accordingly, the deduced equations were applied to predict the transient magnetic field curves of the Quaternary aquifer in three different water zones along the Nile delta, depending on the values of resistivity and chargeability caused by the variations in salinity and clay contents. The calculated transient magnetic curves were then compared with other measured field curves to confirm the validity of the derived equations. Therefore, these curves are expected to be used as guide curves for the transient magnetic field response in this aquifer. Also, these curves are important in estimating the hydro-geophysical characteristics of the main groundwater aquifer in the selected area and in other areas with the same geologic and hydrogeologic settings. Also, a common model is presented for any delta environment in the world in measured and calculated data.

Dependence of coronal mass ejections on the morphology and toroidal flux of their source magnetic flux ropes

Context. Coronal mass ejections (CMEs) stand as intense eruptions of magnetized plasma from the Sun, and they play a pivotal role in driving significant changes of the heliospheric environment. Deducing the properties of CMEs from their progenitors in solar source regions is crucial for space weather forecasting. Aims. The primary objective of this paper is to establish a connection between CMEs and their progenitors in solar source regions, enabling us to infer the magnetic structures of CMEs before their full development. Methods. We created a dataset comprising a magnetic flux rope series with varying projection shapes (S-, Z-, and toroid-shaped), sizes, and toroidal fluxes using the Regularized Biot-Savart Laws (RBSL). These flux ropes were inserted into solar quiet regions with the aim of imitating the eruptions of quiescent filaments. Thereafter, we simulated the propagation of these flux ropes from the solar surface to a distance of 25 R⊙ with our global coronal magnetohydrodynamic (MHD) model COCONUT. Results. Our parametric survey revealed significant impacts of source flux ropes on the consequent CMEs. Regarding the flux-rope morphology, we find that the projection shape (e.g., sigmoid or torus) can influence the magnetic structures of CMEs at 20 R⊙, albeit with minimal impacts on the propagation speed. However, these impacts diminish as source flux ropes become fat. In terms of toroidal flux, our simulation results demonstrate a pronounced correlation with the propagation speed of CMEs as well as the successfulness in erupting. Conclusions. This work builds the bridge between the CMEs in the outer corona and their progenitors in solar source regions. Our parametric survey suggests that the projection shape, cross-section radius, and toroidal flux of source flux ropes are crucial parameters in predicting magnetic structures and the propagation speed of CMEs, providing valuable insights for space weather prediction. On the one hand, the conclusion drawn here could be instructive in identifying the high-risk eruptions with the potential to induce stronger geomagnetic effects (Bz and propagation speed). On the other hand, our findings hold practical significance for refining the parameter settings of launched CMEs at 21.5 R⊙ in heliospheric simulations, such as with EUHFORIA, based on observations for their progenitors in solar source regions.

Iterative Numerical Approximation Technique for 3D Eddy Current Models in Harmonic Regime Based on the Electromagnetic T-Formulation and the Finite Element Method

In this paper, an iterative numerical approximation technique is used for analyzing the distribution of eddy currents density in conductive materials plates by using the electromagnetic T-formulation and the Biot-Savart law, solved by the finite element method. The proposed approach allows for the meshing of the different parts of the studied system separately, including the sensor and the conductive plate, without the need for an air region. Firstly, this approach reduces the number of unknown variables by avoiding the air region of the system's mesh. Secondly, it simplifies the consideration of the sensor's motion without the need to remesh the system. For this purpose, a calculation code has been developed for solving an electromagnetic three-dimensional non-destructive testing model. This latter permits the resolution of JSEAM # 6 Benchmark problem to validate the proposed method. The impedance variation due to the presence of a defect are evaluated. The obtained results are compared with the experimental ones found in the literature. These results reveal a good agreement, which proves the validity of the proposed method.

Quantitative Investigation of Parallel Interactions Between Charged Particles

On the premise that the charged particle is a normal geometry model, an expression of the migration current, displacement current and magnetic induction intensity generated by the charged particle motion is deduced according to the microscopic definition of current intensity, the total current law and the Biot-Savart law. Further calculate the electric field force between two charged particles in vacuum, give the velocity constraint relationship and velocity value criterion of the electric and magnetic field forces, and compare the consistency with the correlation results obtained considering the relativistic effect. It is pointed out that the magnetic field force is comparable to the electric field force only when the charged particle moves near the speed of light.

Properties of the Biot–Savart Operator Acting on Surface Currents

Properties of the Biot–Savart Operator Acting on Surface Currents

Modeling of a six-degree-of-freedom magnetic levitation platform actuated by noncontact Lorentz forces

In this study, the modeling of a six-degree-of-freedom (DOF) magnetic levitation platform actuated by noncontact Lorentz forces was analyzed. First, an analytic model of the actuator forces was studied, and the magnetic flux density in the working gap of the actuators was established using the Image Method, Ampere molecular hypothesis, and Biot–Savart law. The dynamics model of the floater actuated by the actuators was then built using the Newton–Euler method. Moreover, the double-pendulum chaotic nonlinear system of the hoisted floater was simplified as a single-DOF system, in which the interaction effects between two DOFs were considered. Furthermore, modeling of the platform control system was performed, which included an acceleration measuring unit with six uniaxial accelerometers, a position measuring unit with three two-dimensional PSDs, and a control unit that decouples the six-DOF control. Finally, an experimental setup was built to perform ground tests on the platform. The results verified the platform modeling; translational and rotational positioning accuracies were approximately 5 μm and 40 μrad at the six DOFs, respectively, and the vibrational suppression efficiencies were greater than 90% for low frequency disturbances. Moreover, Kalman estimators and disturbance observers were introduced into the controller to estimate the movement of the floater and observe the direct disturbance. Simulation testing demonstrated a significant improvement in the vibration-suppression performance of the platform.

Application Of Gauss's Law and Biot-Savart Law to Charged Bodies with Inhomogeneous Charges

Application Of Gauss's Law and Biot-Savart Law to Charged Bodies with Inhomogeneous Charges

A Magneto-Electric Device for Fluid Pipelines with Vibration Damping and Vibration Energy Harvesting.

This study introduces an innovative energy harvesting system designed for industrial applications such as fluid pipelines, air conditioning ducts, sewer systems, and subsea oil pipelines. The system integrates magneto-electric flow coupling and utilizes a dynamic vibration absorber (DVA) to mitigate the vibrations induced by fluid flow while simultaneously harvesting energy through magnetic dipole-dipole interactions in a vibration energy harvester (VEH). The theoretical models, based on Hamilton's Principle and the Biot-Savart Law, were validated through comprehensive experiments. The results indicate the superior performance of the small-magnet system over the large-magnet system in both damping and power generation. The study analyzed the frequency response and energy conversion efficiency across different parameters, including the DVA mass, spring constant, and placement location. The experimental findings demonstrated significant vibration reduction and increased voltage output, validating the theoretical model. This research offers new avenues for energy harvesting systems in pipeline infrastructures, potentially enhancing energy efficiency and structural integrity.

An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces

We revisit Yudovich’s well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set Ω⊂R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R}^2$$\\end{document} or on the torus Ω=T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega =\\mathbb {T}^2$$\\end{document}. We construct global-in-time weak solutions with vorticity in L1∩Lulp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1\\cap L^p_{ul}$$\\end{document} and in L1∩YulΘ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1\\cap Y^\\Theta _{ul}$$\\end{document}, where Lulp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p_{ul}$$\\end{document} and YulΘ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y^\\Theta _{ul}$$\\end{document} are suitable uniformly-localized versions of the Lebesgue space Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} and of the Yudovich space YΘ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y^\\Theta $$\\end{document} respectively, with no condition at infinity for the growth function Θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta $$\\end{document}. We also provide an explicit modulus of continuity for the velocity depending on the growth function Θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta $$\\end{document}. We prove uniqueness of weak solutions in L1∩YulΘ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1\\cap Y^\\Theta _{ul}$$\\end{document} under the assumption that Θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta $$\\end{document} grows moderately at infinity. In contrast to Yudovich’s energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón–Zygmund theory or Littlewood–Paley decomposition, and actually applies not only to the Biot–Savart law, but also to more general operators whose kernels obey some natural structural assumptions.

Filament eruption by multiple reconnections

Context. Filament eruption is a common phenomenon in solar activity, but the triggering mechanism is not well understood. Aims. We focus our study on a filament eruption located in a complex nest of three active regions close to a coronal hole. Methods. The filament eruption is observed at multiple wavelengths: by the Global Oscillation Network Group (GONG), the Solar Terrestrial Relations Observatory (STEREO), the Solar Upper Transition Region Imager (SUTRI), and the Atmospheric Imaging Assembly (AIA) and Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamic Observatory (SDO). Thanks to high-temporal-resolution observations, we were able to analyze the evolution of the fine structure of the filament in detail. The filament changes direction during the eruption, which is followed by a halo coronal mass ejection detected by the Large Angle Spectrometric Coronagraph (LASCO) on board the Solar and Heliospheric Observatory (SOHO). A Type III radio burst was also registered at the time of the eruption. To investigate the process of the eruption, we analyzed the magnetic topology of the filament region adopting a nonlinear force-free-field (NLFFF) extrapolation method and the polytropic global magnetohydrodynamic (MHD) modeling. We modeled the filament by embedding a twisted flux rope with the regularized Biot-Savart Laws (RBSL) method in the ambient magnetic field. Results. The extrapolation results show that magnetic reconnection occurs in a fan-spine configuration resulting in a circular flare ribbon. The global modeling of the corona demonstrates that there was an interaction between the filament and open field lines, causing a deflection of the filament in the direction of the observed CME eruption and dimming area. Conclusions. The modeling supports the following scenario: magnetic reconnection not only occurs with the filament itself (the flux rope) but also with the background magnetic field lines and open field lines of the coronal hole located to the east of the flux rope. This multiwavelength analysis indicates that the filament undergoes multiple magnetic reconnections on small and large scales with a drifting of the flux rope.

Several Aspects of Human Exposure to Low Frequency Fields; Incident and Internal Field Dosimetry Procedures and Related Legal issues

The paper deals with simple and efficient dosimetry procedures for human exposure to low frequency (LF) electric and magnetic fields at single and multiple frequencies, respectively. The electromagnetic interference (EMI) sources of interest are overhead power lines and transmission substations. Electric and magnetic fields due to EMI sources are either calculated or measured. Theoretical dosimetry is based on scalar potential integral equation (SPIE) for the case of the electric field assessment while the magnetic fields are computed by means of the Biot-Savart law. The internal fields in the human body are determined by using the canonical body models (disk model and cylindrical model). The obtained results are compared to exposure limits proposed by national and international legislation, respectively (National Gazette No 146/2014, 31/2019, 59/2016) and ICNIRP International Commission on Nonionizing Radiation protection). Finally, legal issues pertaining to human exposure to LF fields are addressed.

The architectural application of shells whose boundaries subtend a constant solid angle

Surface geometry plays a central role in the design of bridges, vaults and shells, using various techniques for generating a geometry which aims to balance structural, spatial, aesthetic and construction requirements.In this paper we propose the use of surfaces defined such that given closed curves subtend a constant solid angle at all points on the surface and form its boundary. Constant solid angle surfaces enable one to control the boundary slope and hence achieve an approximately constant span-to-height ratio as the span varies, making them structurally viable for shell structures. In addition, when the entire surface boundary is in the same plane, the slope of the surface around the boundary is constant and thus follows a principal curvature direction. Such surfaces are suitable for surface grids where planar quadrilaterals meet the surface boundaries. They can also be used as the Airy stress function in the form finding of shells having forces concentrated at the corners.Our technique employs the Gauss-Bonnet theorem to calculate the solid angle of a point in space and Newton's method to move the point onto the constant solid angle surface. We use the Biot-Savart law to find the gradient of the solid angle. The technique can be applied in parallel to each surface point without an initial mesh, opening up for future studies and other applications when boundary curves are known but the initial topology is unknown.We show the geometrical properties, possibilities and limitations of surfaces of constant solid angle using examples in three dimensions.

Slender vortex filaments in the Boussinesq approximation

A model for the motion of slender vortex filaments is extended to include the effect of gravity. The model, initially introduced by Callegari and Ting [“Motion of a curved vortex filament with decaying vortical core and axial velocity,” SIAM J. Appl. Math. 35, 148–175 (1978)], is based on a matched asymptotic expansion in which the outer solution, given by the Biot–Savart law, is matched with the inner solution derived from the Navier–Stokes equations. Building on recent work by Harikrishnan et al. [“On the motion of hairpin filaments in the atmospheric boundary layer,” Phys. Fluids 35, 076603 (2023)], the Boussinesq approximation is applied such that the density variations only enter in the gravity term. However, unlike Harikrishnan et al. [“On the motion of hairpin filaments in the atmospheric boundary layer,” Phys. Fluids 35, 076603 (2023)], the density variation enters at a lower order in the asymptotic expansion and, thus, has a more significant impact on the self-induced velocity of the vortex filament. In this regime, which corresponds to the regime studied by Chang and Smith [“The motion of a buoyant vortex filament,” J. Fluid Mech. 857, R1 (2018)], the effect of gravity is given by an alteration of the core constant, which couples the motion of the filament to the motion within the vortical core, in addition to a change in the compatibility conditions (evolution equations), which determine the leading order azimuthal and tangential velocity fields in the vortex core. The results are used to explain certain properties of buoyant vortex rings, as well as qualitatively explore the impact of gravity on tornado-type atmospheric vortices.