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In this innovative study, a unique approach was engaged to simulate the flow characteristics of nanofluid inside a tank featuring a surface subjected to uniform flux. The testing fluid for this investigation was fabricated by incorporating alumina powders with varying shapes into water. The derivation of the final equations involved the application of Darcy’s law and the formulation of the stream function. The container experienced the combined efficacy of both the Lorentz force and gravity forces. The incorporation of additives resulted in a significant enhancement of the Nusselt number (Nu), demonstrating an increase of 19.8% and 40.28%, contingent on the magnitude of the Hartmann number (Ha). Moreover, an elevation in the shape factor led to a notable rise in Nu by 14%. Remarkably, as the Ha increased, there was a substantial reduction in the cooling rate by 51.33%. Furthermore, in the absence of the Ha, an escalation in the Rayleigh number (Ra) caused Nu to surge by 65.8%. This study holds paramount importance as it introduces a novel technique for simulating nanofluid flow with a sinusoidal surface, providing valuable insights into the complex interplay of forces within the container. The utilization of varying shapes of alumina powders adds a layer of sophistication to the experimentation, making this investigation a noteworthy contribution to the existing body of knowledge. The findings not only enhance our understanding of heat transfer dynamics but also offer practical implications for applications involving nanofluids in containers with nonuniform surfaces subjected to heat flux.

Two lattice Boltzmann method (LBM) formulations are possible to account for the effect of the magnetic field on the velocity field in magnetohydrodynamic (MHD) flows. In the body-force formulation (BFF), the magnetic field effects manifest as an external acceleration. In the extended equilibrium formulation (EEF), the effect appears through a modified equilibrium distribution function. Further, for the velocity field itself, the available choices are the single-relaxation time (SRT) and multi-relaxation time (MRT) models. Thus, for MHD-LBM, there are four possible permutations: SRT-BFF, SRT-EEF, MRT-BFF and MRT-EEF. Numerical implementation of the first three have already been presented in the literature. In this work, we, (i) develop the numerical implementation of MRT-EEF and (ii) perform an assessment of the four possible approaches. Our results indicate that the MRT-EEF is the most robust and accurate of the MHD-LBM computational schemes examined.

The field equations in a three-dimensional commutative space based on a set of commutation relations are derived. In this space, the commutation relation of the position and kinematic momentum of a particle is generalized to include a metric tensor field in addition to a vector field. The introduction of a metric tensor is a generalization of the commutation relation for Feynman’s proof of the Maxwell equations. In this paper, as the equations of motion and the field equations are classical, the Poisson bracket and not the commutation relation is used in the calculations.

As the commutative space is defined by the Poisson bracket, the equations of motion for the particle and the field equations for the metric tensor and vector are derived from the Poisson bracket in Hamiltonian mechanics. The Helmholtz conditions, which express the existence of a Lagrangian for a particle in the space, are also derived from the Poisson bracket. Then the field equations are calculated explicitly by two approaches. One is to calculate the Helmholtz conditions using the equations of motion. The other is to calculate the Jacobi identity for the kinematic momentum or velocity of the particle. In addition to the homogeneous Maxwell equations, the generalized field equations are obtained to define the generalized electric and magnetic fields of the tensor field. Just like the usual electric and magnetic fields, the generalized fields are invariant under a local gauge transformation and should play significant roles in physics.

Finally, the homogeneous Maxwell equations of the vector field are seen to exhibit similarities with the generalized field equations for the tensor field. This similarity provides a useful theoretical framework for constructing gravitoelectromagnetism, which is based on analogies between the equations for electromagnetism and relativistic gravitation. It remains to establish the usefulness of the theoretical framework with applications of the field equations.

In classical electrodynamics, the well-known Lorentz force law falls short of providing a satisfactory result for the trajectory of point-like charged particles when considering that particle’s own self-force. While there have been many historical attempts, Gralla, Harte and Wald developed a new model for classical charged particles that is free from pathologies while being consistent with Maxwell’s equations and conserves stress-energy. Expanding upon this approach, we derive a relativistically correct, modified Lorentz force law in vector form, which includes radiation reaction, and spin- and magnetic moment-dependent correction terms, suitable to be included in classical electrodynamics lectures and beam dynamics simulation tools. As by-products we obtain evolution equations for mass, spin angular momentum and the radiated power. We compare the new equations to the classical ones and use the new equations to conduct numerical simulations, showing that the results are free of any nonphysical artifacts, and which might be possible to test in future experiments at particle accelerators. The new equations foster improved insight into beam dynamics.

A mathematical theory was developed to simulate how magnetic force causes hybrid nanopowder to move through space. This numerical approach was created by combining the FEM and FVM, and triangular elements were used to create the grid. Iron oxide and MWCNT mixture were dispersed in water, and the hybrid nanomaterial’s properties were assessed using earlier empirical formulas. Gravity force can participate in the phenomena because there is a hot, wavy wall on the bottom side. Radiation flux impact was added, and different permeability spaces were used. A counterclockwise eddy is formed inside the domain as an output of gravity force and with the increase of Da, it divides into two vortexes. Additionally, increasing Da causes about a 75% increase in eddy power and increases iso-temperature distortions. The velocity decreases by between 82.5% and 90% as Lorentz terms are considered for flow equations, depending on the amount of Da. When Ra is taken into account at $\mathrm{\text{Da}}=10-2$, $\mathrm{\text{Ha}}=60$, Nu increases by about 27.75%. When $\mathrm{\text{Ha}}=0$, Nu increases by about 10.59% as permeability increases. With the use of MHD, Nu decreases by 43.26% when Ra and permeability are at their highest points.

Numerical technique for examining the transport of nanofluid within the porous container has been applied. The tank has one curved hot wall which is located in the center of the outer cylinder. The single-phase approach to deriving properties of nanofluid was applied and Darcy law has been implemented to involve the porous term in the equation. The format of the equation has been converted to stream function format to remove the pressure terms and final equations were solved via CVFEM. The written code was verified according to the previous data of the published work. Outputs showed that loading alumina causes Nu to augment by 26.71% when Ha is zero and it can be increased about 41.22% when Ha = 15. Applying a magnetic field can reduce the Nu around 37.93% when $\mathrm{\text{Ra}}=700$. With increase of strength of rotating cell with the rise of Ra, Nu increases by about 38.22%. Changing the shape of alumina can increase the Nu by about 11.73% when $\mathrm{\text{Ha}}=15$ and $\mathrm{\text{Ra}}=700$.

Both open and closed loop control algorithms have been developed for manipulating wake flows past a solid cylinder in an electrically low-conducting fluid. The intent is to avoid vortex shedding and flow separation from the body, which is achieved through the introduction of localized electromagnetic forces (Lorentz forces) in the azimuthal direction generated by an array of permanent magnets and electrodes on the surface of the circular cylinder. The array of actuators offers the advantage of making the Lorentz force time and space dependent. More specifically, one closed loop control method has been derived from the equations of motion capable of determining at all times the intensity of the Lorentz force in order to control the flow. This is accomplished first, independently of the flow (open loop algorithm) and second, based on some partial flow information measured on the surface of the solid body (closed loop algorithm).

The necessity of the vacuum interrupters (VIs) has been widely recognized on switching and controlling the fault currents in medium voltage level. An axial magnetic field (AMF) electrode has more advantages of the switching capability than other contact designs such as securing higher current value for transferring from the constriction arc to the diffuse. The heat flux and the local temperature on the electrode are increased by arc constriction, which is influenced by Lorentz force. It has undesirable influence on the characteristics of vacuum arc. In this study, we simulated the influence of Lorentz force on vacuum arc behaviors with an AMF electrode by using a commercial FEM package, ANSYS. The vacuum arc has been modeled with the sequential coupling method of two different fields, which are on the electromagnetic and thermal-flow. Arc constriction with various applied currents could be predicted with the results of temperature distribution.

An efficient hybrid modal-molecular dynamics method is developed for the vibration analysis of large scale nanostructures. Using the reduced order method, presented in this paper, linear and nonlinear vibrations of a suspended graphene nanoribbon (GNR) carrying an electric current in a harmonic magnetic field are investigated. The resonance frequencies as well as the nonlinear vibration response obtained by the present technique and direct molecular dynamic simulations are in very good agreement. Also, the obtained results illustrate the hardening behavior of nonlinear vibrations which is diminished by stretching the GNR.

In the traditional Kaluza–Klein theory, the cylinder condition and the constancy of the extra-dimensional radius (scalar field) imply that time-like geodesics on the five-dimensional bundle project to solutions of the Lorentz force equation on spacetime. This property is lost for nonconstant scalar fields, in fact there appears new terms that have been interpreted mainly as new forces or as due to a variable inertial mass and/or charge. Here we prove that the additional terms can be removed if we assume that charged particles are coupled with the same spacetime conformal structure of neutral particles but through a different conformal factor. As a consequence, in Kaluza–Klein theory the proper time of the charged particle might depend on the charge-to-mass ratio and the scalar field. Then we show that the compatibility between the equation of the projected geodesic and the classical limit of the Klein–Gordon equation fixes unambiguously the conformal factor of the coupling metric solving the conformal ambiguity problem of Kaluza–Klein theories. We confirm this result by explicitly constructing the projection of the Klein–Gordon equation and by showing that each Fourier mode, even for a variable scalar field, satisfies the Klein–Gordon equation on the base.

In this work, according to the electromagnetic field tensor in the framework of generalized uncertainty principle (GUP), we obtain the Lorentz force and Faraday’s law of induction in the presence of a minimal length. Also, the ponderomotive force and ponderomotive pressure in the presence of a measurable minimal length are found. It is shown that in the limit $\beta \to 0$, the generalized Lorentz force and ponderomotive force become the usual forms. The upper bound on the isotropic minimal length is estimated.

In this paper, we study applications of fractional Heisenberg antiferromagnetic model associated with the magnetic $n$-lines in the normal direction. Evolution equations of magnetic $n$-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their explicit solutions are examined in terms of magnetic and geometric quantities via the conformable fractional derivative method. Finally, we obtain new numerical fractional solutions for nonlinear fractional Schrödinger system with the inextensible Heisenberg antiferromagnetic flow model.

In this paper, we attempt to introduce a new approach to determine the magnetic helicity and energy dissipation of the normal electromagnetic vortex filaments in magnetohydrodynamics (MHD). We successfully compute the vector potential and magnetic helicity of Frenet–Serret vectors and associated electromagnetic fields in the existence of the Lorentz force. We further describe a set of differential equations relating the magnetic vector potential of the electromagnetic vector fields and the Godbillon–Vey helicity model. Then, we focus on the time evolution of the normal electromagnetic field lines in MHD to evaluate the evolution of the magnetic helicity and energy dissipation theoretically. As a result, we obtain analytical solutions and numerical simulations of the time evolution equations of both geometric quantities as well as the magnetic and electric field vectors.

We find the equations of the magnetic curves corresponding to the magnetic fields defined by the Killing vector fields on the 3-dimensional Heisenberg manifold.

In this paper, we introduce the dual magnetic trajectories traced by the dual Frenet vectors of any dual curves in dual space via the dual Lorentz force, and then we give the characterizations of dual magnetic curves. Moreover, we define the dual flux ruled surfaces corresponding to the Killing vectors on the dual unit sphere and identify the conditions for these surfaces to be minimal or developable. Additionally, we present certain characterizations related to these surfaces. We demonstrate the visual representations of some dual flux ruled surfaces in some examples.

It is well-known that rotating nanobeams can have different dynamic and stability responses to various types of loadings. In this research, attention is focused on studying the effects of magnetic field, surface energy and compressive axial load on the dynamic and the stability behavior of the nanobeam. For this purpose, it is assumed that the rotating nanobeam is located in the nonuniform magnetic field and subjected to compressive axial load. The nonlocal elasticity theory and the Gurtin–Murdoch model are applied to consider the effects of inter atomic forces and surface energy effect on the vibration behavior of rotating nanobeam. The vibration frequencies and critical buckling loads of the nanobeam are computed by the differential quadrature method (DQM). Then, the numerical results are testified with those results are presented in the published works and a good correlation is obtained. Finally, the effects of angular velocity, magnetic field, boundary conditions, compressive axial load, small scale parameter and surface elastic constants on the dynamic and the stability behavior of the nanobeam are studied. The results show that the magnetic field, surface energy and the angular velocity have important roles in the dynamic and stability analysis of the nanobeams.

Bending of bidirectional functionally graded nanobeams under mechanical loads and magnetic force was investigated. The nanobeam is assumed to be resting on the Winkler–Pasternak foundation. Eringen’s nonlocal elasticity theory and Timoshenko beam model are utilized to describe the mechanical behavior of the nanobeam. Material properties of the functionally graded beam are assumed to vary in the thickness and length of the nanobeam. Hamilton’s principle is employed to derive the governing equation and related boundary conditions. These equations are solved using the generalized differential quadrature method. The obtained results are compared with the results presented in other studies, to ensure the validity and versatility of this method. This comparison shows a good agreement between the results. Results are presented and discussed for different values of functionally graded materials indices, different aspect ratios, and different boundary conditions. The effect of the magnetic field and elastic foundation on buckling load has also been studied. The difference in nanobeam behavior for different values of the size-effect parameter is clearly shown.

In this work, we derive the $J$-, $M$- and $L$-integrals of body charges and point charges in electrostatics, and the $J$-, $M$- and $L$-integrals of body forces and point forces in elasticity and we investigate their physical interpretation. Electrostatics is considered as field theory of an electrostatic scalar potential $\phi$ (scalar field theory) and elasticity as field theory of a displacement vector $u$ (vector field theory). One of the basic quantities appearing in the $J$-, $M$- and $L$-integrals is the electrostatic Maxwell–Minkowski stress tensor in electrostatics and the Eshelby stress tensor in elasticity. Among others, it is shown that the $J$-integral of body charges in electrostatics represents the electrostatic part of the Lorentz force, and the $J$-integral of body forces in elasticity represents the Cherepanov force. The $M$-integral between two-point sources (charges or forces) equals half the electrostatic interaction energy in electrostatics and half the elastic interaction energy in elasticity between these two-point sources. The $L$-integral represents the configurational vector moment or torque between two body or point sources (charges or forces). Interesting mathematical and physical features are revealed through the connection of the $J$-, $M$- and $L$-integrals with their corresponding infinitesimal generators in both theories. Several important outcomes arise from the comparison between the examined concepts in electrostatics and elasticity. Differences and similarities, that provide a deeper insight into the $J$-, $M$- and $L$-integrals and the related quantities to them, are pointed out and discussed. The presented results show that the $J$-, $M$- and $L$-integrals are fundamental concepts which can be applied in any field theory.

The following article has been written primarily by the high school students who make up the team “Cryptic Ontics”, one of the two winning teams in the 2018 edition of CERN’s Beamline for Schools (BL4S) competition, and is based on the set of experiments the students endeavoured to conduct over the course of a two-week period at CERN.

Reconstructing influential physical theories from scratch often helps in uncovering hitherto unknown logical connections and eliciting instructive empirical checkpoints within said theory. With this in mind, in the following article, a top-down reconstruction (beginning with the experimental observations and ending at the theoretical framework) of the Lorentz force equation is performed, and potentially interesting questions which come up are explored. In its most common form, the equation is written out as: $F=qE+q(v\times B)$. Only the term that includes the magnetic field $q(v\times B)$ will be dealt with for this article. The independent parameters we use are (i) the momenta of the particles, (ii) the charge (rather, the types) of particles, either positive or negative, and (iii) the current passing through the dipole generating the electromagnetic field. We then measure the angle by which particles get deflected while varying these three parameters and derive an empirical relationship between them.

We describe a laboratory exercise for an introductory calculus-based electricity and magnetism course in which students construct and study the performance of a “rolling railgun” formed by two small coin magnets connected by a ferromagnetic axle which carries current from one rail to the other. This exercise can be scaled up from a simple, mostly qualitative activity to a more comprehensive comparison between theory and experiment that will challenge students’ calculus skills. The required components are small and inexpensive enough to mail to students who are taking the course remotely. We report on our initial success in incorporating this lab into our curriculum at Towson University.