Equations Of Motion For Particles Research Articles

Calculation of electron-optical characteristics of a quadrupole-cylindrical field

The article is devoted to the calculation and analysis of electron-optical characteristics of a electrostatic quadrupole-cylindrical field. The object of the research is the development of an energy analyzer based on a quadrupole-cylindrical field. The structure of electrostatic quadrupole-cylindrical fields is obtained on the basis of the superposition of a basic cylindrical field and axially symmetrical cylindrical quadrupoles. The calculation of electron-optical characteristics of a quadrupole-cylindrical field was fulfilled on the basis of the analytical method for calculating of trajectories of charged particles. The problem of integrating of differential equations of charged particle motion and the analytical description of the trajectory equation in a quadrupole-cylindrical field has been solved. A projection of charged particle trajectory from the source to its image has been calculated. The main aberration coefficients that determine the conditions of secondorder angular focusing are calculated. The most optimal scheme of a quadrupole-cylindrical mirror for energy analysis has been found.

PARTICLE MOTION OVER THE EDGE OF AN INCLINED PLANE THAT PERFORMS AXIAL MOVEMENT IN A VERTICAL LIMITING CYLINDER

Differential equations of a relative material particle motion over the edge of an inclined flat ellipse that rotates around the axis of a vertical limiting cylinder have been deduced. The position of a plane relative to the axis of the rotation is set by the angle ranging from zero to ninety degrees in its value. If the angle is equal to zero, the plane is perpendicular to the axis of rotation and if the angle is equal to ninety degrees, it passes through the axis of rotation. The equations have been solved using numerical methods. Analytical solution has been found for certain angles. The aim of the research is to investigate the transportability of a technological material in a vertical direction by a cascade operating element that rotates in a cylindrical cover. The working part of the operating element is an inclined rigid plane, which is limited by an ellipse — the line of its contact with a cover. The objective of the research is to analytically describe the movement of a single particle of the technological material on two surfaces, namely, an inclined plane and a vertical cover. The research methodology is based on the methods of differential geometry and the theory of surfaces, theoretical mechanics and numerical methods of solving differential equations. The paper presents a first developed analytical description of the relative particle motion in an ellipse — a contact line of an inclined plane and a limiting vertical cylinder, in which the inclined plane rotates. The kinematic characteristics of such motion have been determined.

Weak cosmic censorship conjecture in Kerr black holes of modified gravity

By neglecting the effects of self-force and radiation, we investigate the possibility of destroying the Kerr-MOG black hole through the point particle absorption process. Using the instability of event horizon and equation of particle motion, we get the upper and lower energy bounds allowed for a matter particle to produce the naked singularity. We find that the energy gap always exists between the upper and lower energy bounds for both extremal and near-extremal black holes, which means some tailored particles can actually lead to the violation of the weak cosmic censorship conjecture. However, when considering the effect of the adiabatic process, the result shows that the Kerr-MOG black hole gets more stable instead of a naked singularity, and thus the weak cosmic censorship conjecture can be restored at some level.

GROUP ANALYSIS OF MOTION EQUATION OF CHARGED PARTI-CLE PLACED IN CROSSED FIELDS

The transformation of motion equation of a charged particle placed in crossed longitudinal magnetic field and ra-dial electric one is considered under transition from the laboratory frame of reference to a uniformly rotating coor-dinate system. It is shown that the transformation of equation in a plane transversal to the magnetic field admits a group. Using the method of group analysis, the transformation invariant and canonical variables are found.

The Discreteness-driven Relaxation of Collisionless Gravitating Systems: Entropy Evolution and the Nyquist–Shannon Theorem

Abstract The time irreversibility and fast relaxation of collapsing N-body gravitating systems (as opposed to the time reversibility of the equations of motion for individual stars or particles) are traditionally attributed to information loss due to coarse graining in the observation. We show that this subjective element is not necessary once one takes into consideration the fundamental fact that these systems are discrete, i.e., composed of a finite number, N, of stars or particles. We show that a connection can be made between entropy estimates for discrete systems and the Nyquist–Shannon sampling criterion. Specifically, given a sample with N points in a space of d dimensions, the Nyquist–Shannon criterion constrains the size of the smallest structures defined by a function in the continuum that can be uniquely associated with the discrete sample. When applied to an N-body system, this theorem sets a lower limit to the size of phase-space structures (in the continuum) that can be resolved in the discrete data. As a consequence, the finite N system tends to a uniform distribution after a relaxation time that typically scales as . This provides an explanation for the fast achievement of a stationary state in collapsing N-body gravitating systems such as galaxies and star clusters, without the need to advocate for the subjective effect of coarse graining.

Simulation Research on Nose Cone Shedding Trajectory of Terminal Guided Projectile

The falling off of nose cone during the flight of terminal guidance projectile may cause casualties and property losses, but the nose cone trajectory has not been noticed. In this paper, the differential equations of particle motion for terminal guided projectile are established by using the methods of external ballistic and theoretical mechanics. Three ballistic coefficients corresponding to firing distance, flight time and ballistic height are obtained by solving the ballistic equation in reverse. According to the time from launching to the nose cone and the ballistic coefficient, the initial state of the nose cone is calculated, and then according to the resistance coefficient and the mechanical equations of the nose cone, the trajectory of the nose cone is obtained by simulation, which provides scientific guidance for setting the safe area accurately.

Improving the reliability, as well as energy and environmental efficiency of gas cleaning systems at TPP

Energy is the basic sector of the economy and the largest consumer of primary energy resources of any country, which is why the development of world energy is accompanied by global pressure on the environment. The issues are considered of reducing the atmospheric impact of emissions of thermal power plants, improving the reliability and working life of their units, systems, and plants as a whole. The principles are presented of development and improvement of technologies for processing industrial emissions of thermal power plants, the neutralization of which is currently relevant on a regional and global scale. Analysis is carried out of existing methods of cyclone and filtration treatment. An improved design of a cyclone filter is proposed, which allows to increase the reliability of gas turbine and steam-gas units of TPP, while ensuring the efficiency of separation of the suspended part of the flow at the gas treatment point (GTP) of TPP. Similar devices can also be used to increase the degree of cleaning atmospheric emissions released by the TPP coal dust preparation and flue gas systems at coal generation from fi ne particles of PM10 and PM2,5 classes (coal dust and ash), owing to reduction of the size of caught particles from average values for cyclones and wet scrubbers of the order of 5–10 μm to 0.5 μm. The design of the cyclone filter is improved as a result of research of cyclone filtration by methods of Computational Fluid Dynamics (CFD). A system of Reynolds-averaged equations of a single-phase Navier-Stokes flow is used for mathematical modeling of motion in the cyclone filter. To determine the efficiency of separation of the suspended part of the flow in the cyclone filter, the Rercomplex is used obtained by reducing a set comprising the Navier-Stokes equations and the equation of particle motion based on Newton's law to a dimensionless form. Numerical characteristics of the suspension sedimentation from a multiphase flow in a cyclone separator of specified dimensions are found by means of the Rercomplex. The results of bench tests of the proposed design of the cyclone filter are given.

The theoretical and numerical analysis of water inrush through filling structures

Water inrush through filling structures was divided into three typical types. The IB-SEM numerical method was established based on the coupling of Spectral/hp element method and Rigid immersed boundary method. And the governing equations of particles motion and contact in fluid were constructed. Then the direct water inrush was investigated by the IB-SEM system. The slices method was modified and applied into the slip water inrush research. The calculating equation of minimum safe thickness was deduced by the force balance condition. Finally, the whole process piping of water inrush was simulated.

Motion of a Test Particle According to the Scalar Ether Theory of Gravitation and Application to its Celestial Mechanics

Abstract The standard interpretations of special relativity (Einstein–Minkowski) and general relativity (GR) lead to a drastically changed notion of time: the eternalism or block universe theory. This has strong consequences for our thinking about time and for the development of new fundamental theories. It is therefore important to check this thoroughly. The Lorentz–Poincaré interpretation, which sees the relativistic effects as following from a “true” Lorentz contraction of all objects in their motion through the ether, uses a conservative concept of time and is in the absence of gravitation indistinguishable from the standard interpretation; but there exists currently no accepted gravitation theory for it. The scalar ether theory of gravitation is a candidate for such a theory; it is presented and discussed. The equations of motion for a test particle are derived; the case of a uniformly moving massive body is discussed and then specialized to the case of spherical symmetry. Formulas for the acceleration of test particles are given in the preferred frame of the ether and in the rest frame of the massive body that moves with velocity V with respect to the ether. When the body rests in the ether (V = 0), the acceleration is up to order c−2 identical to GR. The acceleration of a test particle for V ≠ 0 is given; this makes it possible to fit observations in celestial mechanics to ephemerides with V as a free parameter. The current status of such fits (although to ephemerides and not to observations) is presented and discussed.

Influence of gas flow oscillation on the efficiency of particulate solids treatment in a circulating fluidized bed

Apparatuses with fluidized bed are widely used in the solid fuel energetic, chemical and other industries. In particular, fluidized bed furnaces play the increasing role for thermo-chemical treatment and burning the solid fuels including bio-fuels. The use of pulsating gas flows is one of the possible ways to intensify the process of gas-particle interaction. However, the efficiency of the way is not always obvious, and new approaches are needed for preliminary estimation of its advantages. The method of mathematical modeling is used to solve the above problem. The model is based on the set of differential equations of a single particle motion in the upstream gas flow with variable with time velocity, and with variable with time particle properties due to this or that thermo-physical or chemical process of gas-particle interaction. The study of influence of the process parameters on its efficiency is done by numerical methods. The model of the effect of oscillations in the gas flow on the efficiency of particle processing in a circulating fluidized bed has been developed. The necessary time for enough particle treatment is compared for different amplitude and frequency of gas flow oscillation. The most considerable gain in time is being reached for the strong dependence of the reaction rate constant on the velocity of flow around the particle. In this case the optimum frequency of oscillations exists that give the maximum gain in time. The growth of the amplitude leads to the monotonic increase of the gain in time, or gives the optimum amplitude. Transition to the pulsating gas flow in a circulating fluidized bed reactor leads to considerable gain in time of particle treatment only in the case of strong dependence of the reaction rate constant on the velocity of flow around the particle. In this case the optimum frequency and amplitude of oscillation exist that gives the maximum gain in time.

The duality between a non-Hermitian two-state quantum system and a massless charged particle

We show that the equations for the dynamics of a non-Hermitian two-state quantum system are the same as the equations of motion for a massless charged particle in an electromagnetic field. Using simple analytical arguments to prove this unexpected duality between two very different domains in physics, we further exemplify it through a case-study of polarization of light propagating in a dichroic medium with magneto-optic activity.

A parametric cut size prediction model for a turbo air classifier

AbstractA new parametric cut size prediction model is put forward based on the simplified single particle motion rule. According to analysis of the inner flow field of a turbo air classifier and Lagrangian equation of particle motion, the particle's trajectory in the annular region is calculated and simulated using MATLAB. Since particles with different sizes have different trajectories in the same flow field, the cut size of a turbo air classifier can also be obtained. The simulated results show the cut size increases by widening the annular region and it decreases by narrowing the annular region. Talc powder classification experiments are carried out to verify this cut size prediction model. The calculated cut sizes using the proposed prediction model are in agreement with the experimental results. A new way to predict cut size is put forward, which provides a theoretical guidance to optimize operational parameters and the structure of the turbo air classifier.

Visible Light Actuated Efficient Exclusion Between Plasmonic Ag/AgCl Micromotors and Passive Beads.

Insight is provided into the collective behavior of visible-light photochemically driven plasmonic Ag/AgCl Janus particles surrounded by passive polystyrene (PS) beads. The active diffusion of single Janus particles and their clusters (small: consisting of two or three Janus particles and large: consisting of more than ten Janus particles), and their interaction with passive PS beads, are analyzed experimentally and in simulations. The diffusivity of active Janus particles, and thus the exclusive effect to passive PS beads, can be regulated by the number of single Janus particles in the cluster. On the simulation side, the Langevin equations of motion for self-propelled Janus particles and diffusing passive PS beads are numerically solved using Molecular-Dynamics simulations. The complex interactions of both subsystems, including elastic core-to-core interactions, short-range attraction, and effective repulsion due to light-induced chemical reactions are considered. This complex mixed system not only provides insight to the interactive effect between active visible light-driven self-propelled micromotors and passive beads, but also offers promise for implications in light-controlled propulsion transport and chemical sensing.

Numerical Solution of Generalized Smoluchowski Equations for Cylindrical Micelles

The relaxation kinetics of polydisperse cylindrical micelles has been numerically studied under different initial conditions corresponding to fast concentration and dilution of a surfactant solution. The kinetic description has been based on a set of difference Smoluchowski equations, which takes into account the capture and release of surfactant monomers by micelles, as well as the fusion and fission of micelles. The dependences of fusion coefficients of cylindrical aggregates on aggregation numbers have been plotted on the basis of the Burgers−Oseen equations for translational motion of spherocylindrical particles in a viscous liquid. The solution of the generalized Smoluchowski kinetic equations obtained for a nonequilibrium distribution of cylindrical micelles over aggregation numbers has been compared with the numerical solution of the Becker−Doring kinetic equations for cylindrical micelles under the same initial conditions given the mechanism of molecular aggregation via the attachment and detachment of surfactant monomers.

Motion of Stellar Particles and Photons within Spherical Liquid Stars

Objectives: The motion of test particles and photons within a spherical liquid star is studied. Equations of motion of test particles in the star’s interior and equatorial plane are derived. The general relativity interior metric tensor for a spherical liquid star and tensor analysis is used to study the motion of particles and photons within the star. Methods: The obtained geodesic equations of motion for stellar particles are explicitly space coordinate and coordinate proper time derivatives. This differs from Schwarz child’s equations of motion in a gaseous field. Findings: Along the equatorial plane of the liquid star, the equations of motion have additional terms not found in Schwarz child’s field and in Newtonian theory. The equation of motion of a photon in this gravitational field differs from that of Schwarz child’s mass point field by the definition of its Hilbert radius. Applications: This article thus initiates the study of the motion of stellar particles and photons within spherical liquid stars in the framework of the General Theory of Relativity which hitherto has not been exploited. Keywords: Geodesic, Motion, Photons, Spherical Liquid Stars, Stellar Particles

Dynamics of test particles in the five-dimensional Gödel spacetime

We derive the complete set of geodesic equations for massive charged test particle and light motion in the five-dimensional, rotating and charged solution of the Einstein-Maxwell-Chern-Simons field equations in five-dimensional minimal gauged supergravity and present their analytical solutions. We study the polar and radial motion, depending on the spacetime and test particle parameters, and characterize the test particle motion qualitatively by the means of parametric plots and effective potentials. We use the analytical solutions in order to visualize the test particle motion by three-dimensional plots.

Impact of hydrophobic micron ellipsoids on liquid surfaces

HypothesisHydrophobic spheres may exhibit three impact modes after impacting the liquid surface. For the impact of common non-spherical particles in practical processes, particle shape affects the flow of surrounding fluid and forces acting on them, thus the impact behaviors of non-spherical particles may differ from those of spherical particles and remain to be revealed. SimulationThe impact of hydrophobic micron ellipsoidal particles is numerically studied by solving the coupled equations of particle motion and surrounding fluid flow combining VOF method and dynamic meshing technique. The motion characteristics of the ellipsoids and fluids, and the main forces acting on the ellipsoids during impact are analyzed. FindingsThe increase in the axis ratio (AR) of ellipsoid decreases the surface tension and fluid force, resulting in a smaller total force acting on the ellipsoid. Correspondingly, the ellipsoid’s impact behavior changes. An impact-mode phase diagram is presented for the studied ellipsoids. Three impact modes, namely, submergence, rebound, and oscillation, exist when AR > 0.8, while only submergence and oscillation exist when AR ≤ 0.8. The critical velocities decrease with the increase in AR, which is well illustrated by the analysis of energy conversion under critical conditions.

Numerical Simulation of Solid-Fluid 2-Phase-Flow of Cutting System for Cutter Suction Dredgers

Abstract The study of the flow characteristics of the solid-fluid two phase flow in the cutter suction dredger is very important for exploring the slurry formation mechanism and optimizing the operational parameters. In this study, standard k-ε model and Multiple Reference Frame are applied to numerically simulate flow field in and around the cutting system, then with the steady convergent result of the simulation as the initial condition, Discrete Phase Mode is used to solve the particle motion equation by fully coupling the continuous phase and the particles. The influence of suction flow velocity and cutter’s rotating speed on particles suction are analyzed, and effectively suctioned particles numbers are also quantitatively studied. The simulation result shows that the DPM model is able to simulate the movement of particles in and around the cutter suction dredger’s cutting system, in the fluid flow filed velocity vector and pressure distribution on different planes show different characteristics, and under higher suction velocity and lower cutter rotating speed more particles are suctioned into the suction inlet. The results can help better understand flow characteristics of solid-fluid 2-phase-flow of cutter suction dredger’s cutting system, and provide theoretical support for relative system design and operational parameters optimization.

PARTICLE MOTION ON THE SURFACE OF A CONCAVE SOIL-TILLING DISK

Relative particle motion on the internal rough surface of a concave soil-tilling disk, which rotates around horizontal axis under a soil reaction force, has been considered. A disk blade is positioned in a vertical plane, which makes an incidence angle with the direction of the machine movement. This angle has its acceptable limit and when it exceeds the limit, it causes disk dragging. In the paper, it has been assumed that dragging was non-existent, the rotational rate of a disk is stable and it depends on stabilized velocity of the machine movement and on an incidence angle. When a machine is operated, soil particles get onto an internal disk surface and perform a relative motion, which determines the slipping trajectory of a particle on a disk surface. The trajectory of the absolute particle motion relative to a fixed coordinate system allows tracing the rise height of a particle after its gets onto a disk surface. Differential equations of particle motion have been developed and solved using numerical methods. A particle gets onto a disk with specified initial criteria, namely the direction of its entering on a disk and its initial velocity. Two models of particle motion on a disk have been considered. According to the first model, we assume that further particle motion after its getting onto a disk surface is performed due to the kinetic energy of a particle at the moment of its getting onto a disk surface. According to the second model, the backup force of other particles, which breaks the force of particle friction on a surface, is taken into account. Differential equations of particle motion have been developed in projections onto a fixed solid system of axes. Thus, the system includes three differential equations in three unknown functions. One of the unknown functions is the surface reaction force and the dependences, which describe relative particle motion on a disk surface. For the purpose of the quality of soil loosening, we conducted multivariate experiment to determine the area of the soil dissipation. On the basis of field experiments, it has been determined that the production process quality of tillage machine meets agrotechnical requirements.

Palatini formulation of f(R,\xa0T) gravity theory, and its cosmological implications

We consider the Palatini formulation of f(R, T) gravity theory, in which a non-minimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similar to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the f(R, T) gravity in the Palatini formulation. Cosmological models with Lagrangians of the type f=R-alpha ^2/R+g(T) and f=R+alpha ^2R^2+g(T) are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.