On Measuring and Modeling Local Average Particle Velocities for particle ensembles in classifying systems

In most multi-phase flow problems, the particulate phase is exposed to an external field which causes dispersion. Therefore, local velocity measurements of the disperse phase are no longer equivalent with respect to averaging in time and averaging in volume. While the local time-averaged velocity still characterizes the transport of the ensemble in the Eulerian sense, one has to be be careful in modeling this velocity average by considering the ensemble's composition. It is shown for different particle ensembles that the conventional particle velocity average M_1,0 calculated with respect to the dispersion relationship and a particle size number density distribution is far below the measured ensemble average; the deviation depends on the width of the particle size distribution. It is deduced that Eulerian particle velocity values referring to a certain time interval can be modeled by a ratio of velocity moments M_2,0/M_1,0 calculated with particle size number distributions referring to a certain probe volume. This relationship was confirmed by measurements.     

On the Electromagnetic Origin of Inertia and Inertial Mass

We address the problem of inertial property of matter through analysis of the motion of an extended charged particle. Our approach is based on the continuity equation for momentum (Newton’s second law) taking due account of the vector potential and its convective derivative. We obtain a development in terms of retarded potentials allowing an intuitive physical interpretation of its main terms. The inertial property of matter is then discussed in terms of a kind of induction law related to the extended charged particle’s own vector potential. Moreover, it is obtained a force term that represents a drag force acting on the charged particle when in motion relatively to its own vector potential field lines. The time rate of variation of the particle’s vector potential leads to the acceleration inertia reaction force, equivalent to the Schott term responsible for the source of the radiation field. We also show that the velocity dependent term of the particle’s vector potential is connected with the relativistic increase of mass with velocity and generates a longitudinal stress force that is the source of electric field lines deformation. In the framework of classical electrodynamics, we have shown that the electron mass has possibly a complete electromagnetic origin and the obtained covariant equation solves the “4/3 mass paradox” for a spherical charge distribution.     

Force multipole moments for a spherically symmetric particle in solution

We present a general theorem for the force multipole moments of arbitrary order induced in a spherically symmetric particle immersed in a fluid whose motion satisfies the linear Navier-Stokes equation for steady incompressible viscous flow. The multipole moments are expressed in terms of the unperturbed fluid velocity field. It is shown that for a particle with a finite extension only a few terms give rise to fluid perturbations which are not confined to the interior of the particle. We give explicit results for a polymer satisfying the Debye-Bueche-Brinkman equations and for a hard sphere with mixed slip-stick boundary conditions.     

Monte Carlo analysis of repeated overshoot structures

The effectiveness of two recently proposed repeated overshoot structures is investigated using Monte Carlo simulation. Results are presented showing electron velocity, energy and valley occupancy as a function of bias conditions. High local peak velocities sometimes are observed, but for a given mean field across a unit cell the average velocities are always relatively low; less than or at best equal to the steady state velocity in bulk GaAs with the same average fields. The reasons for this are explained in terms of the diffusion process.     

An approximate calculation of the non-adiabatic motion of a charged particle in an inhomogeneous rotationally symmetric magnetic field

A study is made of the motion of a particle injected longitudinally into a rotationally symmetric magnetic field increasing linearly in an arbitrarily sharp way on the axis of symmetry. By expanding the magnetic potential in a series in the vicinity of the line of force from which the particle starts, and by restricting our considerations to linear terms, we arrive at a single linearized equation of motion for the radial motion, as we assume the longitudinal motion to be uniform. The linearized equation can be solved exactly. Numerical evaluation is carried out (with the linearized line of force) for several different slopes of the field, and compared for similar cases with a numerical evaluation of an exact nonlinear problem. The method described in the paper can also be applied to cases with more general initial conditions, when the particle also moves with an azimuthal velocity.The author extends his thanks for valuable advice, discussions and encouragement to Dr. M. Seidl, on whose incentive the work was undertaken and who is the author of equation (13).     

The Faraday law of induction for an arbitrarily moving charge

The Faraday law of electromagnetic induction for an arbitrarily moving charge is generalized and the expression for the force acting on the charge in an alternating magnetic field is obtained. It is shown that besides the Lorentz force perpendicular to the velocity of the particle, the Faraday force parallel to the particle velocity and proportional to it is acting on the charge, too. The equations of motion of the charged particle and the magnetic moment are obtained in the time-varying magnetic field. The problems of induction acceleration of charged particles (betatron) and induction heating of medium (plasma, plasma betatron) are considered.     

Zur Theorie der Bewegung geladener Teilchen in richtungskonstanten Magnetfeldern exponentieller Zeitabhängigkeit. III. Allgemeine Schaltprozesse

The motion of a charged particle in a magnetic field is calculated for the following case: the spatially homogeneous magnetic field having a constant direction is a superposition of a field constant in time and one decreasing exponentially in time; taken into account is the influence of the electric field induced by the time dependent magnetic field and a friction force proportional to the particle velocity. The higher transcendental functions appearing in the exact solutions are approximated in various ways in dependence on the values of argument and parameters. In this manner approximated formulae of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle, and the domain of validity of these formulae is determined. The particle orbits are classified, and their dependence on the initial values, parameters of the magnetic field and on the magnitude of the friction force is studied. A comparison between our results and a rectangular variation of the field is given. It is shown that the electric field induced by the time dependent magnetic field has an important influence on the particle motion.     

湍流边界层对数律区相干结构的Tomo-TRPIV实验研究

应用层析高时间分辨率粒子图像测速技术（tomographic time-resolved particle image velocimetry，Tomo-TRPIV），测量了Re_θ=2 460的平板湍流边界层三维3分量瞬时速度场的时间序列.提出湍流空间多尺度局部平均速度结构函数的概念，应用流向脉动速度沿流向的空间多尺度局部平均速度结构函数的正负过零点法，从瞬时脉动速度场中检测相干结构的喷射和扫掠事件，对检测到的喷射和扫掠事件的瞬态局部速度场、速度梯度场、涡量场、速度变形率场进行空间相位对齐叠加平均，获得喷射和扫掠事件的局部速度场、速度梯度场、涡量场、速度变形率场的典型特征.研究发现，相干结构喷射和扫掠时，速度梯度、速度变形率、涡量均表现为空间反对称分布的4极子结构.特别是流向涡量是沿流向、法向、展向均为反对称分布的法向多层4极子结构，表明法向各层相干结构是紧密联系，互相关联的.这种法向多层的4极子反对称结构导致强烈的动量、质量和能量交换，维持了相干结构的演化和发展过程.      

Zur Theorie der Bewegung geladener Teilchen in richtungskonstanten Magnetfeldern exponentieller Zeitabhängigkeit. II. Ausschaltvorgang ohne und mit Reibung

The motion of a charged particle in a magnetic field is calculated for the following case: the spatially homogeneous magnetic field having a constant direction decreases exponentially in time (switch-off process); taken into account is the influence of the electric field and a friction force proportional to the particle velocity. The higher transcentendal functions appearing in the exact solution are approximated in various ways in dependence on the values of argument and parameters. In this manner approximated formulae of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle. The particle orbits are classified and their dependence on the initial values, parameters of the magnetic field and on the magnitude of the friction force is studied. A comparison between our results and a rectangular variation of the field shows that the latter is not a good approximation for a really exponential decreasing field. A detailed investigation shows that the electric field induced by the time dependent magnetic field has an important influence on the particle motion.     

Study of a generalized Langevin equation with nonlocal dissipative force,harmonic potential and a constant load force

We analyze the motion of a particle governed by a generalized Langevin equation with nonlocal dissipative force, linear external force and a constant load force. We consider the dissipative memory kernel consisting of two terms. One of them is described by the Dirac delta function which represents a local friction, whereas for the second one we consider two types: the exponential and power-law functions which represent nonlocal dissipative forces. For these cases, one can obtain exact results for the relaxation function. Then, we obtain the first moments and variances of the displacement and velocity. The long-time behaviors of these quantities are also investigated.     

二维耦合定向输运模型研究

建立了外部驱动力及噪声作用下的二维耦合定向输运模型, 其中的一个维度上为周期性分段棘齿势, 另一垂直维度上为周期性对称非棘齿势, 外部驱动力及噪声加在周期对称非棘齿势方向上, 而棘齿势方向不加任何驱动, 采用非平衡统计及非线性动力学理论研究了过阻尼情况下耦合系统在两个维度上的输运性质. 结果显示, 棘齿势与非棘齿势方向均可产生定向输运, 其中棘齿势方向的系统平均速度对耦合强度、噪声强度、驱动力强度及粒子数目均有明显的依赖性, 合适的耦合强度、噪声强度、驱动力强度或粒子数目下均可产生最大输运速度. 而非棘齿势方向的系统平均速度受非棘齿势势垒高度影响显著, 但随耦合强度、驱动力强度、驱动力初相位差及粒子数目的变化均出现波动现象, 表现出平均速度对这些参量的依赖性较弱.     

On the motion of charged particles in an alternating nonuniform electric field

The equations of motion for a charged particle in an electric field featuring a stationary and an oscillating component are considered for the case where the force of friction is linear in the particle velocity. The averaging of these equations over the period of field oscillations is legitimate under some specific conditions. The most general expression for an additional stationary force acting on the particle under these conditions is derived, and the limiting values of this force are found. Applications of the results obtained in the present study are considered. Such applications include the use of pulsed currents in the electrochemical dimensional treatment of materials.     

Exact Harmonic Metric for a Uniformly Moving Schwarzschild Black Hole

The harmonic metric for Schwarzschild black hole with a uniform velocity is presented. In the limit of weak field and low velocity, this metric reduces to the post-Newtonian approximation for one moving point mass. As an application, we derive the dynamics of particle and photon in the weak-field limit for the moving Schwarzschild black hole with an arbitrary velocity. It is found that the relativistic motion of gravitational source can induce an additional centripetal force on the test particle, which may be comparable to or even larger than the conventional Newtonian gravitational force.     

A Hamiltonian Model for Linear Friction in a Homogeneous Medium

 We introduce and study rigorously a Hamiltonian model of a classical particle moving through a homogeneous dissipative medium at zero temperature in such a way that it experiences an effective linear friction force proportional to its velocity (at small speeds). The medium consists at each point in a space of a vibration field modelling an obstacle with which the particle exchanges energy and momentum in such a way that total energy and momentum are conserved. We show that in the presence of a constant (not too large) external force, the particle reaches an asymptotic velocity proportional to this force. In a potential well, on the other hand, the particle comes exponentially fast to rest in the bottom of the well. The exponential rate is in both cases an explicit function of the model parameters and independent of the potential. Received: 18 July 2001 / Accepted: 20 April 2002 Published online: 12 August 2002     

Zur Theorie der Bewegung geladener Teilchen in richtungskonstanten Magnetfeldern exponentieller Zeitabhängigkeit. I. Allgemeine Theorie und Einschaltvorgang

The general solution of the equations of motion for a charged particle in a magnetic field is given for the following case: the spatially homogeneous magnetic field having a constant direction is a superposition of a field constant in time and one decreasing exponentially in time; taken into account is the influence of the electric field induced by the time dependent magnetic field and a friction force proportional to the particle velocity. The higher transcendental functions appearing in the exact solution are approximated in various ways in dependence on the values of the argument and parameters. The important case of a switching process without a friction force is investigated in detail. The higher transcendential functions can be approximated by simplier functions in such a way, that the solutions for the switching process, valid for all times, differ from the solutions in the case of a linear increasing magnetic field only by factors consisting of elementary functions. Approximated formulae of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle. The particle orbits are classified and their dependence on the initial values and parameters of the magnetic fields is studied. A comparison between our results and a rectangular variation of the field shows that the latter is not a good approximation for a really exponential increasing field. Finally a detailed investigation shows that the electric field induced by the time dependent magnetic field has an important influence on the particle motion.     

Theorie der Bewegung geladener Teilchen in richtungskonstanten zeitabhängigen Magnetfeldern und elektrischen Zusatzfeldern

The motion of a charged particle in spatially homogeneous electric and magnetic fields is calculated for the case of the magnetic field to have a constant direction and its intensity to vary with an arbitrary power of time. The special case of a linearly increasing magnetic field is treated in detail taking into account a friction force proportional to the particle velocity. Generally, the equations of motion are reduced to a single differential equation of second order which is integrated exactly. The higher transcendental functions appearing in the solution are then approximated by elementary functions. Thus asymptotic approximative equations of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle. The dependence of the particle orbit on the initial values of position and velocity and on the properties of the magnetic field is studied, and it is shown, how the particle motion is a helical motion superposed by a drift. The influence of the electric field induced by the time dependent magnetic field on the particle motion is considered in detail. For an additional electric field being present a drift formula is derived which is a generalization of the well-known ?? × ?? 93 drift for constant fields.     

Mobility in a periodic potential: A simple derivation

A simple derivation is given for the mobility of Brownian particles in a periodic potential in the overdamped regime. The method makes use of the fact that the steady state current density, in response to a uniform external force, is uniform in space and can be expressed as a product of the particle density and mean velocity field. To lowest order in the external force, the particle density is given by the equilibrium density in the absence of the external force.     

Formation of Suprathermal Particle Fluxes in a Strongly Turbulent Plasma

We consider the problem on the formation of suprathermal particle fluxes by electrostatic structures in strongly turbulent cosmic plasmas. It is shown that regions with a strong plasma turbulence can be large accelerators of charged particles. We give solutions of the stationary kinetic equation in a turbulent layer for different acceleration regimes and estimate the efficiency of diffusion over the longitudinal and transverse velocities of particles with respect to the magnetic field. The transverse diffusion in velocity space is more efficient for ions and leads to strong isotropization of ion fluxes. Electrons move almost along the magnetic field. We reveal the conditions under which the regular force in a nonuniform magnetic field influences the stochastic-acceleration process. The average energy of axial motion of the particles and the particle fluxes at large distances from the injection region are estimated. Ions and electrons can be accelerated up to comparable energies. We analyze the characteristic features of the motion of the relativistic-particle beams. It is shown that strong plasma turbulence can form particle beams with specific energies. The proposed mechanism is useful for explanation of the properties of energetic particles in cosmic plasmas with magnetic-field-aligned currents, e.g., in high-latitude regions of planetary magnetospheres, force-free configurations of the solar corona, and the solar wind.     

磁场旋度对磁场漂移的影响

讨论了静态非均匀磁场中的磁场旋度对带电粒子引导中心漂移的影响。运用三维矢量分析的方法,将带电粒子垂直于磁场运动所引起的磁场漂移分为两项,分别由磁场的曲率和磁场的旋度决定。给出了螺旋状环形磁场中由磁场旋度引起的磁场漂移的近似表达式,讨论了该漂移成分对于该磁场中通行粒子轨道和捕获粒子轨道的可能影响。结果表明,带电粒子垂直于磁场运动所引起的磁场漂移主要由磁场的曲率决定,而磁场旋度对该漂移的影响比较微弱。     

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讨论了静态非均匀磁场中的磁场旋度对带电粒子引导中心漂移的影响。运用三维矢量分析的方法,将带电粒子垂直于磁场运动所引起的磁场漂移分为两项,分别由磁场的曲率和磁场的旋度决定。给出了螺旋状环形磁场中由磁场旋度引起的磁场漂移的近似表达式,讨论了该漂移成分对于该磁场中通行粒子轨道和捕获粒子轨道的可能影响。结果表明,带电粒子垂直于磁场运动所引起的磁场漂移主要由磁场的曲率决定,而磁场旋度对该漂移的影响比较微弱。