Equations Of Particle Dynamics Research Articles

Generalized dynamics equation with causal symmetry

ABSTRACT The particle dynamics equation’s formal logic is examined in the context of classical mechanics to investigate the nature of inertial forces. First, a fresh perspective on the dual dependence of the inertial reference frame and inertial force in Newtonian dynamics is presented. The causal relationship between the two sides of Newton’s second law is found to be asymmetrical and inconsistent. Secondly, a more general particle dynamics equation, applicable in any translational frame of reference without additional assumptions, is introduced. Essentially, Newton’s second law is only an extreme case of the newly generalized dynamics equation since in Newton’s second law an entire term of the forces acting on the reference object is omitted. The nature of inertial force is unveiled as the mass-ratio weighted real force acting on the reference object. This qualitative explanation is entirely naturalistic as the reference frame’s acceleration depends directly on the acceleration of its reference object(s), and the acceleration of every reference object depends directly on the forces acting on this reference object.

N-Euler mathematical modelling and numerical simulation of liquid–gas–solid flows

Three phase flows are omnipresent in industrial as well as natural configurations. Understanding the behaviour of solid particles interacting with liquid–air flows can be of great interest for both topics. In this paper we present a new liquid–gas–solid modelling technique based on the hybrid method combining liquid–gas two-fluid phase-average approach and probability density function (PDF) methods for solid dispersed phases. The PDF approach is based on single particle dynamic equation taking into account its interaction with more than one fluid and separate stochastic Lagrangian equations for each fluid phase velocity seen by the particles. This method does not assume the carrier phases typology. The solid particles can interact with bubbles, droplets, pockets and stratified flow through drag only (other interaction forces are supposed to be negligible for our cases). In this work, it is assumed that the total drag applied to the particle is a linear combination of the drags computed if the particles were interacting separately with each fluid phases. Finding the linear combination coefficient represent the main modelling step. Particle collisions and particle turbulent dispersion are taken into account in the model. Dispersed bubbles can undergo coalescence and fragmentation. The effect of all these sub-models is studied in the comparison with experimental data.The fluid and particle equations are first derived before introducing the appropriate coupling terms. Then, the closure terms for turbulence and multiphase drag are introduced. The model is implemented in neptune_cfd, a finite volume solver developed by EDF, CEA, IRSN and Framatome which allows for the numerical resolution of separate Eulerian equations (mass, momentum and energy) for n phases coupled by interfacial transfer terms. The model is then compared against experimental data obtained in a bubble column charged with particles. The results obtained with it are satisfactory and enable us to study more in details its hypothesis and to confirm comments made in the first section. Eventually, the model is tested on an industrial case which shows its efficiency.

Identification of swirling air flow velocity by non-neutrally buoyant tracer particle based on machine learning

In the non-intrusive measurement of swirling air flow, helium-filled soap bubbles (HFSBs) are ideal neutrally buoyant tracer particles However, there are some researchers that do not use HFSBs in the non-intrusive measurement of swirling air flow, leading to some kind of measurement inaccuracy. Since the flow velocity data has been implicitly included in the physical equations of any kind of tracer particles, it is possible to extract such hidden flow velocity from particle trajectory. In this study we propose a physics-informed procedure of adopting SINDy algorithm to identify the hidden physical equations of non-neutrally buoyant particle dynamics, so that the implicit flow velocity can be discovered. First of all, the numerical experiment is conducted to generate particle trajectory in a 2D swirling air flow in small cyclone separator. Based on the numerical experiment trajectory data, the input variables for SINDy algorithm are properly constructed. The output of SINDy algorithm, which are the identified physical equations, are evaluated and validated on two different-density particle trajectory data. Our results show that the physical equations of tracer particle dynamics can be identified and the discovered flow velocity data has a maximum deviation of 1.4% from the truth (R2≥0.999). The proposed method may remove the requirement of NB tracer particle in non-intrusive measurement of swirling air flow, and may be applied to recognize the physical equations of complex particle laden flow.

Path tracking control of solar-powered UAV based on energy management strategy

To coordinate the conflict between the high-altitude task and long-term energy efficiency of high-altitude long endurance solar UAV, it is one of the core issues of UAV control. A strategy combining the energy management with the route tracking is presented. Firstly, based on the three-dimensional particle motion model, energy storage battery model and solar energy acquisition model for the solar UAV, according to the aerodynamic parameters of the solar UAV and the typical horizontal flight, climb, descent and other motion processes of the solar UAV, the energy management strategy of the solar UAV is designed, and the allocation mechanism and optimal flight parameters of the energy acquisition, storage and consumption of the solar UAV in different flight stages are determined. The surplus solar energy is stored by using the gravitational potential energy, and then the energy management based on height adjustment is carried out to realize the longitudinal tracking control of the solar UAV under different lighting and energy conditions. Then based on the task requirements of the lateral movement of the solar UAV, a track tracking control method based on the feedback linearization method is established by decoupling the particle dynamics equation of the solar UAV, and the track tracking control of the solar UAV in the lateral direction is realized. Finally, a simulation throughout 24 hours is implemented and illustrated the effectiveness of the energy management strategy and route tracking control law.

Global existence and decay rates of solutions for Vlasov–Navier–Stokes–Fokker–Planck equations with magnetic field

In this paper, we consider the two-phase flow model coupled by the incompressible magnetohydrodynamic equations and the discrete-phase (particle) dynamics equation Vlasov–Fokker–Planck in [Formula: see text]. When the initial value is a small perturbation of an equilibrium state, the global well-posedness of the strong solution in the three-dimensional global space is established by using the optimal decay estimation and energy estimation. In particular, we are able to show the algebraic convergence rate of the solution to the equilibrium state. While for periodic domain, the same result still holds, but the rate of convergence is exponential.

Relativistic Hydrodynamic Interpretation of de Broglie Matter Waves

We present a classical hydrodynamic analog of free relativistic quantum particles inspired by de Broglie’s pilot wave theory and recent developments in hydrodynamic quantum analogs. The proposed model couples a periodically forced Klein–Gordon equation with a nonrelativistic particle dynamics equation. The coupled equations may represent both quantum particles and classical particles driven by the gradients of locally excited Faraday waves. Exact stationary solutions of the coupled system reveal a highly nonlinear mechanism responsible for the self-propulsion of free particles, leading to the onset of unsteady motion. Although the model is essentially nonrelativistic, a stabilizing mechanism for any particle traveling close to the wave signaling speed emerges through the coupling with the wavefield. Consequently, inline particle oscillations comparable to de Broglie’s wavelength are realized through this fully-classical model, suggesting a new classical interpretation for the motion of relativistic quantum particles.

Mori-Zwanzig projection operator formalism: Particle-based coarse-grained dynamics of open classical systems far from equilibrium.

We present a generalized Langevin equation (GLE) of motion that governs exactly the time evolution of phase-space observables in finite open systems described by classical Hamiltonians with explicitly time-dependent potentials. This formalism is based on the Mori-Zwanzig projection operator (PO) method with a time-independent Zwanzig PO within a Heisenberg (Lagrangian) picture and reduced description of Hamiltonian systems in terms of canonical relevant and irrelevant coordinates. We demonstrate that, similarly to closed systems, GLE dynamics in Hamiltonian systems in the presence of time-dependent potentials is determined by conservative, dissipative memory, and projected force fields, and that the memory functions relate to the projected force, which is a two-time process, in a way that is reminiscent of the equilibrium second fluctuation-dissipation relation. We further show that, in the most general case, the memory kernel depends on the relevant momentum gradients of the (Boltzmann) entropy of the irrelevant subsystem. Using two Zwanzig operators which are, respectively, functionals of the canonical and generalized canonical probability densities, we then derive what we call canonical and generalized canonical GLEs. Further, we can formulate the particle-based, coarse-grained (CG) GLE dynamics by transitioning to Jacobi coordinates which corresponds to a particle set partitioning of the Hamiltonian system. The obtained canonical CG GLE of motion for the relevant momenta is a generalization of the CG equation of motion known for closed systems. Also, using a Markovian approximation of the canonical CG GLE, we can extend the dissipative particle dynamics equation to open systems. A distinctive feature of our extension is a use of explicitly time-dependent frictions, which reflect the changes in the dissipation rate caused by time-dependent coupling to an external bath. Our GLE formalism and workflow constitute a general and viable framework that can be readily used as a starting point to rigorously formulate microscopically informed CG treatments for a variety of phenomena in externally forced systems far from equilibrium.

Clustering behaviour and settling velocity of bidisperse inertial particles in turbulent open channel flow

The clustering behaviour and settling velocity of bidisperse inertial particles with sub-millimetre diameters travelling in a turbulent open channel flow are investigated using the simultaneous particle image velocimetry and particle tracking velocimetry measurement technique. Instantaneous images of flow field and inertial particles are separated based on different wavelengths of light emitted by fluorescent seeding particles and inertial particles. Four flow configurations of turbulence are generated by turbulence grids with different sizes in the open channel. The clustering behaviour of monodisperse and bidisperse inertial particles and the discrepancy between them in the different flow cases are studied. We found that bidisperse inertial particles have weaker clustering phenomenon as compared with corresponding monodisperse inertial particles and this weak clustering phenomenon is almost consistent when the Taylor microscale Reynolds number Reλ changes from 90 to 267. The relationship between settling velocity and local concentration of inertial particles is then investigated. The results support the preferential concentration as the mechanism for the enhancement of settling velocimetry when the normalized particle concentration is in the range of 0.1~10. However, the anisotropy property and large eddy effect in turbulence will greatly change the interaction between inertial particles and turbulent flows and further alter their settling velocity, such as the average settling velocity decreasing with increasing Stokes number which depends on the dissipation time scale. Accordingly, we propose a new dimensionless multiscale parameter SvηReL1/2 to take into account the effect of large eddies and turbulence anisotropy on the settling velocity. With the simultaneous measurement of particle settling velocities and fluid velocities, we also analyse the unsteady force term in the particle dynamic equation. When the turbulence intensity is small and/or the particle inertia related to particle diameter and density ratio between inertial particle and fluid is small, the unsteady term can have a large contribution and it should not be neglected in the particle dynamic equation for the point-particle method in the numerical simulations of particle-laden flows.

Particles in turbulent separated flow over a bump: Effect of the Stokes number and lift force

Particle-laden turbulent flow that separates due to a bump inside a channel is simulated to analyze the effects of the Stokes number and the lift force on the particle spatial distribution. The fluid friction Reynolds number is approximately 900 over the bump, the highest achieved for similar computational domains. The presence of the bump creates a complex background flow with a recirculating region and a strong shear layer. A range of particle Stokes numbers are considered, each simulated with and without the lift force in the particle dynamic equation. The effect of the lift force on the particle concentration is dominant in regions of high spanwise vorticity, particularly at the walls and in the shear layer. The concentration change is of the order of thousands when compared to cases where the lift force is omitted. At a low Stokes number, the particles segregate at both top and bottom walls and are present in the recirculating region. As the Stokes number increases, particles bypass the recirculating region and their redistribution is mostly affected by the strong shear layer. Particles segregate at the walls and particularly accumulate in secondary recirculating regions behind the bump. At higher Stokes numbers, the particles create reflection layers of high concentration due to their inertia as they are diverted by the bump. The fluid flow is less influential, and this enables the particles to enter the recirculating region by rebounding off walls and create a focused spot of high particle concentration.

Simulations of dust dynamics around a cone hood in updraft conditions

A cone hood is an efficient device for capturing dust releases generated by a variety of process equipment. For stationary airflow conditions, a circular cone hood with a round flange is the most efficient design. The goal of this article is to determine the effect that inflow velocity, suction velocity, and terminal settling velocity of dust particles have on the aspiration coefficient in combination with hood length and inclination angle. No studies have yet addressed the efficiency of an exhaust hood facing an updraft flow of air with suspended dust particles. To simulate the moving fluid, we used the discrete vortices method accounting for flow separation at sharp edges of the cone hood. A custom test bench was built to validate the velocity field distribution around the exhaust hood. To evaluate capture efficiency, we determined the aspiration coefficient using plotted limiting trajectories of dust particles by solving equations of particle dynamics numerically in view of gravity and streamlining airflow patterns. In order to validate our estimate of the aspiration coefficient, we compared our findings with regularities identified by earlier researchers for a simpler problem of dust-air mixture approaching a circular exhaust opening. The following conditions were considered: the ratio of updraft velocity to the exhaust hood suction velocity varying between 0.01 and 0.5; the ratio of dust particle terminal velocity to the suction velocity varying between 0.000625 and 0.2; flange angle varying between 0° and 90°; and the ratio of flange length to the exhaust opening radius varying between 1 and 4. Using regularities discovered by us, exhaust hood designs can be tailored to a variety of application conditions in terms of dust release capture efficiency.

A new method to calculate unsteady particle kinematics and drag coefficient in a subsonic post-shock flow

We introduce a new method (piecewise integrated dynamics equation fit, PIDEF) that uses the particle dynamics equation to determine unsteady kinematics and drag coefficient (CD) for a particle in subsonic post-shock flow. The uncertainty of this method is assessed based on simulated trajectories for both quasi-steady and unsteady flow conditions. Traditional piecewise polynomial fitting (PPF) shows high sensitivity to measurement error and the function used to describe CD, creating high levels of relative error (1) when applied to unsteady shock-accelerated flows. The PIDEF method provides reduced uncertainty in calculations of unsteady acceleration and drag coefficient for both quasi-steady and unsteady flows. This makes PIDEF a preferable method over PPF for complex flows where the temporal response of CD is unknown. We apply PIDEF to experimental measurements of particle trajectories from 8-pulse particle tracking and determine the effect of incident Mach number on relaxation kinematics and drag coefficient of micron-sized particles.

Modeling of Particle Trajectory and Erosion of Large Rotor Blades

When operating in hostile environments, engines components are facing a serious problem of erosion, leading to a drastic drop in aerodynamic performance and life-cycle. This paper outlines the modeling and simulation of particle trajectory and erosion induced by sand particles. The governing equations of particle dynamics through the moving of large rotor blades are introduced and solved separately from the flow field by using our in-house particle tracking code based on the finite element method. As the locations of impacts are predicted, the erosion is assessed by semiempirical correlations in terms of impact conditions and particle and target surface characteristics. The results of these computations carried out for different concentrations of suspended dust (sand) cloud generated at takeoff conditions reveal the main areas of impacts with high rates of erosion seen over a large strip from the blade suction side, around the leading edge and the pressure side of blade. The assessment of the blade geometry deterioration reveals that the upper corner of blade suffers from an intense erosion wear.

Cosmological evolution of statistical system of scalar charged particles

In the paper we consider the macroscopic model of plasma of scalar charged particles, obtained by means of the statistical averaging of the microscopic equations of particle dynamics in a scalar field. On the basis of kinetic equations, obtained from averaging, and their strict integral consequences, a self-consistent set of equations is formulated which describes the self-gravitating plasma of scalar charged particles. It was obtained the corresponding closed cosmological model which also was numerically simulated for the case of one-component degenerated Fermi gas and two-component Boltzmann system. It was shown that results depend weakly on the choice of a statistical model. Two specific features of cosmological evolution of a statistical system of scalar charged particles were obtained with respect to cosmological evolution of the minimal interaction models: appearance of giant bursts of invariant cosmological acceleration Ω at the time interval 8⋅103–2⋅104 t Pl and strong heating (3–8 orders of magnitude) of a statistical system at the same times. The presence of such features can modify the quantum theory of generation of cosmological gravitational perturbations.

Modeling of aggregation kinetics by a new moment method

A new flocculation dynamics model is established based on the proposed Taylor expansion moment method, for the fine particle aggregation and floc formation. Mechanisms of hydrodynamic shear and differential sedimentation dominated flocculation are studied in the framework of Smoluchowski mean field theory. The Taylor expansion technique is applied and the fractal dimension is introduced into the system of particle dynamics equations. The effects of particle diffusion and vertical transport have been included. The influence of collision efficiency and the combination with fractal dimension on the calculation results are studied. The formula for estimating the floc time accounting for the fractal dimension is derived. The proposed model is applicable to all the three types of floc mechanisms: Brownian motion, differential settlement and orthokinetic flocculation. Increase of fractal dimension number results in smaller critical floc size and reduced floc structure coefficient and smaller standard deviation of floc size distribution, which reflects the self-holding. Results of the present research show that particle flocculation and fragmentation can be balanced in a certain scope, to achieve the so-called self-retention.

Biomechanical Analysis of Contemporary Throwing Technique Theory

Based on the movement process of throwing and in order to further improve the throwing tech- nique of our country, this paper will first illustrate the main influence factors which will affect the shot distance via the mutual combination of movement equation and geometrical analysis. And then, it will give the equation of the acting force that the throwing athletes have to bear during throwing movement; and will reach the speed relationship between each arthrosis during throwing and batting based on the kinetic analysis of the throwing athletes' arms while throwing. This paper will obtain the momentum relationship of the athletes' each arthrosis by means of rotational inertia analysis; and then establish a restricted particle dynamics equation from the La- grange equation. The obtained result shows that the momentum of throwing depends on the momentum of the athletes' wrist joints while batting.

Optimizing performance of the deconvolution model reduction for large ODE systems

We investigate the numerical performance of the regularized deconvolution closure introduced recently by the authors. The purpose of the closure is to furnish constitutive equations for Irving-Kirkwood-Noll procedure, a well known method for deriving continuum balance equations from the Newton’s equations of particle dynamics. A version of this procedure used in the paper relies on spatial averaging developed by Hardy, and independently by Murdoch and Bedeaux. The constitutive equations for the stress are given as a sum of nonlinear operator terms acting on the mesoscale average density and velocity. Each term is a “convolution sandwich” containing the deconvolution operator, a composition or a product operator, and the convolution (averaging) operator. Deconvolution is constructed using filtered regularization methods from the theory of ill-posed problems. The purpose of regularization is to ensure numerical stability. The particular technique used for numerical experiments is truncated singular value decomposition (SVD). The accuracy of the constitutive equations depends on several parameters: the choice of the averaging window function, the value of the mesoscale resolution parameter, scale separation, the level of truncation of singular values, and the level of spectral filtering of the averages. We conduct numerical experiments to determine the effect of each parameter on the accuracy and efficiency of the method. Partial error estimates are also obtained.

Numerical Analysis and Experiment for Microparticle Collector Using Dielectrophoretic Force

The motion of dielectric microparticles in the proposed particle collector is modeled and analyzed with coupled equations of electromagnetics, fluidics, and particle dynamics. The dielectrophoretic and drag forces on microparticles are obtained using distributions of electric field and fluid velocity field. The two fields are numerically calculated by the FEM. All the forces on the particles are employed for Newton's dynamic equation for particle motion analysis. To evaluate the proposed new method, an experimental system for the particle collector is designed and built to measure the trapping performance, which is estimated using standard microparticles and a particle counter under a given air flow.

Microscopic derivation of particle-based coarse-grained dynamics

In this paper we revisit the derivation of equations of motion for coarse-grained (CG) particles from the microscopic Hamiltonian dynamics of the underlying atomistic system in equilibrium. The derivation is based on the projection operator method and time-convolution equation. We demonstrate that due to the energy exchange between CG and intraparticle phase space coordinates in the microscopic system, the choice of projection operator is not unique, leading to different CG equations of motion that have the form of the nonlinear generalized Langevin equation (GLE). We derive the idempotence properties for the projection operators along the system trajectories and show that these properties result in streaming terms of the respective GLEs that are conservative forces and allow the expression of the non-conservative forces explicitly through thermodynamic averages, which can be measured from the microscopic simulations. The difference between GLEs that are presented herein lies in how the non-conservative forces are partitioned into dissipative and projected contributions. We compute the projected force and analyze conditions under which the projected (stochastic) force is orthogonal to (uncorrelated with) the momenta of CG particles, therefore justifying a transition to a framework of stochastic differential equations. We show that a position- and momentum-independent memory function appears only if the projected force is fully decoupled from the past CG positions and momenta, respectively. In the case of non-vanishing correlations between the projected force and the CG coordinates in past times, we derive explicitly the position- and momentum-dependent memory function in a form of projection onto a space spanned by N-order Hermite polynomials. The expressions presented herein can be used to construct a hierarchy of thermodynamically consistent CG models with momentum-dependent memory functions. They can also be used to design computational schemes for obtaining the parameters for GLEs and their variants such as dissipative particle dynamics equations from the microscopic data. We illustrate these applications by presenting the GLE with a memory function that is quadratic in the particle momenta.

Effect of spark discharges on the trajectories of insulating particles in roll-type corona-electrostatic separators. Experimental and numerical study

Corona-electrostatic separation is a multi-variable process that has been thoroughly studied in connection with its various applications in the recycling industry. The aim of the present paper is to point out one parasitic phenomenon that adversely affects the efficiency of the separation: the sparks generated at the passage of conductive particles through the electric field zone. The experiments were carried out on a laboratory roll-type corona electrostatic separators, and the sparks were generated by introducing 16 calibrated copper pins in 40-g samples of granular insulating material (PVC; typical granule size: 1.5 mm) that were fed at a constant rate onto the surface of the grounded rotating roll electrode. The distribution of the PVC granules in the 14 boxes of the collector was altered by the occurrence of the spark discharges, as they were accompanied by the annealing of the electric field between the electrodes. The numerical simulation of insulating granules charging and movement under the action of the electric field enabled a better understanding of the interactions between the spark discharges and the other factors that influence their trajectories and affect the efficiency of the separation: roll-speed, particle size and ambient humidity. The particle dynamics equations were solved using an iterative scheme by using the electric field calculated in any point with the commercial software TRICOMP. The good agreement between the predictions made by these simulations and the experimental findings confirms the ability of the mathematical model to reflect the complexity of the physical phenomena. ► Sparks occur between the electrodes at the passage of conductive particles. ► Sparks aneal the electric field and limit the maxim charge of insulating particles. ► Sparks have a stronger effect on the trajectories of coarse particles. ► Purity of the separated products is poorer in the presence of sparks. ► Numerical modeling can provide an accurate description of the phenomenon.

Semi-bottom-up coarse graining of water based on microscopic simulations

The generalized dissipative particle dynamics (DPD) equation derived from the generalized Langevin equation under Markovian approximations is used to simulate coarse-grained (CG) water cells. The mean force and the friction coefficients in the radial and transverse directions needed for DPD equation are obtained directly from the all atomistic molecular dynamics (AAMD) simulations. But the dissipative friction forces are overestimated in the Markovian approximation, which results in wrong dynamic properties for the CG water in the DPD simulations. To account for the non-Markovian dynamics, a rescaling factor is introduced to the friction coefficients. The value of the factor is estimated by matching the diffusivity of water. With this semi-bottom-up mapping method, the radial distribution function, the diffusion constant, and the viscosity of the coarse-grained water system computed with DPD simulations are all in good agreement with AAMD results. It bridges the microscopic level and mesoscopic level with consistent length and time scales.