Nonlinear Particle Dynamics Research Articles

Inertial particle focusing in fluid flow through spiral ducts: dynamics, tipping phenomena and particle separation

Small finite-size particles suspended in fluid flow through an enclosed curved duct can focus to points or periodic orbits in the two-dimensional duct cross-section. This particle focusing is due to a balance between inertial lift forces arising from axial flow and drag forces arising from cross-sectional vortices. The inertial particle focusing phenomenon has been exploited in various industrial and medical applications to passively separate particles by size using purely hydrodynamic effects. A fixed size particle in a circular duct with a uniform rectangular cross-section can have a variety of particle attractors, such as stable nodes/spirals or limit cycles, depending on the radius of curvature of the duct. Bifurcations occur at different radii of curvature, such as pitchfork, saddle-node and saddle-node infinite period (SNIPER), which result in variations in the location, number and nature of these particle attractors. By using a quasi-steady approximation, we extend the theoretical model of Harding et al. (J. Fluid Mech., vol. 875, 2019, pp. 1–43) developed for the particle dynamics in circular ducts to spiral duct geometries with slowly varying curvature, and numerically explore the particle dynamics within. Bifurcations of particle attractors with respect to radius of curvature can be traversed within spiral ducts and give rise to a rich nonlinear particle dynamics and various types of tipping phenomena, such as bifurcation-induced tipping (B-tipping), rate-induced tipping (R-tipping) and a combination of both, which we explore in detail. We discuss implications of these unsteady dynamical behaviours for particle separation and propose novel mechanisms to separate particles by size in a non-equilibrium manner.

Investigation of dual-frequency effect on the ion energy and flux in Torr-regime plasma by a two-dimensional GPU-PIC simulation

Dual frequency (DF) capacitively coupled plasma (CCP) equipment has been widely used in semiconductor processing and requires particle-in-cell (PIC) simulations to investigate the nonlinear particle dynamics. It is believed that DF driving controls the electron density and ion energy separately, but a strong coupling exists between the two frequencies in reality. The two-dimensional (2D) PIC simulation of DF CCPs under deposition process conditions has not yet been reported due to the heavy computational load that can be resolved with a graphics processing unit (GPU) computation. This study quantitatively investigates the Torr-regime DF CCP using a 2D GPU-PIC simulation. The increase of the low-frequency (LF) power enhances ion transport, which results in the rise of plasma potential, the increase of sheath thickness, and the reduction of the electron density. The ions exhibit the Maxwellian distribution due to the frequent charge-exchange collisions. The ion flux keeps uniformity while electron density becomes abruptly nonuniform with the increase of the LF power, which is suitable to maintain the quality of deposited films.

Nonlinear trapping in wave–particle interactions in tokamaks

The physics and transport consequences of the nonlinear trapping of the wave–particle interactions on the transit and bounce time scale are investigated by solving drift kinetic equation including collisions. The nonlinear particle dynamics is governed by a new set of coupled evolution equations for the phase of the wave, the toroidal component of the canonical momentum (radial motion), and the particle energy. The resonant particles are trapped nonlinearly in the phase space in a pendulum-like motion to resolve the linear resonance. The trapping mechanism is similar to that of the superbananas in the neoclassical theory. The nonlinear particle dynamics is employed to facilitate the solution of the drift kinetic equation. It is shown analytically that the particle distribution in the nonlinearly trapped region is flattened when a term of the order of is neglected. Here, δA is the typical magnitude of the normalized perturbed wave field. The trapped and barely circulating particles diffuse collisionally in the phase space. Transport fluxes, wave–particle energy transfer rate and nonlinear damping rate are calculated. The transport coefficients have a new analytic scaling U in the context of the wave–particle interaction, and can be significant even for a small amplitude wave in tokamaks. Here, ν is the collision frequency, r is the minor radius, and U is a function of the wave frequency, aspect ratio, magnetic shear, and mode numbers. It is estimated that when δA ∼ 10−4 for the magnetic perturbation, alpha particle energy loss rate is comparable to that of the neoclassical theory. The energy transfer rate is proportional to part of the particle flux, a new relation, and scales as ν ; the nonlinear damping or growth rate ν .

Nonlinear dynamics of inertial particles in the ocean: from drifters and floats to marine debris and Sargassum

Buoyant, finite-size or inertial particle motion is fundamentally unlike neutrally buoyant, infinitesimally small or Lagrangian particle motion. The de-jure fluid mechanics framework for the description of inertial particle dynamics is provided by the Maxey-Riley equation. Derived from first principles - a result of over a century of research since the pioneering work by Sir George Stokes - the Maxey-Riley equation is a Newton-type-law with several forces including (mainly) flow, added mass, shear-induced lift, and drag forces. In this paper we present an overview of recent efforts to port the Maxey-Riley framework to oceanography. These involved: 1) including the Coriolis force, which was found to explain behavior of submerged floats near mesoscale eddies; 2) accounting for the combined effects of ocean current and wind drag on inertial particles floating at the air-sea interface, which helped understand the formation of great garbage patches and the role of anticyclonic eddies as plastic debris traps; and 3) incorporating elastic forces, which are needed to simulate the drift of pelagic Sargassum. Insight on the nonlinear dynamics of inertial particles in every case was possible to be achieved by investigating long-time asymptotic behavior in the various Maxey-Riley equation forms, which represent singular perturbation problems involving slow and fast variables.

Average nonlinear dynamics of particles in gravitational pulses: Effective Hamiltonian, secular acceleration, and gravitational susceptibility

Particles interacting with a prescribed quasimonochromatic gravitational wave (GW) exhibit secular (average) nonlinear dynamics that can be described by Hamilton's equations. We derive the Hamiltonian of this "ponderomotive" dynamics to the second order in the GW amplitude for a general background metric. For the special case of vacuum GWs, we show that our Hamiltonian is equivalent to that of a free particle in an effective metric, which we calculate explicitly. We also show that already a linear plane GW pulse displaces a particle from its unperturbed trajectory by a finite distance that is independent of the GW phase and proportional to the integral of the pulse intensity. We calculate the particle displacement analytically and show that our result is in agreement with numerical simulations. We also show how the Hamiltonian of the nonlinear averaged dynamics naturally leads to the concept of the linear gravitational susceptibility of a particle gas with an arbitrary phase-space distribution. We calculate this susceptibility explicitly to apply it, in a follow-up paper, toward studying self-consistent GWs in inhomogeneous media within the geometrical-optics approximation.

Current sheets in planetary magnetospheres

In this article we aim to highlight the problems related to the structure and stability of the comparatively thin current sheets that were relatively recently discovered by space missions in the magnetospheres of the Earth and planets, as well as in the solar wind. These magnetoplasma structures are universal in collisionless cosmic plasmas and can play a key role in the processes of storage and release of energy in the space environment. The development of a self-consistent theory for these sheets in the Earth’s magnetosphere, where they were first discovered, has a long and dramatic history. Solution of the problem of the thin current sheet structure and stability become possible in the framework of a kinetic quasi-adiabatic approach required to explain their embedding and metastability properties. It was found that the structure and stability of current structures are completely determined by the nonlinear dynamics of plasma particles. Theoretical models have been developed to predict many properties of these structures and interpret many experimental observations in planetary magnetospheres and the heliosphere.

Simultaneous Control of Multispecies Particle Transport and Segregation in Driven Lattices.

We provide a generic scheme to separate the particles of a mixture by their physical properties like mass, friction, or size. The scheme employs a periodically shaken two-dimensional dissipative lattice and hinges on a simultaneous transport of particles in species-specific directions. This selective transport is achieved by controlling the late-time nonlinear particle dynamics, via the attractors embedded in the phase space and their bifurcations. To illustrate the spectrum of possible applications of the scheme, we exemplarily demonstrate the separation of polydisperse colloids and mixtures of cold thermal alkali atoms in optical lattices.

Nonlinear dynamics of resonant electrons interacting with coherent Langmuir waves

We study the nonlinear dynamics of resonant particles interacting with coherent waves in space plasmas. Magnetospheric plasma waves such as whistler-mode chorus, electromagnetic ion cyclotron waves, and hiss emissions contain coherent wave structures with various discrete frequencies. Although these waves are electromagnetic, their interaction with resonant particles can be approximated by equations of motion for a charged particle in a one-dimensional electrostatic wave. The equations are expressed in the form of nonlinear pendulum equations. We perform test particle simulations of electrons in an electrostatic model with Langmuir waves and a non-oscillatory electric field. We solve equations of motion and study the dynamics of particles with different values of inhomogeneity factor S defined as a ratio of the non-oscillatory electric field intensity to the wave amplitude. The simulation results demonstrate deceleration/acceleration, thermalization, and trapping of particles through resonance with a single wave, two waves, and multiple waves. For two-wave and multiple-wave cases, we describe the wave-particle interaction as either coherent or incoherent based on the probability of nonlinear trapping.

Edge States and Chiral Solitons in Topological Hall and Chern–Simons Fields

The multi-component extension problem of the (2+1) D -gauge topological Jackiw–Pi model describing the nonlinear quantum dynamics of charged particles in multi-layer Hall systems is considered. By applying the dimensional reduction (2 + 1) D → (1 + 1) D to Lagrangians with the Chern–Simons topologic fields , multi-component nonlinear Schrodinger equations for particles are constructed with allowance for their interaction. With Hirota‘s method, an exact two-soliton solution is obtained, which is of interest in quantum information transmission systems due to the stability of their propagation. An asymptotic analysis t →±∞ of soliton-soliton interactions shows that there is no backscattering processes. We identify these solutions with the edge (topological protected) states – chiral solitons – in the multi-layer quantum Hall systems. By applying the Hirota bilinear operator algebra and a current theorem, it is shown that, in contrast to the usual vector solitons, the dynamics of new solutions (chiral vector solitons) has exclusively unidirectional motion. The article is published in the author’s wording.

Symbolic and Numerical Modeling of Nonlinear Dynamics of Particles in Accelerators

This paper is devoted to one of the methods of symbolic computation, which based on perturbation theory and constructed in the framework of matrix formalism. This method can be used for solving optimization problems and preliminary modeling in accelerator physics. There are presented the main theoretical positions and demonstration the result of this method on example of one of classical problems.

Liquid-based stationary phase for deterministic lateral displacement separation in microfluidics.

Deterministic lateral displacement (DLD) is a promising separation scheme in microfluidic systems. In traditional DLD, a periodic array of solid posts induces the separative migration of suspended particles moving through the system. Here, we present a radical departure from traditional DLD systems and use an array of anchored liquid-bridges as the stationary phase in the DLD device. The liquid-bridges are created between two parallel plates and anchored to the bottom one by cylindrical wells. We show that the non-linear particle dynamics observed in traditional DLD systems is also present in the anchored-liquid case, enabling analogous size-based separation of suspended particles. The use of liquid-bridges as the stationary phase presents additional possibilities in separation technologies, potentially eliminating or significantly reducing clogging, enabling renewable and/or reconfigurable systems, allowing a different set of fabrication methods and providing alternative ways to separate particles based on their interaction with liquid-liquid interfaces. Some of these advantages could also extend to filtration methods based on similar liquid-based stationary phases.

Thermally induced passage and current of particles in a highly unstable optical potential.

We discuss the statistics of first-passage times of a Brownian particle moving in a highly unstable nonlinear potential proportional to an odd power of position. We observe temperature-induced shortening of the mean first-passage time and its dependence on the power of nonlinearity. We propose a passage-time fraction as both a simple and experimentally detectable witness of the nonlinearity. It is advantageously independent of all other parameters of the experiment and observable for a small number of trajectories. To better characterize the stochastic passage in the unstable potential, we introduce an analogy of the signal-to-noise ratio for the statistical distribution of the first-passage times. Interestingly, the upper bound for the signal-to-noise ratio is temperature independent in the unstable potential. Finally, we describe the nonequilibrium steady state of the particle cyclically passing through unstable odd nonlinearity. The maximum of the steady-state probability distribution shifts against the directions of the current and this counterintuitive effect increases with temperature. All these thermally induced effects are very promising targets for experimental tests of highly nonlinear stochastic dynamics of particles placed into optical potential landscapes of shaped optical tweezers.

Feasibility study of a periodic arc compressor in the presence of coherent synchrotron radiation

The advent of short electron bunches in high brightness linear accelerators has raised the awareness of the accelerator community to the degradation of the beam transverse emittance by coherent synchrotron radiation (CSR) emitted in magnetic bunch length compressors, transfer lines and turnaround arcs. Beam optics control has been proposed to mitigate that CSR effect. In this article, we enlarge on the existing literature by reviewing the validity of the linear optics approach in a periodic, achromatic arc compressor. We then study the dependence of the CSR-perturbed emittance to beam optics, mean energy, and bunch charge. The analytical findings are compared with particle tracking results. Practical considerations on CSR-induced energy loss and nonlinear particle dynamics are included. As a result, we identify the range of parameters that allows feasibility of an arc compressor for driving, for example, a free electron laser or a linear collider.

Analytical expressions for fringe fields in multipole magnets

Fringe fields in multipole magnets can have a variety of effects on the linear and nonlinear dynamics of particles moving along an accelerator beam line. An accurate model of an accelerator must include realistic models of the magnet fringe fields. Fringe fields for dipoles are well understood and can be modeled at an early stage of accelerator design in such codes as mad8, madx, gpt or elegant. Existing techniques for quadrupole and higher order multipoles rely either on the use of a numerical field map, or on a description of the field in the form of a series expansion about a chosen axis. Usually, it is not until the later stages of a design project that such descriptions (based on magnet modeling or measurement) become available. Furthermore, series expansions rely on the assumption that the beam travels more or less on axis throughout the beam line; but in some types of machines (for example, Fixed Field Alternating Gradients or FFAGs) this is not a good assumption. Furthermore, some tracking codes, such as gpt, use methods for including space charge effects that require fields to vary smoothly and continuously along a beam line: in such cases, realistic fringe field models are of significant importance. In this paper, a method for constructing analytical expressions for multipole fringe fields is presented. Such expressions allow fringe field effects to be included in beam dynamics simulations from the start of an accelerator design project, even before detailed magnet design work has been undertaken. The magnetostatic Maxwell equations are solved analytically and a solution that fits all orders of multipoles is derived. Quadrupole fringe fields are considered in detail as these are the ones that give the strongest effects. The analytic expressions for quadrupole fringe fields are compared with data obtained from numerical modeling codes in two cases: a magnet in the high luminosity upgrade of the Large Hadron Collider inner triplet, and a magnet in the nonscaling FFAG EMMA. In both examples, the analytical expressions provide a good approximation to the numerical field maps.

Effects of elasticity on the nonlinear collective dynamics of self-propelled particles

Some rodlike self-propelled particles form collective structures because of hydrodynamic interactions characterized by large-scale self-driven flows. Using a continuum fluid dynamics approach, the effects of elasticity on the pseudo-steady state dynamics of the collective patterns are studied. The Oldroyd-B and Jeffery constitutive models are used to represent the suspending fluid as examples of non shear-thinning viscoelastic fluids. A coupling between the fluid and particle dynamics leads to self-driven structures that are shorter and evolve irregularly in comparison with the long periodic correlated structures in Newtonian fluids. Inspired by turbulent drag reduction by polymers, a mechanism has been suggested for how elasticity restricts the formation of the self-driven collective structures. Viscoelasticity forces the system to transition from an “active” state to a “suppressed” state.

Measurement of nonlinear observables in the Large Hadron Collider using kicked beams

The nonlinear dynamics of a circular accelerator such as the Large Hadron Collider (LHC) may significantly impact its performance. As the LHC progresses to more challenging regimes of operation it is to be expected that the nonlinear single particle dynamics in the transverse planes will play an increasing role in limiting the reach of the accelerator. As such it is vital that the nonlinear sources are well understood. The nonlinear fields of a circular accelerator may be probed through measurement of the amplitude detuning: the variation of tune with single particle emittance. This quantity may be assessed experimentally by exciting the beam to large amplitudes with kicks, and obtaining the tunes and actions from turn-by-turn data at Beam Position Monitors. The large amplitude excitations inherent to such a measurement also facilitate measurement of the dynamic aperture from an analysis of beam losses following the kicks. In 2012 these measurements were performed on the LHC Beam 2 at injection energy (450 GeV) with the nominal magnetic configuration. Nonlinear coupling was also observed. A second set of measurements were performed following the application of corrections for ${b}_{4}$ and ${b}_{5}$ errors. Analysis of the experimental results, and a comparison to simulation are presented herein.

Nonlinear dynamics of particles excited by an electric curtain

The use of the electric curtain (EC) has been proposed for manipulation and control of particles in various applications. The EC studied in this paper is called the 2-phase EC, which consists of a series of long parallel electrodes embedded in a thin dielectric surface. The EC is driven by an oscillating electric potential of a sinusoidal form where the phase difference of the electric potential between neighboring electrodes is 180°. We investigate the one- and two-dimensional nonlinear dynamics of a particle in an EC field. The form of the dimensionless equations of motion is codimension two, where the dimensionless control parameters are the interaction amplitude (A) and damping coefficient (β). Our focus on the one-dimensional EC is primarily on a case of fixed β and relatively small A, which is characteristic of typical experimental conditions. We study the nonlinear behaviors of the one-dimensional EC through the analysis of bifurcations of fixed points in the Poincaré sections. We analyze these bifurcations by using Floquet theory to determine the stability of the limit cycles associated with the fixed points in the Poincaré sections. Some of the bifurcations lead to chaotic trajectories where we then determine the strength of chaos in phase space by calculating the largest Lyapunov exponent. In the study of the two-dimensional EC, we independently look at bifurcation diagrams of variations in A with fixed β and variations in β with fixed A. Under certain values of β and A, we find that no stable trajectories above the surface exists; such chaotic trajectories are described by a chaotic (strange) attractor, for which the largest Lyapunov exponent is found. We show the well-known stable oscillations between two electrodes come into existence for variations in A and the transitions between several distinct regimes of stable motion for variations in β.

Nonlinear dynamics of adhesive micro-spherical particles on vibrating substrates

The coupled (rocking and out-of-plane) vibrational dynamics of micro-spherical particles on vibrating flat substrates is studied experimentally, analytically, and computationally. In the experimental spectral responses of some of vibrating particles, in addition to their predicted rocking resonance frequencies, extra resonance peaks at the doubles of these predicted frequencies have been previously observed and reported. To understand and explain the rocking resonance frequency doubling effect, a two-dimensional mathematical model describing the coupled dynamics of a micro-spherical particle in the out-of-plane and rocking coordinates is developed and reported. Previously, the doubling effect was leading to ambiguity in adhesion characterization. It is determined that the frequency doubling observed in the spectral domain responses of particles is caused by nonlinear coupling between the out-of-plane and in-plane modes of motion while the particle is experiencing a whirling-like motion taking place around a nearly stationary axis tilted with respect to the substrate normal (as opposed to the substrate normal itself). In current study, the leaning angles of the rocking motion with whirling of a set of polystyrene latex particles are approximated by utilizing the coupled dynamic model. Also, the work-of-adhesion values extracted from the experimental resonance frequencies of a set of particles using the developed model are compared to those reported in the literature and a good agreement is found. The detection and analysis of the reported motion is relevant to adhesion bond characterization and particle manipulation.

Nonlinear dynamics of spherical particles in Poiseuille flow under creeping-flow condition

We study the nonlinear dynamics of spherical colloids under the influence of a pressure driven flow at vanishing Reynolds number. The colloids are confined between two parallel planar walls with a distance comparable to the particle diameter and they interact hydrodynamically via the solvent. We show that the bounded Poiseuille flow gives rise to new classes of trajectories resulting in cross-streamline migration. Two particles moving on these new trajectories exhibit either bound or unbound states. In the first case they oscillate on closed trajectories in the center-of-mass frame. In the second case, they exhibit cross-swapping trajectories in addition to swapping trajectories which were already observed in unbounded or bounded linear shear flow. The different classes of trajectories occur depending on the initial positions of the two particles and their size. We present state diagrams in the lateral positions, where we categorize the trajectories and color code the oscillation frequencies of the bound states. Finally we discuss how the results on the two-particle system help to understand the stability of particle trains composed of several particles.

Magnetization reversal of small particles in an oscillating magnetic field

The nonlinear behavior of single spherical particles and many-particle ensembles under an external oscillating magnetic field was studied using the Landau-Lifshitz-Gilbert approach. The exact analytical formulae for calculating field frequency and the dependence of the configuration of the initial moment on the system mean magnetization were obtained. Using asymptotic decomposition we obtained simple expressions for describing the nonlinear dynamics of spherical particles in the cases of large and small frequencies. These solutions are in good agreement with numerical calculations of the Landau-Lifshitz-Gilbert equation for the considering systems.