Infinitesimal Particle Research Articles

A Definition of Particle Rolling in a Granular Assembly in Terms of Particle Translations and Rotations

The paper presents a definition of particle rolling for the interactions of two and three-dimensional particles of arbitrary shape, in case of infinitesimal particle translations and rotations. The definition is based on a purely kinematical analysis, and it is shown to satisfy the objectivity condition.

Bifurcations and equilibria in the extended N-body ring problem

We consider the motion of an infinitesimal particle under the gravitational field of ( n+1) bodies in ring configuration, that consist of n primaries of equal mass m placed at the vertices of a regular polygon, plus another primary of mass m 0= βm located at the geometric center of the polygon. We analyze the phase flow, determine the equilibria of the system, their linear stability and the bifurcations depending on the mass of the central primary (parameter β). This study is extended to the case when the central body is an ellipsoid or a radiation source. In this case, the topology of the problem is modified.

On the viscous motion of a small particle in a rotating cylinder

The dynamics of a non-neutrally buoyant particle moving in a rotating cylinder filled with a Newtonian fluid is examined analytically. The particle is set in motion from the centre of the cylinder due to the acceleration caused by the presence of a gravitational field. The problem is formulated in Cartesian coordinates and a relevant fractional Lagrangian equation is proposed. This equation is solved exactly by recognizing that the eigenfunctions of the problem are Mittag–Leffler functions. Virtual mass, gravity, pressure, and steady and history drag effects at low particle Reynolds numbers are considered and the balance of forces acting on the particle is studied for realistic cases. The presence of lift forces, both steady and unsteady, is taken into account. Results are compared to the exact solution of the Maxey–Riley equation for the same conditions. Substantial differences are found by including lift in the formulation when departing from the infinitesimal particle Reynolds number limit. For particles lighter than the fluid, an asymptotically stable equilibrium position is found to be at a distance from the origin characterized by X ≈ −Vτ/Ω and Y/X ≈ (CS/3π√2) Res1/2, where Vτ is the terminal velocity of the particle, Ω is the positive angular velocity of the cylinder, Res is the shear Reynolds number a2Ω/v, and CS is a constant lift coefficient. To the knowledge of the authors this work is the first to solve the particle Lagrangian equation of motion in its complete form, with or without lift, for a non-uniform flow using an exact method.

Burgers turbulence initialized by a regenerative impulse

In this article we look at a one-dimensional infinitesimal particle system governed by the completely inelastic collision rule. Considering uniformly spread mass, we feed the system with initial velocities, so that when time evolves the corresponding velocity field fulfils the inviscid Burgers equation. More precisely, we suppose here that the initial velocities are zero, except for particles located on a stationary regenerative set for which the velocity is some given constant number. We give results of a large deviation type. First, we estimate the probability that a typical particle is located at time 1 at distance at least D from its initial position, when D tends to infinity. Its behaviour is related to the left tail of the gap measure of the regenerative set. We also show the same asymptotics for the tail of the shock interval length distribution. Second, we analyse the event that a given particle stands still at time T as T tends to infinity. The data to which we relate its behaviour are the right tail of the gap measure of the regenerative set. We conclude with some results on the shock structure.

A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system

We focus on the dynamics of a small particle near the Lagrangian points of the Sun-Jupiter system. To try to account for the effect of Saturn, we develop a specific model based on the computation of a true solution of the planar three-body problem for Sun, Jupiter and Saturn, close to the real motion of these three bodies. Then, we can write the equations of motion of a fourth infinitesimal particle moving under the attraction of these three masses. Using suitable coordinates, the model is written as a time-dependent perturbation of the well-known spatial Restricted Three-Body Problem. Then, we study the dynamics of this model near the triangular points. The tools are based on computing, up to high order, suitable normal forms and first integrals. From these expansions, it is not difficult to derive approximations to invariant tori (of dimensions 2, 3 and 4) as well as bounds on the speed of diffusion on suitable domains. We have also included some comparisons with the motion of a few Trojan asteroids in the real Solar system.

Doubly-Symmetric Periodic Solutions of the Spatial Restricted Three-Body Problem

The existence of two new families of periodic solutions to the spatial restricted three-body problem is shown. These solutions are independent of the mass ratio of the primaries, have large inclinations and are symmetric with respect to two coordinate planes. In one family the infinitesimal particle is very far from the primaries and in the other family the infinitesimal is very close to one of the primaries.

Clustering statistics for sticky particles with Brownian initial velocity

We establish a connection between two different models of clustering: the deterministic model of sticky particles which describes the evolution of a system of infinitesimal particles governed by the dynamic of completely inelastic shocks (i.e. clustering occurs upon collision with conservation of masses and momenta), and the random model of the so-called additive coalescent in which velocities and distances between clusters are not taken into account. The connection is obtained when at the initial time, the particles are uniformly distributed on a line and their velocities are given by a Brownian motion.

Bridging the Continuum–Discontinuum Gap in the Theory of Diffuse Reflectance

A system of equations described by Benford relate the absorption and remission properties of a layer of a material to the properties of any other thickness of the material. R, the fraction of light remitted from an infinitely thick sample, may be calculated from Benford's equations by increasing the sample thickness until the total remission converges to its upper limit. The fractions of light absorbed ( a0) and remitted ( r0) by a very thin layer may be similarly calculated. The relationship A( r,t) = [(1 – r)2 – t2] / r = (2 – a – 2 r) a / r = 2 a0 / r0 = (1 – R)2 / R describes an Absorption/Remission Function for the material as a function of a, r and t, the fractions of light absorbed, remitted and transmitted by a specified layer. This is a more general expression than the widely used Kubelka–Munk equation, but gives results equivalent to it for the case of infinitesimal particles.

The molecule maxim: Guidelines for an evolved enterprise culture in Hungary

Problems arising from the transformation of Hungary from a monolithic society into a market economy are addressed. In the past, under communism, the worker was a faceless digit in a mass proletariat. He had no rights, but also no sense of obligation to society and the community. How can the business community instil in its worker a sense of their own unique individuality and role in their company and in society, a psychological awakening that will benefit not only the works himself but his employer and the company's end product? This new corporate culture must trickle down from the management who must undertake the task of re‐educating their workers in the interests of improved production and a better society. The author calls this the Molecule Maxim, in that management must stop thinking from the top downwards and start at the bottom of the pyramid, radiating outwards and upwards from this infinitesimal particle—the molecule—around which to build the concept, aim the creativity and thus the ultimate success and profitability of the company.

A NOTE ON PARAMETER VALUES FOR STABLE LOW-INCLINATION PERIODIC MOTION IN THE RESTRICTED THREE-BODY PROBLEM WITH RADIATION

The intervals of the mass parameter (µ) values for possible stability of the basic families of 3D periodic orbits in the restricted three-body problem determined elsewhere are now extended into regions of theµ - q1 parameter space of the photogravitational restricted three-body problem, where q1 is the radiation factor of m1 and it is assumed that m2 does not radiate. Several 3D periodic orbits corresponding to these regions are computed and tested for stability and seven regions, corresponding to the vertical-critical orbits l1v, l'1v, l6v, m1v, m2v and i1v, survive this stability test, emerging as the regions allowing the simplest types of stable low inclination 3D motion of the infinitesimal particle.

Chaotic particle motion under linear surface waves.

We investigate the motion of infinitesimal particles in the flow field inside the fluid under a traveling surface wave. It is shown that, even for two-dimensional waves, a superposition of two or more traveling harmonic waves is enough to generate chaotic particle motion, i.e., Lagrangian chaos. (c) 1996 American Institute of Physics.

On regularity of superprocesses

Three theorems on regularity of measure-valued processesX with branching property are established which improve earlier results of Fitzsimmons [F1] and the author [D5]. The main difference is that we treatX as a family of random measures associated with finely open setsQ in time-space. Heuristically,X describes an evolution of a cloud of infinitesimal particles. To everyQ there corresponds a random measureXτ which arises if each particle is observed at its first exit time fromQ. (The stateXt at a fixed timet is a particular case.) We consider a monotone increasing familyQt of finely open sets and we establish regularity properties of\(\bar X_t = X_{\tau _t } \) as a function oft. The results are used in [D6], [D7] and [D10] for investigating the relations between superprocesses and non-linear partial differential equations. Basic definitions on Markov processes and superprocesses are introduced in Sect. 1. The next three sections are devoted to proving the regularity theorems. They are applied in Sect. 5 to study parts of superprocess. The relation to the previous work is discussed in more detail in the concluding section. It may be helpful to look briefly through this section before reading Sects. 2–5.

On Mixed Problems of Transport Equations of Polar Fluids with Infinitesimal Particle Surface Deformation. 3rd Report. Unique Existence Theorem for a Weak Solution under Imperfect Reflective Barrier Boundary Conditions, Positive-Valued Property and Comparison Theorem.

In a previous report, we proposed a statistical theory for powder surface modification technology by means of a mechanical method using the transport equations of polar fluids with infinitesimal particle surface deformation. In this paper, we demonstrate the unique existence theorem for a weak solution on a mixed problem under imperfect reflective barrier boundary conditions (∥MB∥<1) by a priori evalution and the successive approximation method. We consider the relationship between weak solutions and other improper solutions, such as strong solutions and almost-everywhere solutions. The positive-valued property and the comparison theorem for weak solutions are proven by positive operators to assign the physical meaning to improper solutions.

On Mixed Problems of Transport Equations of Polar Fluids with Infinitesimal Particle Surface Deformation. 2nd Report, Unique Existence Theorem of a Weak Solution on Perfect Absorptive Barrier Boundary Condition.

A clear design principle has not been established for powder surface modification technology by mechanical and physial methods, such as microencapsulation. In this paper, the transport equations of polar fluids proposed by the authors in a previous report are applied to microencapsulation. Using the functional analysis, we show mathematical properties of mixed problems under perfect absorptive barrier boundary condition. The unique existence theorem of a weak solution to mixed problems is obtained by a priori evalution and a successive approximation method. Since this weak solution tends to be a sufficient classical solution, these mixed problems are wellposed in the sense of Hadamard.

Orbital stability zones about asteroids

We have numerically investigated a three-body problem consisting of the Sun, an asteroid, and an infinitesimal particle initially placed about the asteroid. We assume that the asteroid has the following properties: a circular heliocentric orbit at R = 2.55 AU, an asteroid/Sun mass ratio of μ = 5 × 10 −12, and a spherical shape with radius R A = 100 km; these values are close to those of the minor planet 29 Amphitrite. In order to describe the zone in which circum-asteroidal debris could be stably trapped, we pay particular attention to the orbits of particles that are on the verge of escape. We consider particles to be stable or trapped if they remain in the asteroid's vicinity for at least 5 asteroid orbits about the Sun, or about 20 years. Applying this criterion to particles started on circular orbits around the asteroid, we find that, as the starting distance from the asteroid is varied, a fairly abrupt transition between trapped and untrapped objects occurs. We define the distance where the transition occurs to be the critical distance. Our orbital plots for Amphitrite can be scaled for application to other asteroids using the same functional dependence as the Hill radius which is r H = ( μ 3 ) 1 3 R ; for Amphitrite r H ∼ 450 R A. We find empirically that initially circular prograde orbits remain bound out to a critical distance of about ( r H)/2 = 225 R A, while initially circular retrograde orbits remain bound out to nearly twice that distance. Particle orbits that start out circular and are inclined with respect to the asteroid's orbital plane have critical distances between these two extremes. Note that our choice of initially circular orbits is arbitrary; different initial conditions would generally lead to different critical distances. This study explores the three-dimensional aspects of stability more thoroughly than previous studies. To first order, particles that are on stable orbits reside within a region that is approximately spherical for angles | θ| < 35° (θ is the latitude in a spherical coordinate system) such that | z| < 285 R A, but is fairly flat and parallel to the x− y plane at z = ±285 R A over the poles. The radius of the spherical surface is roughly 480 R A. These distances would be reduced if the model included the asteroid's orbital eccentricity or other perturbations such as those from Jupiter. This result does not address whether any mechanisms exist to populate such orbits.

On the Mixed Problems for the Transport Equations of Polar Fluids with the Infinitesimal Particle Surface Deformation. 1st Report: Transport Equation.

The transport equation has become an extremely important topic in physics and engineering. In this paper, to describe the motions of polar fluids microscopically, the mixed problems for the transport equations of polar fluids are considered from a mathematical point of view. The transport equations of a polar fluid consisting of two particle components, a solvent and a polar particle, are obtained by the extended Carlemann's method. To consider the infinitesimal particle surface deformation, the component particles are hypothetically divided into the following parts: (i) an open sphere with rigid body, and (ii) a boundary part with elastic body. This modelling is more important from the standpoint of systematizing the statistical theory for cluster phenomena of magnetic fluids and molecular aerothermodynamics.

The Fractal Character of Localizable Measure-Valued Processes, II. Localizable Processes and Backward Trees

Regard a large population of infinitesimal particles (i.e. a measure) in the case, when the particles evolve (i.e. move, branch, die) independently of each other. Those evolutions we will call localizable. In the present part of this paper we answer at first the question about the structure of localizable evolutions, which take place in one step (independent clustering of measures). In localizable evolutions one can follow the path of any particle (and their predecessors) surviving up to some fixed time. The totality of these paths is the backward tere corresponding to this time point. In the case of continuous time localizable Markov evolutions the random backward tree measure is constructed via finite-dimensional distributions using the extension theorem from part I.

The Fractal Character of Localizable Measure‐Valued Processes. I — Random Measures on Product Spaces

AbstractSummary. In part II the evolution of large populations of infinitesimal particles is studied, in which one can follow the path of any particle surviving up to time t. To construct the distribution of the totality of these paths — the so‐called backward tree —, we need a general extension theorem for random measures. The present part gives such a theorem under very natural conditions, and its variante for product spaces.

The Fractal Character of Localizable Measure-Valued Processes, III. Fractal Carrying Sets of Branching Diffusions

Regard a large population of infinitesimal particles (i.e. measures) in the case, when the particles evolve (i.e. move, branch, die) independently of each other. Those evolutions we called localizable. In the present part of this paper we study branching diffusion processes, which result from high frequent alternations of pure macroscopic motion and a pure branching mechanism. The main aim is to compute the HAUSDORFF-BESICOVITCH dimension of random sets carrying the population.

Stability at potential maxima: The L4 and L5 points of the restricted three-body problem

The restricted three-body problem, which treats the motion of an infinitesimal particle due to the gravitational attraction of two massive primaries moving on circular orbits about one another, provides an example of motion which is stable at potential maxima. In a reference frame rotating with the two primaries’ orbital angular velocity, the potential felt by a test particle in the plane of the primaries’ orbit has maxima at the two points which form equilateral triangles with the primaries. This potential is the sum of the gravitational potential and a term representing the position-dependent centrifugal force. The maxima, called L4 and L5, are stable locations for the test particle thanks to the velocity-dependent Coriolis force, which is not incorporated in the potential function. Any energy-dissipating process would tend to drive the test particle away from one of these stable points. These phenomena may run counter to common experience and physical intuition.