Grazing Trajectory Research Articles

Inertial hydrodynamic particle–bubble interaction in flotation

The flotation rate of particles of different size and with different contact angles, provided by surface modification of the surface of broken quartz particles, with single monodisperse bubbles was investigated. A monotonic increase of collection efficiency was observed with increasing particle size at high particle hydrophobicity where attachment and stability are at a maximum. At smaller particle hydrophobicity the dependence of collection efficiency on particle diameter for E col is interpreted as a manifestation of particle rebound. At the moment of inertial impact a particle deforms the bubble surface, creates a water layer between particle and bubble and causes the particle to recoil from the bubble surface. These phenomena as well as the influence of centrifugal force were incorporated in a model of particle–bubble inertial hydrodynamic interaction complicated by bubble surface mobility. The purpose of this chapter is an experimental test of the model based on the Generalised Sutherland Equation (GSE), incorporating the angular dependence of attachment efficiency as well as the influence of hydrophobicity in the model. Attachment is impossible at the bubble front pole (angle θ=0) vicinity because of particle rebound and at θ> θ t ( θ t angle of tangency for the particle grazing trajectory) because of the influence of centrifugal forces. With increasing particle diameter and decreasing hydrophobicity, the attachment area confined between angles θ t and θ 1 ( θ 1< θ t) shrinks, causing the collection efficiency E col to decrease and determining the slope of the descending branch of the collection efficiency–particle diameter dependence. This is quantified by means of a modification of the GSE accounting for the shrinkage of the attachment area and attachment at repetitive collision.

Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators

The transition from stable periodic non-impacting motion to impacting motion is analysed for a mechanical oscillator. By using local methods, it is shown that a grazing impact leads to an almost one-dimensional stretching in state space. A condition can then be formulated, such that a grazing trajectory will be stable if the condition is fulfilled. If this is the case, the bifurcation will be continuous and the motion after the bifurcation can be understood by a one-dimensional mapping. This mapping is known to exhibit chaotic solutions as well as arbitrary long stable cycles, depending on parameters.

Surface wake in the random-phase approximation.

The scalar-electric-potential distribution set up by an ion traveling in the vicinity of a plane solid-vacuum interface, that is, the surface-wake potential, is investigated with the specular-reflection model to describe the response of the surface and with the random-phase approximation for the dielectric function of the bulk material. This permits us to address the study of the low-velocity surface wake: the static potential is found to have a dip at the position of the ion; that dip is shifted towards the direction opposite to the velocity vector for velocities smaller than the threshold of creation of plasmons (\ensuremath{\approxeq}1.3${\mathit{v}}_{\mathit{F}}$). Extensive numerical calculations are presented for an ion both inside and outside aluminum. Comparison to the results obtained with the plasmon-pole dielectric function indicates excellent agreement for velocities larger than \ensuremath{\approxeq}1.3${\mathit{v}}_{\mathit{F}}$. On the other side, the possibility of surface-wake riding is suggested, by analogy with bulk-wake riding postulated in the past. In it, the electron would be bound in the first trough of the surface-wake potential set up when the ion describes a grazing trajectory. The main feature introduced by the surface with respect to the bulk consists of allowing the use of ions of higher charge, reducing in this way the relative importance of the electron self-energy, and in addition, giving rise to larger binding energies. When the ion beam is directed along a special direction of an oriented crystal surface, the mechanism of resonant coherent excitation could provide a way for experimentally detecting this phenomenon through the emission of the bound electron with well-defined energy and around a preferential direction.

Competition between static and dynamical effects in adsorption: sticking of Ar on Ar-covered Ru(001)

The sticking of Ar on a cold Ru(001) surface covered with an Ar monolayer is studied as an example of a surface with significant static corrugation of the gas-surface interaction potential, and also substantial dynamical corrugation due to the deformability of the overlayer. Classical molecular dynamics with an empirical pairwise additive potential is used to simulate the system. The results do not fit the usual E cos n Θ dependence of the sticking probability on the incident energy E and angle Θ to the normal (where n = 2 is the smooth surface limit). Instead the angular dependence of sticking is a strong function of E, reflecting a changing competition between static corrugation and dynamical effects. At low energies total energy scaling ( n = 0) is observed, and attributed (with the aid of a simple model) to static corrugation, a generic effect which may well apply to other adsorbate/surface systems. At higher incident energies, dynamical effects during the gas-surface collisional event are responsible for the breakdown of total energy scaling, which results in enhanced sticking for grazing relative to normal trajectories. A grazing trajectory may possess enough in-plane velocity to allow it to move away from the adatom it initially strikes before the adatom recoils from the substrate.

Properties of the optimal trajectories for coplanar, aeroassisted orbital transfer

The optimization of trajectories for coplaner, aeroassisted orbital transfer (AOT) from a high Earth orbit (HEO) to a low Earth orbit (LEO) is examined. In particular, HEO can be a geosynchronous Earth orbit (GEO). It is assumed that the initial and final orbits are circular, that the gravitational field is central and is governed by the inverse square law, and that two impulses are employed, one at HEO exit and one at LEO entry. During the atmospheric pass, the trajectory is controlled via the lift coefficient in such a way that the total characteristic velocity is minimized. First, an ideal optimal trajectory is determined analytically for lift coefficient unbounded. This trajectory is called grazing trajectory, because the atmospheric pass is made by flying at constant altitude along the edge of the atmosphere until the excess velocity is depleted. For the grazing trajectory, the lift coefficient varies in such a way that the lift, the centrifugal force due to the Earth's curvature, the weight, and the Coriolis force due to the Earth's rotation are in static balance. Also, the grazing trajectory minimizes the total characteristic velocity and simultaneously nearly minimizes the peak values of the altitude drop, dynamic pressure, and heating rate. Next, starting from the grazing trajectory results, a real optimal trajectory is determined numerically for the lift coefficient bounded from both below and above. This trajectory is characterized by atmospheric penetration with the smallest possible entry angle, followed by flight at the lift coefficient lower bound. Consistently with the grazing trajectory behavior, the real optimal trajectory minimizes the total characteristic velocity and simultaneously nearly minimizes the peak values of the altitude drop, the dynamic pressure, and the heating rate.

Dynamical deformations in nucleon momentum distributions of colliding ions

We analyze the effects of nucleon momentum distributions of colliding ions arising from excitation of collective vibrational motions. Phonon states are described in RPA and are excited through the action of the mean field of the partner nucleus. The 40Ca + 40Ca system is considered at E A =10 and 44 MeV, around a grazing trajectory. With increasing energy we have clear transition in the dominant collective states, from low-lying modes to the giant quadrupole resonance. Correspondingly we have a noticeable enhancement of dynamical distortions in momentum space, particularly around the Fermi momentum. The importance of nonlinear terms, describing the coupling of different phonon states, is stressed. Some clear effect on energetic particle production is expected.

Debye expansion and JWKB approximation for the elastic scattering of heavy charged particles

By using the Debye expansion of the scatteringS-function and the JWKB solutions of the Schrodinger equation for the spherical-well nuclear potential in the presence of the Coulomb interaction (both for a point charge and for a uniformly charged sphere), it is shown that regions of the complex λ-plane exist in which the multireflection expansion of Knoll and Schaeffer coincides with the JWKB approximation of the Debye expansion. This expansion also provides the correct analytical contribution of all the terms of the multireflection expansion in the regions of the λ-plane where the Knoll and Schaeffer technique cannot be applied, because of the absence of the subdominant exponential. The JWKB approximation gives a reasonable estimate of the exactS-function except in the neighbourhood of the λ value corresponding to the classical grazing trajectory and allows one to interpret the scattering phenomenon in terms of classical trajectories. Furthermore, the Debye expansion allows one to overcome the mathematical difficulties one meets with in the evaluation of the scattering amplitude by the saddle point method in the cases in which the deflection function shows strong oscillations (false rainbows) produced by the superposition of contributions coming from the rays directly reflected by the nuclear surface on the multireflected ones.

Aerosol filtration by fibrous filters at intermediate Reynolds numbers (≤ 100)

The collision efficiencies for spherical aerosol particles impacting on a cylindrical fibre are reported in the intermediate range of Reynolds numbers (≤ 100). The complete Navier-Stokes equations are solved using the successive overrelaxation method to obtain the flow field around the fibre. The calculated flow field is used in computing the particle trajectories in order to obtain the grazing trajectory and thereby the collision efficiency. The effect of the finite size of the particles is considered in terms of the direct interception of the particles on the fibre. Calculations are performed for Reynolds numbers of 1, 10 and 60, Stokes numbers of 1, 2.81, 7.14 and 20, and the particle to fibre radii ratios of 0, 0.05, 0.1 and 0.15. The calculated collision efficiencies for finite Re are found to lie between the reported limiting values for the viscous and the potential flows, as expected. The collision efficiencies for Re = 10 are also compared with those reported by Davies and Peetz (1956) and Landahl and Hermann (1949).

Theoretical predictions of raindrop impaction on a slab type building

The turbulent air flow around a slab type building has been measured by Wise, Sexton and Lillywhite using a scale model in a wind tunnel. Using their data procedures are developed for calculating raindrop trajectories for droplets having radius r in the range 0 ≦ r ≦ 2500 μ with gradient wind speeds up to 26·8 m/s. Grazing trajectories, for fixed droplet size, are obtained and from these the effective collision cross-section of the building is deduced. Employing the droplet size distribution of Best, for a given rainfall intensity, raindrop impaction rates over the building are evaluated. Approximate analytical methods are developed to predict impaction rates and their dependence on the building aspect ratio.

Cloud droplet collision efficiency in electric fields

A mathematical model describing the effects of forces acting on two spherical droplets immersed in a viscous medium is described. The model includes the interaction of the droplets with an externally applied electric field. The collision efficiencies between pairs of droplets ranging in size from 5 to 70 microns in radius are given as results of computations of the grazing trajectories of the smaller droplets relative to the larger drops in electric fields up to 10,000 volts per centimeter. The collision efficiency for a given pair of droplets increases as the applied electric field increases. For example the collision efficiency of a 30 micron drop in relation to a 5 micron droplet increases 34.5 times when the horizontal field is changed from 0 to 3600 volts per centimeter. Results of calculations are given to show how collision efficiencies vary as the orientation of the electric field is varied in relation to the axis of droplet motion. The results show that the maximum and minimum collision efficiencies occur with field orientations of 90 and 42 degrees respectively.

Semiclassical Analysis of the Multinucleon Transfer Reactions between Complex Nuclei

The angular distributions of reactions in which several nucleons are exchanged between complex nuclei are examined in a semiclassical way. It is assumed that since the projectile interacts weakly with the target in a grazing trajectory, it passes through a target along the elastic scattering orbit, while exchanging several nucleons, and finally shows itself as a product of the transfer reaction. The distortion of the Coulomb scattering orbit due to nuclear potential is taken into account. Then for grazing collisions the nuclear and Coulomb forces counteract each other, emitting a particle in a forward direction as observed in experimental data. (auth)