Trajectories of particles

Next, we denote the line between the centres of the two particles at the point of closest approach by the unit vector k. In figure A3.1.7 it can also be seen that the vectors -g and g are each other s mirror images in the direction of kin the plane of the trajectory of particles ... 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Z-spray. Z refers to the approximate shape of the trajectory of particles formed by electrospray ionization... 

Under these circumstances, the settling motion of the particles and the axial motion of the Hquid phase are combined to determine the settling trajectory of these particles. The trajectory of particles just reaching the bowl wall near the point of Hquid discharge defines a minimum particle size that starts from an initial radial location and is separated in the centrifuge. A radius ris chosen to divide the Hquid annulus in the bowl into two equal volumes initially containing the same number of particles. Half the particles of size i present in the suspension are separated the other half escape. This is referred to as a 50% cutoff. 

Suppose that the vector field u(f) is a continuous function of the scalar variable t. As t varies, so does u and if u denotes the position vector of a point P, then P moves along a continuous curve in space as t varies. For most of this book we will identify the variable t as time and so we will be interested in the trajectory of particles along curves in space. 

The difficulty will not go away. Wave-particle duality denies the possibility of specifying the location if the linear momentum is known, and so we cannot specify the trajectory of particles. If we know that a particle is here at one instant, we can say nothing about where it will be an instant later The impossibility of knowing the precise position if the linear momentum is known precisely is an aspect of the complementarity of location and momentum—if one property is known the other cannot be known simultaneously. The Heisenberg uncertainty principle, which was formulated by the German scientist Werner Heisenberg in 1927, expresses this complementarity quantitatively. It states that, if the location of a particle is known to within an uncertainty Ax, then the linear momentum, p, parallel to the x-axis can be known simultaneously only to within an uncertainty Ap, where... 

In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

Figure 3.19 The collision trajectories of particles in a shear field, r0 = hQ + 2a...

By eliminating t between equations 3.89, 3.90 and 3.91, a relation between the displacements in the X- and Y-directions is obtained. Equations of this form are useful for calculating the trajectories of particles in size-separation equipment. 

Thus, the trajectories of particles with various diameters and moisture contents, after leaving the impingement zone, can be determined by solving simultaneously Eqs. (6.5). (6.6) and (6.10) with the initial conditions, Eq. (6.7). 

The results calculated for the flight trajectories of particles with various diameters and moisture contents are shown in Fig. 6.22. The figure indicates that the radial gas flow exhibits certain classification effect for particles with various diameters. However, the moisture content of the particle has almost no effect on the flying distance. These theoretical results illustrate that the arrangement of the upper overflow discharging port is totally unfeasible. 

Figure 6.22 Result calculated for the flying trajectories of particles of various diameters and...

Rgure 12.16 (a) The multistage liquid impinger. (Reproduced from G. W. Hallworth. Br. J. Clin. Pbarm., 4, 689 (1977).) (b) The principal features affecting the performance of inertial impactors air flow, the dimensions 5 and T, the diameter of the jet, D, the density, p, and the diameter of the particles, d. The trajectory of particles which are too small to deposit is shown. 

An inevitable consequence of de Broglie s standing-wave description of an electron in an orbit around the nucleus is that the position and momentum of a particle cannot both be known precisely and simultaneously. The momentum of the circular standing wave shown in Figure 4.18 is given exactly hj p = h/, but because the wave is spread uniformly around the circle, we cannot specify the angular position of the electron on the circle at all. We say the angular position is indeterminate because it has no definite value. This conclusion is in stark contrast with the principles of classical physics in which the positions and momenta are all known precisely and the trajectories of particles are well defined. How was this paradox resolved ... 

Critical objects exist in nature. Jean Perrin mentions salted soapy water and Brownian trajectories of particles suspended in a fluid. There are many others, and some of them have aroused the interest of physicists as, for example, the liquid vapour system at the critical point, the magnetic system at the Curie point, and turbulent systems in the inertial range. 

The liquid flow envelops the bubble surface, and the particles are entrained to a greater or a lesser extent by the liquid. The smaller the particles and the less different their density relative to that of the medium, the weaker are the inertia forces acting upon them and the more closely the particle trajectory coincides with the liquid streamlines. Thus, at the same target distance fairly large particles move almost linearly (Fig. 10.1, line 1), while fairly small particles move essentially along the corresponding liquid flow line (line 2). The trajectories of particles of intermediate size are distributed within lines 1 and 2 as the size of particles decreases, the trajectories shift from line 1 to line 2 and the probability of collision decreases. 

Figure 10.2 Continuous lines illustrate the concept of the grazing trajectory of particles, dashed lines indicate the trajectories of the particle at b < b and b > b. ...

Figure 10.3 The influence of the finite dimension of particles in inertia-free flotation on their trajectory in the vicinity of a floating bubble. The liquid flow lines corresponding to target distances b(a,) and are indicated by dashed lines. The continuous lines are characteristic of the deviation of the trajectory of particles from the liquid flow lines under the influence of short-range hydrodynamic interaction...

Fig. 10.13. Diagram of grazing trajectories of particles taking into account inertia forces and SHRI (near hydrodynamic interaction) 1 - grazing trajectory in terms of Sutherland 2 - liquid stream-line coinciding with grazing trajectory 2 - trajectory branching out from stream-line 2 under the effect of inertia force 2" - trajectory branching out from trajectory 2 under the effect of SHRI 0( collision angle 6 - angle characterising the boundary of the part of trajectory controlled by SHRI.

Thus, Eqs. (10.103) and (10.104) describe trajectories of particles Up flowing around the cylinder surface, including the limiting trajectory, at the conditions 1 and h/ttp 1. 

In the general case, trajectories of particles entrained by the flow are determined from the equation of motion, which can be written as... 

In many microfluidic applications, the number concentration of the bioparticles can be high, and the particle-particle interaction may be important when predicting the trajectory of particles. Introduction of many particles in a microchannel is a straightforward extension of the current model. Again with the... 

The motion of large molecules in microfluidic flows is important because the trajectories of particles in shear flows do not always follow the local flow field. Therefore, a knowledge of the fluid dynamics is not sufficient to completely describe the motion of the particles. When a suspended particle does not track the flow, the... 

For central forces the relative motion of the two particles takes place in a fixed plane that contains both the z axis, in the direction of V2-V1, and the trajectory of particle 2 before its center intersects the sphere of radius a (the... 

In flow around an object, the trajectories of particles suspended in the stream will deviate from the streamlines because of particle inertia. Hence, the particles pass through the boundary layer and settle on the object. The deposition and adhesion of particles on the front side of the object is determined in the... 

The effect of an electric field on the processes of deposition and adhesion of particles can be arbitrarily classified as either weak or strong. The effect of an electric field is considered to be weak if it is not able to change the trajectory of particle flight. If the field has a strong effect, the movement of the particles near an obstacle and the particle adhesion will be determined mainly by the presence of the electric field. If a field with a strong effect is present, the particle adhesion will depend on the relationship between the parameters G and H. The lack of any adhesion under the influence of an electric field is shown in Fig. IX.9 as cases c, e, and f adhesion to part of the surface is shown as cases b and d and adhesion across the entire surface of the obstacle is shown as case a. 

Untrapped trajectories Unbounded trajectories of particles that are under the influence of some external force. Examples include paraboUc and hypetboUc trajectories of bodies subject to a cerrtral gravitational force or the imrestrained motion of a circular pendulum. 

---