Particle trajectory crossing example

For gas-particle flows, fhe mosf obvious manifestation of non-equilibrium behavior is particle trajectory crossing (PTC) at KUp = oo (i.e. no collisions). PTC occurs in the very-dilute-flow regime ( p c 1) and is most easily identified when fhe granular femper-afure is null (Map = c ). An example of PTC is shown in Figure 1.4. The panel on fhe... 

Figure 8.1. An example of particle trajectory crossing in a 2D domain. The fully developed particle number density moo is shown together with the mean particle velocity vector (mio/moo, moi/moo). This example was computed using QBMM with a KBFVM (Yuan Fox, 2011) and realizable second-order fluxes (Vikas et al, 2011a). Note that the mean particle velocity is not defined in regions where the density is exactly zero. Owing to a smali amount of numerical diffusion, the density is very small, but nonzero, outside of the two jets.

If one introduces the panicles into the flow with a negligible initial velocity, Vpo, the particle trajectories are straight lines if the flow around the particles is laminar. One therefore obtains a fan of settling rate dependent panicle trajectories. Cross flow systems, contrary to the classical sedimentation methods.therefore already have the substantial advantage of permitting the measurement of individual settling rate classes. Since these settling rate classes are represented by their trajectories, the amounts present in each class can be determined independently, for example, from the particles deposited and collected at the wall opposite the particle entrance. 

If one introduces particles into a gas or a liquid flow of, for k example, rectangular cross section, the particles are fanned out in the flow. With the system as schematically shown in Fig. 12., the particles enter the flow with a high initial velocity, VpQ, and travel perpendicular to the flow. They are continuously decelerated within the flow and finally come to a halt. The distance travelled is the stop distance, Sq, already explained and shown in Fig. 10. One again obtains a fan of settling rate dependent particle trajectories. Due to the fact, that the fluid and the initial particle velocities are comparatively high, in gas, for example, both, v and VpQ, should be higher than 20 m/s, the residence times of the particles within the classification zone are small. The fan of particles is again Fig.l2 The principle of cross flow classifier stationary and can be used in the technical classification of... 

Prof. Troe has presented to us the capture cross sections for two colliding particles, for example, an induced dipole with a permanent dipole interacting via the potential V(r,0) = ctq/2rA - ocos 0/r2 (see Recent Advances in Statistical Adiabatic Channel Calculations of State-Specific Dissociation Dynamics, this volume). The results have been evaluated using classical trajectories or SAC theory. But quantum mechanically, a colliding pair of an induced dipole and a permanent dipole could never be captured because ultimately they have to dissociate after forming some sort of a collision complex. I would therefore like to ask for the definition of the capture cross section. ... 

For an isolated polymer chain, the problem is purely geometrical. Indeed, the spatial shape of an ideal chain resembles the path of a randomly wandering Brownian particle (see Chapter 6). What new features will the shape of the chain acquire, if we allow for the excluded volume Clearly, since the private space of each monomer is not available to the rest, the chain cannot possibly cross itself at any stage. This sort of behavior can be described as self-avoiding. For example, if there were an equivalent Brownian particle, it would not be allowed to cross its own track. A two-dimensional version of such a trajectory is sketched in Figure 8.2. Thus, we have made it a purely geometrical problem of self-avoiding random walks. 

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