Fictitious particle

In the LB technique, the fluid to be simulated consists of a large set of fictitious particles. Essentially, the LB technique boils down to tracking a collection of these fictitious particles residing on a regular lattice. A typical lattice that is commonly used for the effective simulation of the NS equations (Somers, 1993) is a 3-D projection of a 4-D face-centred hypercube. This projected lattice has 18 velocity directions. Every time step, the particles move synchronously along these directions to neighboring lattice sites where they collide. The collisions at the lattice sites have to conserve mass and momentum and obey the so-called collision operator comprising a set of collision rules. The characteristic features of the LB technique are the distribution of particle densities over the various directions, the lattice velocities, and the collision rules. 

Stokes s law and the equations developed from it apply to spherical particles only, but the dispersed units in systems of actual interest often fail to meet this shape requirement. Equation (12) is sometimes used in these cases anyway. The lack of compliance of the system to the model is acknowledged by labeling the mass, calculated by Equation (12), as the mass of an equivalent sphere. As the name implies, this is a fictitious particle with the same density as the unsolvated particle that settles with the same velocity as the experimental system. If the actual settling particle is an unsolvated polyhedron, the equivalent sphere may be a fairly good model for it, and the mass of the equivalent sphere may be a reasonable approximation to the actual mass of the particle. The approximation clearly becomes poorer if the particle is asymmetrical, solvated, or both. Characterization of dispersed particles by their mass as equivalent spheres at least has the advantage of requiring only one experimental observation, the sedimentation rate, of the system. We see in sections below that the equivalent sphere calculations still play a useful role, even in systems for which supplementary diffusion studies have also been conducted. 

When a bond is broken and a free electron is removed, a hole or electron vacancy is created. When this vacancy is filled by an electron from a neighboring bond, the result is a hole moving in a direction opposite to that of the electron. Thus the hole may be considered as a (fictitious) particle with... 

The relative motion can now be represented by the Brownian motion of a fictitious particle of reduced mass mr(=mp/2), with the friction coefficient f, experiencing the same force F. Consequently,... 

As pointed out earlier, the Fokker-Plank equation [18] describes the motion of the fictitious particle only outside a sphere of radius 7 s + Xr. where RS(=2RV) is the radius of the sphere of influence. The motion of the fictitious particle in the region of thickness... 

Fkj. 1. Different regions for the motion of the fictitious particle of reduced mass. 

Region I. Since the motion of the fictitious particle in this region takes place in the absence of the overall interaction potential between particles (see Fig. 1), the particle distribution function is governed by the steady-state Fokker-PlanV equation... 

Therefore, only those fictitious particles whose velocity vector at the interface of Regions I and II lies in the cone of angle (jt — wi) (see Fig. 1) will enter into the region III. In other words, for a particle to enter into the Region III,... 

The distribution function / is therefore constant along the trajectory of the fictitious particle. 

Since the fictitious particle moves in a central force field described by a spherically symmetric potential function U(r), its angular momentum is conserved. Therefore, the motion of the fictitious particle will be in a plane defined by the velocity and the radius vectors. The Lagrangian may then be conveniently expressed in polar coordinates as... 

In the above equations, e, n, and u are the internal energy, the number density, and the average velocity of the fictitious particle, re-... 

Because the motion of the fictitious particle is considered to be collisionless, in the Regions II and III, the distribution function of the incoming particles is conserved along their trajectories, i.e.,... 

Here, c, the effective particle Knudsen number, is the ratio of the relaxation time f 1 for Brownian motion of the particle to the time needed for the fictitious particle to traverse the distance Rs with its average velocity, 6 is the dimensionless surface temperature, and is the dimensionless depth of the potential well. 

We have thus far obtained the distribution functions for the incoming and reflected particles of reduced mass in different regions in terms of the unknown constants A and B and the unknown dimensionless surface temperature 8. The surface temperature was then related to the internal energy and the number density of the fictitious particle at the reflecting surface. Now, in order to determine the constants A and B, wc must specify the boundary conditions for the mass and the energy flux at the sphere of influence. 

Since the motion of the fictitious particle in Regions II and III is assumed to be collisionless, a reflected particle of reduced mass /Mr with kinetic cncigy less than the depth of the potential well 4>0 will be captured by the potential well. Therefore, the boundary condition for the mass flux is given by... 

As already pointed out earlier, the potential well for the large particles is very deep. In addition, the decay length of the interaction potential is small compared to the radius of the particles. Consequently, the interaction potential can be replaced by a sink at the sphere of influence. Therefore, Regions III and IV in Fig. 1 will collapse into a sink at the sphere of influence and only the motion of the fictitious particle in Regions I and II need to be considered. The equations governing the motion of the fictitious particle in these regions are the same as those discussed earlier in Section 2. 

The motion of the fictitious particle (denoted hereafter as particle ) of reduced mass can be divided into three regions, as shown in Fig. 1. In region I, the particle experiences negligible interaction force since the interaction potential in this region is negli-... 

The parameter nr in Eq. [13] is determined from the condition that every pair of colliding particles are reflected, i.e., the net flux of the fictitious particles at the reflecting spherical surface is zero. Therefore,... 

Using the zero flux condition [13b] and the boundary condition [13a], the parameter r can be expressed in terms of the flux of the fictitious particles incident upon the reflecting surface A, and the surface temperature Tr as,... 

In view of the fact that the correlation function for the random force, as given by Eq. [16], is a Dirac 8 function, the strict Langevin equation (Eq. [15]) is not amenable to computer simulation. In order to circumvent the above difficulty, it is convenient to describe the motion of the fictitious particle by the generalized Langevin equation. The generalized Langevin equation, which can be derived from the Liouville equation (11), along with the supplementary conditions are... 

The random Brownian motion of the fictitious particle of reduced mass m, in terms of its momentum p = mrv is described by the Langevin equation ... 

In order to simulate the motion of the fictitious particle by an equation which constitutes a close approximation of the strict Langevin equation [15], we choose in the generalized Langevin equation ts 4 tp (=l/ )... 

Let us consider the fictitious particle of reduced mass mr for which the location and the momentum at time t = nts are x and p , respectively, and the random force BlJ ] has a constant value in the time interval tn < t < tn+1. The external force acting on the fictitious particle is given by,... 

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