Post-collision velocities

As usual, f denotes the distribution function / evaluated at the post collision velocity v of the collision process v —>- v. ... 

In this case no detailed collision kinetics are involved. The collision parameters rnm and the parameters in the Maxwellian post-collision velocity distribution f m are derived from experimentally determined gas viscosity or diffusivity, and the collisional invariants, respectively. Usually this term is negligible in present experiments, but exceptions exist [16]. In particular for ITER, and the high collisionality there, these terms are expected to become more relevant. However, due to the BGK-approximations made, their implementation into the models does not require further discussion here. 

The nine-fold integration for the gain term is over both pre-collision velocities and over the second (all but the first) post-collision velocity. Both pre-collision states are folded with their corresponding distribution function. 

Another method that introduces a very simplified dynamics is the Multi-Particle Collision Model (or Stochastic Rotation Model) [130]. Like DSMC particle positions and velocities are continuous variables and the system is divided into cells for the purpose of carrying out collisions. Rotation operators, chosen at random from a set of rotation operators, are assigned to each cell. The velocity of each particle in the cell, relative to the center of mass velocity of the cell, is rotated with the cell rotation operator. After rotation the center of mass velocity is added back to yield the post-collision velocity. The dynamics consists of free streaming and multi-particle collisions. This mesoscopic dynamics conserves mass, momentum and energy. The dynamics may be combined with full MD for embedded solutes [131] to study a variety of problems such as polymer, colloid and reaction dynamics. 

Execution of the method requires the physical domain to be divided into a distribution of conqiutational cells. The cells provide geometric boundaries and volumes, which are used to sample macroscopic properties. Also, only molecules located within the same cell, at a given time, are allowed to collide. The DSMC simulation proceeds from a set of prescribed initial condition. The molecules randomly populate the computational domain. These simulated molecules are assigned random velocities, usually based on the equilibrium distribution. The simulated representative particles move for a certain time step. This molecule motion is modeled deterministically. This process enforces the boundary conditions. With the simulated particles being appropriately indexed, the molecular collision process can be performed. The collision process is modeled statistically, which is different from deterministic simulation methods such as the molecular dynamics methods. In general, only particles within the same computational cell are considered to be possible collision partners. Mthin each cell, collision pairs are selected randomly and a representative set of collisions is performed. The post-collision velocities are determined. There are several... 

For finite differences in the scattering velocity of NO, we have a relationship between the post-collision velocity spread, Avfjoi nd the velocity spread in the NO molecules before collision ... 

Our typical NO-Ar system has 2 atm of Ar backing both valves with one doped with 5% NO. Two laser beams are used to ionize NO, one laser beam is resonant with a transition in the (0,0) band of the A-X system near 226 nm, thereby selecting a particular rotational state which is then ionized near threshold with a 300 nm laser beam. This technique is known as (1 + l )REMPI. Images similar to Fig. 8.2 are taken for each product NO rotational state. By measuring the velocity profile of the molecules scattered in the direction of the laboratory origin, the crossing point of the two beams, we can measure the post collision velocity distributions for the side scattered molecules as a function of rotational state, see Fig. 8.4. 

In this equation, r is a unit vector separating the particles 1 and 2, so that (r v)u(r v) is the velocity of approach of the particles together and is zero if the particles are separating. The first delta function describes the post -collision motion, which leads to a gain of the density f2(vl5 v2,rlT r2, f)> while the second delta function term describes the pre -collison motion which leads to a loss of the density f2 (v, v2, r , r2, f), and (a = 1, b = 2)... 

N number of particles, a range of inter-particle forces). If the collision process is binary and non-reactive (post-collision species i, j remain the same as pre-collision species i and j), these indices do not appear in the collision integral, and we can adopt the standard notations of a binary collision turning the two velocities v, v into v, v, with the corresponding abbreviations for the distribution functions /, /i and /, /, respectively. Let W(v, vi —> v, v ) denote the probability for such a transition, then... 

In our first experiments on the kinematic cooling process, we chose to scatter a molecular beam of NO from an atomic beam of argon. The energetics were selected to provide a velocity vector cancellation that results in the the post collision NO7 5 (NO in the j = 7.5 rotational state) being stationary in the laboratory frame. We will show below why this quantum state of NO is essentially stationary in the laboratory reference frame,... 

The recoil velocity of the NO molecule is equal in magnitude to the velocity of the system s center of mass. The post collision energy of the NO molecule... 

The determination of post-agglomeration velocities of the resulting agglomerate is based on the laws of a perfectly inelastic collision considering conservation of both translational and angular momentum. Therefore, the new translational... 

In fact (98) is model independent to order u, since only the linear term in u c, contributes. To close the hydrodynamic equations for the mass and momentum densities [(74) and (75)] we need expressions for the pre-collision and post-collision momentum fluxes, and n . From (76) we can obtain an expression for in terms of the velocity gradient. 

Equations (135)-(138) can be solved to relate the pre- and post-collision stresses to the velocity gradient ... 

The starting point of the statistical mechanical development in [102] is the notion of a GLG. We define V (r,t) in (30) as the number of particles with velocity C at site r at time t. In contrast with the standard LB model, v, is a (positive) integer in contrast with lattice-gas models, v, > 1. The state at a particular lattice site, v(r,t), is modified by the collision process, subject to the constraints of mass and momentum conservation the post-collision state, v (r,t), is then propagated to the neighboring sites (30). 

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