Instantaneous momenta

The components of these forces are of course determined by the magnitude of the instantaneous momenta of the species B and C at the moment of collision. F is proportional to the component of relative momentum of the pair B and C along the line of centers BC. A more complex case, in which we assume rough spheres, can be taken. In this case the components of momentum tangent to the spheres B and C at contact also 2ontribute to changes in rotation and vibration. 

Subtract the time-averaged momentum equation from the instantaneous momentum equations. 

Differentiate the instantaneous momentum equations with respect to xi. 

The Brownian force is the well-known force that becomes important in the case of very small particles suspended in a continuous phase. The Brownian force can be defined as the instantaneous momentum exchange due to collisions between the molecules of the continuous phase with a suspended particle. When the particle is small enough to perceive the molecular nature (and motion) of the continuous phase (i.e. when the particle Knudsen number is large enough), it exhibits a random motion, which was observed as early as 2000 years ago by the Roman Lucretius. The Brownian force is typically described as a stochastic process (Gardiner, 2004), and it can be modeled as a Wiener process " ... 

It is now well established that in a turbulent boundary layer flow the transport of momentum to the wall is intermittent. The events responsible for this locally large instantaneous momentum transport are called sweeps and ejections depending on whether the transport is towards or away from the wall. A variety of models describing the flow field in such events as well as their overall organisation in the main flow have been proposed. Visualisation of ejections show that they are associated with a jet-like and a vortex-like flow field. The relation between the strengthes of these fields are not well known since in order to measure the vorticity in such events,one would need probes which are not available to-day. 

When a small particle is suspended in a fluid, it is subjected to the impact gas or liquid molecules. For ultrafine (nano) particles, the instantaneous momentum imparted to the particle varies at random, which causes the particle to move on an erratic path now known as Brownian motion. Figure 19 illustrates the Brownian motion process. 

The collisional pressure tensor represents the instantaneous momentum transfer at binary particle collisions, over the distance separating the centers of the two colliding particles. The pressure tensor closure is derived based on an extension of the kinetic theory of dense gases. The collisional pressure tensor is thus the second out of the two pressure tensor components that is calculated by use of the KTGF. 

The LB fluid and the solid particles are coupled by an instantaneous momentum transfer at the half-time step, which is therefore presumed to be conservative ... 

The assumption of instantaneous momentum exchange brings forward an important limitation of the DSMC approach the real duration of a particle-particle collision must be small relative to the mean free time between two consecutive collisions. Therefore, the method should not be applied to very dense systems in which prolonged particle—particle contacts take place such as in hopper flows or dead zones in fluidized systems. On the other hand, the method is extremely efficient for dilute and semi-dilute systems such as pneumatically conveyed powders, riser flows, and liquid sprays. 

After impact the first bead assumes a velocity 2v, due to its rigid elastic response. This is the instantaneous particle velocity that the bead acquires. The first bead travels across the gap d and impacts the second bead. The only way by which momentum and energy can be simultaneously conserved is for the first bead to come to rest at the instant the second bead acquires a velocity... 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

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. 

The phase space for three-dimensional motion of a single particle is defined in terms of three cartesian position coordinates and the three conjugate momentum coordinates. A point in this six-dimensional space defines the instantaneous position and momentum and hence the state of the particle. An elemental hypothetical volume in six-dimensional phase space dpxd Pydpzdqxdqydqz, is called an element, in units of (joule-sec)3. For a system of N such particles, the instantaneous states of all the particles, and hence the state of the system of particles, can be represented by N points in the six-dimensional space. This so-called /r-space, provides a convenient description of a particle system with weak interaction. If the particles of a system are all distinguishable it is possible to construct a 6,/V-dimensional phase space (3N position coordinates and 3N conjugate momenta). This type of phase space is called a E-space. A single point in this space defines the instantaneous state of the system of particles. For / degrees of freedom there are 2/ coordinates in /i-space and 2Nf coordinates in the T space. 

Release momentum. For jet releases, the amount of air entrained in an unobstructed jet is proportional to the jet velocity. Depending on the orientation of the jet relative to nearby obstructions, the momentum of a jet can be dissipated without significant air entrainment. The degree of initial air entrainment can be an important determinant of the hazard extent, particularly for flammable hazards. It would be (possibly overly) conservative to assume the source momentum is dissipated without air dilution. Explosive releases are high-momentum, instantaneous releases. For explosive releases, a rough first approximation is to assume that the mass of contaminant in the explosion is mixed with 10 times that mass of air. 

In the last term of equation 1.23, the averages are taken over the fixed volume V of the section. This term is simply the rate of change of the momentum of the fluid instantaneously contained in the section. It is clear that accumulation of momentum may occur with unsteady flow even if the flow is incompressible. In general, the mass flow rates M and M2 into and out of the section need not be equal but, by continuity, they must be equal for incompressible or steady compressible flow. 

Writing the instantaneous velocity components vx, vy as the sums of the mean values and fluctuations, and taking the time average gives the mean momentum flux as ... 

In the instantaneous approximation, see [26], the formfactor of the nonlocal interaction depends on the three-momentum only and the Matsubara summation in Eq. (7) can be performed analytically using standard methods. 

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