Instantaneous addition, simulation

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. 

We have seen that the instantaneous faradaic current at an electrode is related to surface concentrations and charge transfer rate constants, and exponentially to the difference of the electrode potential from the E° of the electrochemical couple. This is represented in Figure 5.1c by Zf. With very few exceptions, this leads to intractable nonlinear differential equations. These systems have no closed form solutions and are treatable only by numerical integrations or numerical simulations (e.g., cyclic voltammetry). In addition, the double-layer capacitance itself is also nonlinear with respect to potential. 

In percussive sounds, such as a bell, gong, drum and other nonperiodic sounds, spectral components are typically aharmonic and can be simulated by forming an irrational relation between ooc and b)m (e.g., (Om = /2C0c). In addition, the envelope is characterized by a sharp (almost instantaneous) attack and rapid decay, and the bandwidth moves from wide to narrow. Bell-like sounds, for example, can be made by making the modulation index proportional to an amplitude envelope which has exponential decay. For a drum-like sound, the envelope decay is even more rapid than the bell, and also has a quick overshoot giving a reduced initial bandwidth, followed by a widening and then narrowing of the bandwidth. 

This chapter focused on the sensitivity of instantaneous LES fields to multiple parameters such as number of processors, initial condition, time step, changes in addition ordering of cell residuals for cell vertex methods. The baseline simulation used for the tests was a fully developed turbulent channel. The conclusions are the following ... 

This paper proposes a phenomenological analysis, based on laboratory experimental work, of the effects of adsorption properties on pol3nner slug propagation. The adsorption properties studied include kinetic aspects, i.e. instantaneous adsorption, reorganization of macromolecules inside adsorbed layer, exchanges between free and adsorbed polymer, desorption as well as properties at thermodynamic equilibrium which can be described by a partially reversible adsorption isotherm. The conditions for hydro-dynamic retention are also discussed. In addition, an analysis of the effects of polymer polydispersity on each of these adsorption phenomena shows that these effects cannot be neglected in a predictive simulator. 

Figure 16 Chain configurations from the KG MD simulations with additional bending energy /tb=3. (a) and (c) show 27 instantaneous and mean path configurations, respectively (b) shows mean paths over shorter time for soft MD model and (d) shows KG MD with shorter chain fl/= 128 but over a time interval of order of reptation time. First and last mean paths are shown by larger spheres. The total trajectory time not and the averaging time for the mean paths lav are shown in the table. End monomers are shown by larger spheres, as well as the middle monomer in (a)-(c).

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