Kinetic model inelastic

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

For elastic collisions, several different kinetic models have been proposed in order to close the Boltzmann hard-sphere collision term (Eq. 6.9). For inelastic collisions (e < 1), one must correctly account for the dependence of the dissipation of granular energy on the value of e. One method for accomplishing this task is to start from the exact (unclosed) collision integral in Eq. (6.68). From the definition of if given in Eq. (6.60), it can be... 

At equilibrium, where the yelocity distribution is Maxwellian, it is straightforward to show that < > = 4 J p/7T, where 0p is the granular temperature. We should note that Eq. (6.109) corresponds to an inelastic Maxwell particle (Maxwell, 1879), and, most importantly, it still contains the exact dependence on tu = (1 + e)/2. We will therefore refer to this kinetic model as the inelastic Maxwell collision model. 

Eor inelastic collisions, the coefficient of restitution will appear explicitly in the kinetic model as seen above in the monodisperse case. We will consider a binary case with inelastic... 

Chapter 6 is devoted to the topic of hard-sphere collision models (and related simpler kinetic models) in the context of QBMM. In particular, the exact source terms for integer moments due to collisions are derived in the case of inelastic binary collisions between two particles with different diameters/masses, and the use of QBMM to overcome the closure problem is illustrated. 

At the intermediate wavelengths no useful analytic forms of solution are known. On the other hand, (138) yields readily to numerical solutions. Such results show a smooth interpolation between characteristic hydrodynamic and free-particle behavior. Notice, however, that in this region it is essential to treat the collisions and molecular flow on equal footing for this reason it would be inappropriate to apply either (154) or (155). Since the intermediate k range is particularly relevant to neutron inelastic scattering studies of liquids and related computer molecular dynamics simulations, the validity of our kinetic model solutions is of interest. 

Rg. 1. Dynamic structure factor 5(ik, various values off = fta .The solid curves are present kinetic model calculations the points are coherent neutron inelastic scattering data/ ... 

In conclusion, we were able to reproduce the optical response of GFP with a novel photodynamical model which includes VER, an energy-dependent ESPT, and an additional decay pathway leading to internal conversion of the protonated chromophore. In particular, the non-exponentiality of the kinetics is traced back to VER dynamics which are slower than the primary ESPT. This might be attributed to the highly rigid tertiary structure of the protein which protects its chromophore from inelastic collisions with the aqueous surroundings. 

When the gas-solid flow in a multiphase system is dominated by the interparticle collisions, the stresses and other dynamic properties of the solid phase can be postulated to be analogous to those of gas molecules. Thus, the kinetic theory of gases is adopted in the modeling of dense gas-solid flows. In this model, it is assumed that collision among particles is the only mechanism for the transport of mass, momentum, and energy of the particles. The energy dissipation due to inelastic collisions is included in the model despite the elastic collision condition dictated by the theory. 

To model the particle velocity fluctuation covariances caused by particle-particle collisions and particle interactions with the interstitial gas phase, the concept of kinetic theory of granular flows is adapted (see chap 4). This theory is based on an analogy between the particles and the molecules of dense gases. The particulate phase is thus represented as a population of identical, smooth and inelastic spheres. In order to predict the form of the transport equations for a granular material the classical framework from the kinetic theory of... 

We consider the orientational dynamics only and ignore the spatial coordinates of interacting rods (an analog of the Maxwell model of binary collisions in kinetic theory of gases, see e.g. [15]). Since the motor residence time on microtubules (about 10 sec) is much smaller than the characteristic time of pattern formation (10 min or more), we model molecular motor - microtubule inelastic interaction as an instantaneous colUsion in which two rods change... 

The simplified-kinetic-theory treatment of reaction rates must be regarded as relatively crude for several reasons. Numerical calculations are usually made in terms of either elastic hard spheres or hard spheres with superposed central attractions or repulsions, although such models of molecular interaction are better known for their mathematical tractability than for their realism. No account is taken of the internal motions of the reactants. The fact that every combination of initial and final states must be characterized by a different reaction cross section is not considered. In fact, the simplified-kinetic-theory treatment is based entirely on classical mechanics. Finally, although reaction cross sections are complicated averages of many inelastic cross sections associated with all possible processes by which reactants in a wide variety of initial states are converted to products in a wide variety of final states, the simplified kinetic theory approximates such cross sections by elastic cross sections appropriate to various transport properties, by cross sections deduced from crystal spacings or thermodynamic properties, or by order-of-magnitude estimates based on scientific experience and intuition. It is apparent, therefore, that the usual collision theory of reaction rates must be considered at best an order-of-magnitude approximation at worst it is an oversimplification that may be in error in principle as well as in detail. 

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