Pair correlation function velocity

Kuwabara function for particle deposition in porous media particle radial distribution function pair correlation function for particles of types a and / constant appearing in the pair correlation function velocity parameters used in conjunction with EQMOM mean velocity difference used to approximate vi - Vp I... 

The pair correlation function of the velocities and the pair correlation functions of some time derivatives of the velocity are sometimes taken into account.75 However, the validity of this description in the nonadiabaticity regions also has to be proved. The dynamic description or the description using the differentiable random process is more rigorous in this region.76... 

In order to evaluate the collisional integrals Pc, qc, and y, explicitly, it is important to know the specific form of the pair distribution function /(2)(vi, ri V2, ry. t). The pair distribution function /(2) may be related to the single-particle velocity distribution function f by introducing a configurational pair-correlation function g(ri, r2). In the following, we first introduce the distribution functions and then derive the expression of /(2) in terms of f by assuming /(1) is Maxwellian and particles are nearly elastic (i.e., 1 — e 1). 

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. 

The next step is to provide a closure for the pair correlation function appearing in the collision source and collisional-flux terms. For moderately dense flows, the collision frequency for finite-size particles is known to be larger than that found using the Boltzmann Stofizahlansatz (Carnahan Starling, 1969 Enksog, 1921). In order to account for this effect, the pair correlation function can be modeled as the product of two single-particle velocity distribution functions and a radial distribution function ... 

Note that this assumption simply transforms the problem of modeling the pair correlation function into the new problem of modeling o-The usual model for go assumes that the radial distribution function depends neither explicitly on the collision angle (i.e. on X12) nor explicitly on x. The former amounts to assuming that the particle with velocity V2 has no preferential spatial direction relative to the particle with velocity vi. The radial distribution function can then be modeled as a function of the disperse-phase volume fraction. For example, a typical model is (Carnahan Starling, 1969)... 

In general, gap depends on the volume fractions of each particle type and on the particle diameters. However, it can also depend on other moments of the velocity distribution function. For example, if the mean particle velocities Uq. and Vp are very different, one could expect that the collision frequency would be higher on the upstream side of the slower particle type. The unit vector Xi2 denotes the relative positions of the particle centers at collision. If we then consider the direction relative to the mean velocity difference, (Uq, - U ) xi2, we can model the dependence of the pair correlation function on the mean velocity difference as °... 

Here, ko and k are the neutron velocities before and after the scattering process. Looking at Eq. (2.75) immediately reveals the relation to G(r, t). The double differential scattering cross-section is directly related to the Fourier transform of the van Hove pair correlation function, which is given by... 

In a semi-dilute solution the pair correlation function is given by equation (11) and the velocity correlation function is given by the screened Oseen tensor... 

In homogeneous turbulence, the velocity spectrum tensor is related to the spatial correlation function defined in (2.20) through the following Fourier transform pair ... 

Figure 1 Structural (left column) and dynamical (right column) properties of the systems investigated. Upper left centre-of-mass radial pair distribution function gooo( ) lower left spherical harmonic expansion coefficient g2oo(r) upper right angular velocity correlation function lower right orientational correlation function. Dotted lines CO, 80 K, 1 bar thin lines CS2, 293 K, 1 bar thick lines CS2, 293 K, 10 kbar.

Before concluding this section on the implications of the pair kinetic theory for configuration space descriptions, we show that the kinetic equation may also be used to obtain the kinetic theory result for the rate kernel. This can be accomplished by projecting out the position and velocity dependence of the pair phase-space correlation function ab,ab( 2> 1 2 /) to obtain an equation for... 

The ( ) denotes the ensemble average, V is the volume, T is the temperature, c, and are the total energy and velocity of the ith particle, is the force between particles / and j, and cp is the pair potential function. For a two-phase system, the mean partial enthalpy also comes into the equation of J (Eapen et al. 2007). Long runs are necessary to achieve reliable values, because a single value of J is obtained in one time step. One is also faced with the problem of integrating a correlation function that may have a long time tail (Allen and Tildesley 1989 Rapaport 1995). 

In many respects, at a superficial level, the theory for the chemical reaction problem is much simpler than for the velocity autocorrelation function. The simplifications arise because we are now dealing with a scalar transport phenomenon, and it is the diffusive modes of the solute molecules that are coupled. In the case of the velocity autocorrelation function, the coupling of the test particle motion to the collective fluid fields (e.g., the viscous mode) must be taken into account. At a deeper level, of course, the same effects must enter into the description of the reaction problem, and one is faced with the problem of the microscopic treatment of the correlated motion of a pair of molecules that may react. In the following sections, we attempt to clarify and expand on these parallels. 

Although the analysis in terms of the propagators for independent motion gL is convenient for displaying the content of the kinetic theory expression for the rate kernel, calculations based on (10.4), which contains the propagator for the correlated motion of the AB pair, are probably more convenient to carry out. In kinetic theory, such rate kernel expressions are usually evaluated by projections onto basis functions in velocity space. (We carry out such a calculation in Section X.B). Hence the problem reduces to calculation of matrix elements of (coupled AB motion in a nonreactive system) and subsequent summation of the series. This emphasizes the point that a knowledge of the correlated motion of a pair of molecules for short distance and time scales is crucial for an understanding of the dynamic processes that contribute to the rate kernel. 

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