Closure approximations

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

Standard closure approximations may be used for the wall-PRISM equation. For example, the PY closure is... 

O Brien, E. E. (1966). Closure approximations applied to stochastically distributed second-order reactions. The Physics of Fluids 9, 1561-1565. 

In Equation 12.13, N is the number density of molecules in the beam of radiation (and is thus inversely proportional to the molar volume, Vm), and o is the permittivity of the vacuum. A useful and widely employed method to evaluate the sum in Equation 12.13 leads via the closure approximation to a one-term equation commonly known as the dispersion relation,... 

The source term describes the formation of the m + 1 reactants at time t — t0 with these initial positions at rA° and r ls r 2... etc. The integral in eqn. (220) describe the reduction in the density of the quencher and fluorophor distribution if the quenching process is very slow. Unfortunately, within the integral is n( r0, f°), which is the very density that is sought. As a first approximation, ne<1 could be used. Wilemski and Fixman suggested a more satisfactory (closure) approximation... 

The second high-frequency term involves a sum over all discrete states and an integration over the continuum states the difficulties involved have been outlined before. Little is known about the continuum states, but what few calculations there are for simple systems92 suggest that they may be at least as important as the discrete states. For this reason early calculations were done in the closure approximation, notably by Van Vleck in the 1930 s. The difficulties of calculating xHF have been reviewed by Weltner.93 Experimentally xHF may be obtained from rotational magnetic moments. For linear molecules these can be obtained from molecular-beam experiments, which also measure the anisotropy x Xi- directly. The anisotropies may also be derived from crystal data, the Cotton-Mouton effect and, recently, Zeeman microwave studies principally by Flygare et al.9i... 

This equation is not closed in the unknown, second-moment tensor due to the presence of the (RRRR) term. One solution procedure often used, is to invoke a closure approximation, of which the form (RRRR) = (RR) (RR) is the simplest. This approximation, however, is only quantitatively accurate in the limit of nearly perfect orientation of the dumbbells, but is able to offer correct, qualitative responses for many purposes. 

Another closure approximation relies on the use of higher order moments, but this is almost a deadlock as for only two reacting components, a 13-equation model is required (3). 

Martin101 has calculated the long-range dipole moment for systems of three identical atoms. For the H H H system he has used a pseudo-spectral perturbation method, while for He-"He "He he has used the closure approximation. These quantities could be experimentally accessible via a pressure-induced i.r. absorption whose intensity varied as the cube of the pressure. 

If Eq. (11-3) is multiplied by uu and integrated over the unit sphere, one obtains an evolution equation for the second moment tensor S (Doi 1980 Doi and Edwards 1986). In this evolution equation, the fourth moment tensor (uuuu) appears, but no higher moments, if one uses the Maier-Saupe potential to describe the nematic interactions. Doi suggested using a closure approximation, in which (uuuu) is replaced by (uu) (uu), thereby yielding a closed-form equation for S, namely. 

A closure approximation must also be invoked to express the rotary diffusivity Dr in terms of S. Doi chose the following approximation ... 

Figure 11.17 Dimensionless shear stress and first normal stress difference as functions of dimensionless shear rate predicted by the Doi with the closure approximation for the fourth moment, (uuun) — (uu) (uu) and U — 2U = 6.

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