Isotropy local

The final result is very well known in experimental studies and is usually used to evaluate a32 from known and s values. Other studies in this field include the work by Shinnar and Church (S7) who used the Kolmogoroff theory of local isotropy to predict particle size in agitated dispersions, and an analysis by Levich (L3) on the breakup of bubbles. Levich derived an expression for the critical bubble radius (the radius at which breakup begins) ... 

A second simplihcation results from introducing the Born-Oppenheimer separation of electronic and nuclear motions for convenience, the latter is most often considered to be classical. Each excited electronic state of the molecule can then be considered as a distinct molecular species, and the laser-excited system can be viewed as a mixture of them. The local structure of such a system is generally described in terms of atom-atom distribution functions t) [22, 24, 25]. These functions are proportional to the probability of Ending the nuclei p and v at the distance r at time t. Building this information into Eq. (4) and considering the isotropy of a liquid system simplifies the theory considerably. 

At high Reynolds numbers, the scalar gradient and the velocity gradient will be uncorrelated due to local isotropy. Thus, Gf will be negligible. Likewise, the mean-scalar-curvature term Cf, defined by... 

In addition to the Reynolds number, local isotropy for the scalar field will depend on the Schmidt number Sc must be large enough to allow for a inertial-convective sub-range in the scalar energy spectmm. 

Combining this expression with (3.153), and using the continuity equation for (U), yields S"p = 0. Note that local isotropy does not exclude the possibility that scalar gradient vectors are collinear as is often the case when l = Vp. 

The third constraint has to do with local isotropy of the scalar and velocity fields (Fox 1996b Pope 1998). At high Reynolds numbers, the small scales of the velocity and scalar... 

Invoking local isotropy, the first term on the right-hand side is null. The second term corresponds to molecular transport in real space, which is negligible at high Reynolds numbers. Thus, local isotropy requires... 

Note, however, that in the presence of a mean scalar gradient the local isotropy condition is known to be incorrect (see Warhaft (2000) for a review of this topic). Although most molecular mixing models do not account for it, the third constraint can be modified to... 

HI) The molecular mixing model must yield the correct local scalar isotropy. 

III) Velocity and scalar gradients must be uncorrelated (III ) Local scalar isotropy must be correct... 

As this chapter is primarily concerned with single-drop performance, it seems best to omit consideration of drop sizes in highly turbulent liquid fields. The work of Shinnar and Church (S7), utilizing Kolmogo-roff s hypothesis of local isotropy, seems to bear excellent promise from a fundamental viewpoint. Correlating equations for predicting drop size in stirred tanks and mixers have been given by Treybal (T3). 

The work by Shinnar and Church (S7) on the use of the concept of local isotropy should be pursued and extended. 

One important difference between the solid and liquid phases is that the isotropy of the liquid is replaced by the anisotropy of the solid. The geometrical correlations that exist in a solid lattice may be expected to play an important part when reaction centers are localized and can react with their neighboring environment. In this section we point out some of these effects in the radiolysis of solid n-hexane the trans/cis hexene yields, the isomeric dimer pattern, and the fragmentation processes. 

Accepting Kolmogoroff s theory of local isotropy, they find for the critical drop size the relation... 

The relation between the local held and the external held depends on the symmetry of the lathee structure of the solid. In practical applications almost exclusively isotropy is assumed, in which case one has E oc = (e + 2)/3 E. One can solve, then, for e (co) with the result... 

Herring, invoking arguments of local isotropy and using kinetic theory as a guide, propose using Eq. (26) with... 

The eddy viscosity as given earlier for bubble columns has also been correlated (M27) according to Kolmogoroflf s local-isotropy concept, to... 

Approximately stated Kolmogorov s hypothesis of local isotropy yields ([83] see also [121], p. 184) At sufficiently high Reynolds number, the small scale turbulent motions are statistically isotropic. 

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