Self similarity behavior solution

Chapter 4 deals with methods for the solution of population balance equations. It also probes into Monte Carlo simulation techniques. In Chapter 5, the self-similarity behavior of solutions to the population balance equations is considered with various examples. The subject of inverse problems for the identification of population balance models from experimental data on dynamic particle distributions is treated in Chapter 6. The exploitation of self-similar solutions in inverting experimental data is of particular interest. 

Wang (1966) has considered the sum kernel (a(x, y) = x y) and the product kernel (a(x, y) = xy) for their self-similar forms and found them to be generalized functions, viz., Dirac delta functions thus ruling out the possibility of observable self-similar behavior. However, this conclusion was clearly in error, as it is now known that both the sum and product kernels have the respective self-similar solutions... 

When the silver nanocrystals are organized in a 2D superlattice, the plasmon peak is shifted toward an energy lower than that obtained in solution (Fig. 6). The covered support is washed with hexane, and the nanoparticles are dispersed again in the solvent. The absorption spectrum of the latter solution is similar to that used to cover the support (free particles in hexane). This clearly indicates that the shift in the absorption spectrum of nanosized silver particles is due to their self-organization on the support. The bandwidth of the plasmon peak (1.3 eV) obtained after deposition is larger than that in solution (0.9 eV). This can be attributed to a change in the dielectric constant of the composite medium. Similar behavior is observed for various nanocrystal sizes (from 3 to 8 nm). 

In order to understand better what happens when a nucleation point, say x = Xo, is selected, let us focus on the small time behavior of the nontrivial self-similar solution. Consider a solution (2.5) at time t = At. It is convenient to parametrize the functions w(x,Ar) and v x,At) by x and present them as a curve in the (w,v) plane. It is not hard to see that one then obtains a loop, beginning and ending in a point (Wo,0) (see Fig. 8b) the details of the loop depend, of course, on the fine internal structures of shocks and kinks (see Fig. 8a). 

In view of the self-similar character of the solution, the loop does not change as A t —> 0 even though the strain and velocity fields converge to the constant initial data everywhere outside the point x = Xo. This means, that by selecting the point Xq we have supplemented constant initial data with a singularpartrepresentedby a parametric measure (in the state space) located at x = Xo. We conclude that, contrary to the behavior of, say, genuinely nonlinear systems (o w) 0) (see Di Perna, 1985), the choice of a short time... 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

Another polar solvent that has been used in SDS-stabilized microemulsions is glycerol. Hexanol or decanol have been used as cosurfactants and systems both with and without oil have been studied. The ternary system with hexanol as cosurfactant was examined with SANS and NMR self-diffusion measurements by two different groups and both found the microemulsions to be structureless solutions [130,131], Similar behavior was found from a self-diffusion study of the quaternary systems with p-xylcnc or decane as the oil component [131,132],... 

Our concern in this chapter is of certain similarity properties of the solution of population balance equations. These properties are of considerable value not only to the characterization of experimental data, but also to the identification of key model parameters associated with system behavior, and frequently in the elucidation of behavior at the particle level from population data. The property of similarity manifests in the form of what is often described as a self-similar or self-preserving solution associated with the behavior of many partial differential and integro-partial differential equations. 

The choice of the basis functions depends crucially on the behavior of the self-similar distribution (see footnote 5). For example, suppose that the self-similar distribution z6 (z) has the asymptotic behavior (ju < 1) in the region of z close to zero. Then it is possible to show (see Appendix of Sathyagal et al, 1995) that the function g u) is approximated by for u close to zero. In other words, g(u) is of order 0(m ). Consequently, g u) is not analytic at w = 0, and a very large number of basis functions in the expansion (6.1.9) are required to describe adequately the behavior near the origin. This problem can be overcome by choosing basis functions that have the same dependence on u near w = 0, as g u) does. Incorporating as much known analytical information as possible about the nature of the solution is an important aspect of the solution of inverse problems. Let us see how... 

We now return to the issue of the choice of basis functions for solution of the inverse problem (6.2.8). The behavior of the aggregation frequency that relates to the small- / behavior of the function (j)(rj) is the issue of specific interest. We choose to fit with y-distributions that can accommodate either a singular or nonsingular nature of the self-similar distribution near the origin and accordingly set... 

Vasudevan M, Shen A, Khomami B, Sureshkumar R (2008) Self-similar shear thickening behavior in ctab/nasal surfactant solutions. J Rheol 52(2) 527-550... 

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