Self similarity behavior

The self-similar behavior is most obvious when it occurs in this form, i.e. if the exponent is negative and the self-similar region is extensive. G(t), G (co), G"(co), and H(X) all have power law format and they have been used interchangeably in the literature. Less obvious is the self-similar behavior for positive exponent values. 

There is a class of flow situations, first identified by Jeffery [201] and Hamel [163], for which the flow has self-similar behavior. To realize the similar behavior leading to ordinary-differential-equation boundary-value problems, the analysis is restricted to steady-state, incompressible, constant property flows. After first discussing the classic analysis,... 

An interesting question is whether we can demonstrate that such a self-similar behavior should exist as the film-rupture singularity is approached. We consider the combined equations (6-83) and (6-85) with the gravitational term neglected, as already explained. Using the notation developed earlier in this section, we can write this equation in the form... 

Size of the chains between cross-links The randomly coiled chains exhibit self-similar behavior and the transition from the self-similar critical gel to the self-similar chain (between network junctions) is difficult to detect experimentally. 

Fig. 4. Loss tangent of a chemically cross-linking polybutadiene as function of reaction time (48). Data were taken at several frequencies. The GP is marked by the instant at which the loss tangent is independent of frequency. All data are taken at low frequencies where the self-similar behavior prevails.

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. 

During self-similar behavior, we may invoke Eq. (5.1.2) to extract a unique... 

Kapur (1972) has presented evidence of self-similar behavior of comminuted solids from experimental data. The model formulation is precisely as elaborated previously. The similarity variable was chosen to be 80% fines which corresponds to = 0.8. His results show that self-similarity is preserved over a notably long period of time. In an interesting earlier communication, Kapur (1970) also shows that such self-similar behavior leads to the well-known laws of grinding due to Rittinger and Bond (Orr, 1966) by attributing different values to the exponent a. 

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... 

Self-similar behavior from numerical calculations have been shown for the case of Brownian motion by Friedlander and Wang (1966) for which the aggregation kernel is given by... 

Self-similar behavior has also been observed in computer simulation of aggregation processes. Thus aggregates of colloidal particles in diffusion-limited aggregation processes have been found to display self-similar behavior (Meakin, 1983). 

Computational demonstrations have been made of the existence of the similarity distribution t) by Meakin and Ernst (1988) (see footnote 9). The importance of this form of self-similar behavior does not appear to have been realized in experiments. A particularly fruitful area of application lies in the experiments of Wright and Ramkrishna (1994) with liquid droplets in a stirred liquid-liquid dispersion without the restriction imposed by these authors to purely coalescing dynamics, i.e., with the inclusion of droplet breakup as well. 

Instances of self-similarity in the presence of particle growth have been relatively rare in the literature. However, the author has discussed the possibility of self-similar behavior in the dynamics of microbial populations (Ramkrishna, 1994 Ramkrishna and Schell, 1999). We present a simple example to illustrate the broad ideas from the foregoing development. 

FIGURE 6.1.2 Plot of arc length versus In (particle volume) showing a single smooth curve in confirmation of self-similar behavior (from Sathyagal et al, 1995. Reprinted with permission from Elsevier Science.)... 

SO that c is the same as the average value of the scaled aggregation frequency (or the average aggregation rate during self-similar behavior) which we have denoted previously by . The estimation of c by fitting transient data on the scaling particle size to the dynamic behavior represented in (6.2.7) thus directly determines the value of . In the rest of the discussion we shall dispense with the notation c and instead deal only with . 

FIGURE 6.2.11 Self-similar behavior of aggregating fractal clusters obtained by scaling. The scaling size is obtained by taking the ratio of the second moment to the first (i) mass-independent diffusion (ii) mass-dependent diffusion. Notice different properties of the self-similar distribution at the origin for the two cases. (From Wright et al, 1992.)... 

The inverse problems discussed in Sections 6.1 and 6.2 were addressed in the absence of nucleation and growth processes. In this section we investigate inverse problems for the recovery of the kinetics of nucleation and growth from experimental measurements of the number density. It is assumed, however, that particle break-up and aggregation processes do not occur. Determination of nucleation and growth rates is of considerable practical significance since the control of particle size in crystallization and precipitation processes depends critically on such information. We will dispense with the assumption of self-similar behavior, as it is often not observed in such systems. Also, we provide here only a preliminary analysis of this problem, as it is still in the process of active investigation by Mahoney (2000). 

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