Numerical evaluation of self-preserving spectra in smoluchowski's coagulation theory

Abstract

The cluster size distribution Ck(t) in aggregation and coagulation phenomena for large cluster sizes k and large times t approaches a scaling form of self-preserving spectrum ck(t)'~ s -2 ~o(k/s), where s(t) is the mean cluster size. In a mean field approach the scaling form ~o(x) is described by a nonlinear integrodifferential equation, obtained from Smoluchowski's coagulation equation. To verify some theoretical predictions and to provide quantitative information on the scaling form we develop a fixed point method to determine numerical solutions of this equation for aggregation rate constants K(x, y) = x~y ~ + y~x ~ with a >/3 and fl < 0 (class III models), where ~o(x) is bell-shaped, and we study the crossover to class II models (/3 = 0), where ~o(x) ~ x -~ as x ~ 0, and calculate the exponent r.

Nanoemulsion stability: Experimental evaluation of the flocculation rate from turbidity measurements

“…The continuous distribution of particle sizes is a non-linear integro-differential equation whose kernel depends on the size of the colliding aggregates. Meesters and Ernst [151] demonstrated that the use of a constant kernel in this equation leads to an exponential frequency distribution of particle sizes. Smoluchowski result approaches this exponential distribution at long times.…”

Liapunov Functionals for Smoluchowski’s Coagulation Equation and Convergence to Self-Similarity

“…Here, the particle mean volume s(t) and the profile g S are to be determined and depend on the coagulation kernel a but not on the "details" of the initial data. Several computational studies have been performed to check the dynamical scaling hypothesis (1) and seem to support its validity [10,12,14,20]. Also, rather precise information on the profile g S have been obtained by formal arguments in [6,19].…”

“…The formulae (18) and (19) also formally follow from (20) with ψ(y) = y and ψ(y) = y 2 , respectively. Since these functions do not belong to W 1,∞ , an approximation argument has to be used, first with ψ(y) = min {y, R} and then with ψ(y) = y min {y, R}.…”

On the self-similar solution of fragmentation equation: Numerical evaluation with implications for the inverse problem

“…In the latter one attempts to obtain the breakage functions from experimental PSD data. The present work (which may be considered the breakage analogue of coagulation study in [7]) is focused on the development of a general numerical method for problems involving a self-similar PSD and on the study of its shape for several forms of breakage kernel for which no analytical solutions exist.…”