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Instantaneous gelation in Smoluchowski’s coagulation equation revisited

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Ball, Robin, Connaughton, Colm, Stein, Thorwald H. M. and Zaboronski, Oleg V.. (2011) Instantaneous gelation in Smoluchowski’s coagulation equation revisited. Physical Review E, Vol.84 (No.1). Article no. 011111. ISSN 1539-3755

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Official URL: http://dx.doi.org/10.1103/PhysRevE.84.011111

Abstract

We study the solutions of the Smoluchowski coagulation equation with a regularization term which removes clusters from the system when their mass exceeds a specified cutoff size, M. We focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularization. Numerical simulations demonstrate that for such kernels with monodisperse initial data, the regularized gelation time decreasesas M increases, consistent with the expectation that the gelation time is zero in the unregularized system. This decrease appears to be a logarithmically slow function of M, indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cutoff. We also study the case when a source of monomers is introduced in the regularized system. In this case a stationary state is reached. We present a complete analytic description of this regularized stationary state for the model kernel, K(m1,m2)=max{m1,m2}ν, which gels instantaneously when M→∞ if ν>1. The stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent ν for large cluster sizes. The total particle density in the stationary state slowly vanishes as [(ν-1)logM]-1/2 when M→∞. The approach to the stationary state is nontrivial: Oscillations about the stationary state emerge from the interplay between the monomer injection and the cutoff, M, which decay very slowly when M is large. A quantitative analysis of these oscillations is provided for the addition model which describes the situation in which clusters can only grow by absorbing monomers.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science > Centre for Complexity Science Faculty of Science > Mathematics Faculty of Science > Physics
Library of Congress Subject Headings (LCSH):	Coagulation -- Mathematical models
Journal or Publication Title:	Physical Review E
Publisher:	American Physical Society
ISSN:	1539-3755
Official Date:	11 July 2011
Volume:	Vol.84
Number:	No.1
Page Range:	Article no. 011111
Identification Number:	10.1103/PhysRevE.84.011111
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	COST Action
Grant number:	MP0806 (COST Action)
Related URLs:	Other Repository
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URI:	http://wrap.warwick.ac.uk/id/eprint/35735