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Instantaneous gelation in Smoluchowski’s coagulation equation revisited
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Ball, R. C., 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 15393755

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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 [(ν1)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:  15393755 
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:  
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URI:  http://wrap.warwick.ac.uk/id/eprint/35735 
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