Instantaneous gelation in Smoluchowski's coagulation equation revisited.
Dr. Colm Connaughton,
(Mathematics Institute and Centre for Complexity Science, University of Warwick)
jueves 25 de noviembre de 2010
Abstract: We study the solutions of a regularised Smoluchowski coagulation equation
with instantaneously gelling kernels. Regularisation is done by introducing a
cut-off, $M_{\rm max}$, which physically corresponds to the removal from the
system of clusters having mass greater than $M_{\rm max}$. Careful numerical
simulations demonstrate that, for monodisperse initial data, the gelation
time for $\nu>1$ {\em decreases} as $M_{\max}$ increases. This
decrease is the signature of
instantaneous gelation. It is, however, very slow (probably
logarithmic) explaining
previous difficulties in characterising the instantaneous gelation
transition in simulations and justifying the use of instantaneously
gelling kernels as physical models. We also consider solutions with a source
of monomers which ultimately reach a stationary state. Approach to the
stationary state is non-trivial. Oscillations result from the
interplay between the monomer injection and the cut-off which decay very
slowly when the cut-off is large.