Coagulation equations with gelation

Hendriks, E.M.; Ernst, M. H.; Ziff, Robert M.

doi:10.1007/bf01019497

Cited by 221 publications

(156 citation statements)

References 24 publications

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“…One is similar to random percolation models (Herrmann, Stauffer, and Landau 1983;Manneville and de Seze 1981) and uses stochastic simulations on a lattice to evaluate statistics of the evolving system. Another uses differential equations describing the concentrations of polymers of different sizes (Hendriks, Ernst, and Ziff 1983). We employ this second type of model in this work because it is readily adapted to include source and sink terms of polymer and analytic methods can be used to predict gel formation.…”

Section: Gelationmentioning

confidence: 99%

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Fibrin gel formation in a shear flow

Guy

Fogelson²,

Keener³

2007

Mathematical Medicine and Biology: A Journal of the IMA

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Blood clots are made up of platelets and fibrin gel, and the relative amount of fibrin is strongly influenced by the shear rate. In order to explore this phenomenon, this paper presents a model of fibrin gel formation over the surface of an injured blood vessel in a shear flow. A condition for gelation including source and sink terms of polymer is derived. A simplified model of coagulation, involving activation and inhibition of the enzyme thrombin and thrombinmediated production of fibrin monomer, is combined with the model of gelation to explore how the shear rate and other parameters control the formation of fibrin gel. The results show that the thrombin inhibition rate, the gel permeability, and the shear rate are key parameters in regulating the height of the fibrin gel.

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Section: Gelationmentioning

confidence: 99%

“…In the remainder of this paper, we review an analysis of an existing model of gelation for the case that the total amount of polymers is fixed (Hendriks, Ernst, and Ziff 1983;Ziff and Stell 1980). If polymer is removed from this system, gelation may not occur (Hendriks 1984).…”

mentioning

confidence: 99%

Fibrin gel formation in a shear flow

Guy

Fogelson²,

Keener³

2007

Mathematical Medicine and Biology: A Journal of the IMA

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“…is gelling [25,39,53]. Following the contributions [53,88] the conjecture for non-gelling coagulation kernels (1.8) is completely solved in [4] for the discrete coagulation equations (1.1)-(1.2), an alternative proof being given in [46].…”

Section: Introductionmentioning

confidence: 99%

Weak Compactness Techniques and Coagulation Equations

Laurençot

2014

Lecture Notes in Mathematics

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Abstract. Smoluchowski's coagulation equation is a mean-field model describing the growth of clusters by successive mergers. Since its derivation in 1916 it has been studied by several authors, using deterministic and stochastic approaches, with a blossoming of results in the last twenty years. In particular, the use of weak L 1 -compactness techniques led to a mature theory of weak solutions and the purpose of these notes is to describe the results obtained so far in that direction, as well as the mathematical tools used.

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“…A formal calculation in ref. 8 showed that solutions with time-dependent densities do not exist for these kernels (cf. Theorem 3.6 for a rigorous proof of this).…”

Section: Examplesmentioning

confidence: 99%

“…For compact clusters in d dimensions e= 1-d -a, but other values of e are also of interest. (8) The multiplicative form (1.3) might apply to situations in which bond linking was the dominant mechanism. Note that, for the kernel (1.2), the rates for large-large and large-small interactions have the same order of magnitude (i.e., a/,k ~ aj.j for large j and small k), whereas for (1.3), large-large interactions dominate.…”

Section: Introductionmentioning

confidence: 99%

The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation

1990

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The discrete coagulation-fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. These models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys. Our results relate to existence, uniqueness, density conservation and continuous dependence and they generalise the corresponding results in [ref. 2] for the Becker-Doring equations for which the processes are restricted to clusters gaining or shedding one particle. Examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation-fragmentation equations.

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Coagulation equations with gelation

Cited by 221 publications

References 24 publications

Fibrin gel formation in a shear flow

Fibrin gel formation in a shear flow

Weak Compactness Techniques and Coagulation Equations

The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation

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