Analysis of the spatial distribution of aerosol dispersions

Giménez, M.; Schlamp, M.; Clausse, Alejandro

doi:10.1016/0306-4549(94)e0020-e

Cited by 4 publications

(5 citation statements)

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“…3. The next two sections deal with the existence and the uniqueness of the solutions to (2). Not unexpectedly, (2) has a unique non-negative solution for all time, although the multi-dimensional nature of the problem makes the proofs considerably more intricate than the familiar single-vertex system.…”

Section: Sec 2)mentioning

confidence: 99%

“…We begin by proving that the solutions to (4) satisfy Va.i = xQ. Thus, the yaj represent a set of solutions to (2).…”

Section: Sec 2)mentioning

confidence: 99%

“…To make calculations more manageable, assume further that ga0tk = gag is independent of k for all a,/3 6 V, the vertex set of F. The system considered in this a.k -.) / , xajXaj %a,k %a,i ~t~ ^ ' 90a%{3,k ^ ' 9ot3%cv.k ~t~ &a,ki (2) SPATIALLY NON-HOMOGENEOUS COAGULATION EQUATIONS WITH SOURCE TERMS 653 paper is, therefore, oc E i+j=k J = 1 subject to the conditions that for all a € V, k ^ 1, and t ^ 0, wc have sa^(t) ^ 0, Za,k(0) > 0, a:a(0) = a',,. *•(()) < oc, with the sa,k and sa = Efcli sa.k locally integrable.…”

Section: Sec 2)mentioning

confidence: 99%

“…for all a £V, subject to the initial conditions xQ(0) = < 00• Corresponding to (2) we have the system Va,k -^ ^ ' ya.iUa.j ^cv^/a.fc ~~t~" ^ \ 9 dot U fi. k ^ ^ Ocy.JVc*.…”

Section: Sec 2)mentioning

confidence: 99%

“…The physical shape of the cloud of particles, as well as factors such as spatial drift, are ignored. (For numerical work and discussion of some of the relevant features, see for example [2], [4], and [5]. )…”

Section: Introductionmentioning

confidence: 99%

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Spatially non-homogeneous coagulation equations with source terms

Shirvani¹,

Roessel²

2004

Quart. Appl. Math.

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Abstract.The physical process of coagulation or coalescence of particles is often modelled by Smoluchowski's coagulation equation, an infinite system of nonlinear differential equations governing the binary interactions of particles of different sizes. One of the physical assumptions underlying the coagulation equation is spatial homogeneity, in particular, the assumption that coagulation of particles is governed only by particle type or size. The physical shape of the cloud of particles, as well as the relative position of particles of various type, are ignored.One way to model spatial inhomogeneities is to consider several localized clusters of particles distributed within the cloud, with each cluster being a microcosm of coagulating particles. That is to say that coagulation occurs among particles within a cluster but not among particles from different clusters.Interaction between clusters is then accommodated by allowing for a migration, or drift, of particles between clusters. In this article, we model this cloud of localized clusters by a directed graph, with a vertex located at each cluster of particles and a directed edge between vertices representing the migration of particles between clusters, and investigate the effect of the shape of the graph on the properties of the solutions and their evolution.For the case of a constant coagulation kernel, we prove the existence, uniqueness, and global stability of the solutions, as well as mass conservation, in this more general setting.We also give examples of some of the novel features arising from the structure of the graph, such as the possibility of oscillatory behavior, and the much greater degree of control over the creation of large particles, a problem of some importance in practical applications.

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Section: Sec 2)mentioning

confidence: 99%

“…We begin by proving that the solutions to (4) satisfy Va.i = xQ. Thus, the yaj represent a set of solutions to (2).…”

Section: Sec 2)mentioning

confidence: 99%

Section: Sec 2)mentioning

confidence: 99%

“…for all a £V, subject to the initial conditions xQ(0) = < 00• Corresponding to (2) we have the system Va,k -^ ^ ' ya.iUa.j ^cv^/a.fc ~~t~" ^ \ 9 dot U fi. k ^ ^ Ocy.JVc*.…”

Section: Sec 2)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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