Author(s)

C. D. Calin, M. Shirvani & H. J. van Roessel

Abstract

Numerical approaches for the coagulation equationwith source and removal terms, and kernels that are bounded independently of the particle size, are investigated. A few classes of exact solutions are provided. Keywords: coagulation, polymers, exact solution, source, removal, numerical solution. 1 Introduction There is considerable literature on the mathematical theory of coagulation, deterministic and stochastic, discrete and continuous, beginning with the pioneering work of Smoluchowski in 1917 on modelling binary coalescence of particles. For a very comprehensive survey of work up to 1970, including applications, different derivations of the equation from physical assumptions, and discrete versions of the equation, see Drake [1]. The pioneering works of Melzak [2] (on cloud formation) include some of the earliest applications of the theory, and more applications can be found in Drake [1], Lee [3], Krivitsky [4]. The presence of external particle sources, and the removal of particles from the system, however, has not received a great deal of mathematical attention, the work of Simons [5] being a notable recent exception. In Shirvani and Van Roessel [6], the discrete version with constant kernel and source terms is investigated. Laurenc¸ot [7] and Norris [8] are two comprehensive recent studies of coagulation/ fragmentation for unbounded kernels (butwithout source and removal effects). An example of application to the coagulation equation is in the manufacturing of aluminiumalloys. Here, molten metal is kept in a holding furnace for several hours while particles of titanium diboride are added for further solidification and casting.

Keywords

coagulation, polymers, exact solution, source, removal, numerical solution.