J. Phys. A: Math. Gen. zyxwvuts 18 (1985) zyxwvuts 321-326. Printed in Great Britain New exactly solvable models of Smoluchowski’s equations of coagulation F Leyvraz Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48 109, USA Received 23 May 1984, in final form 4 September 1984 Abstract. The Smoluchowski equations of coagulation are solved analytically in two cases involving a finite cut-off of the system: the constant kernel set to zero for any zy j> N on the one hand, and the general three-particle case on the other. Both are seen to exhibit rather unusual large-time behaviour. The first model can be used to account for large particles precipitating out of a system and its behaviour is therefore of particular interest. 1. Introduction The kinetics of irreversible coagulation have been the object of considerable study. In particular the following rate equations for the concentrations zyxw cj of clusters of size j: (where Rjk is the reaction rate between j-clusters and k-clusters) have been quite extensively studied. However, while much work of qualitative or numerical character exists, the only exactly solved case is: zyxwv Rk, zyxwvu = zyxwv A + B( k + zyxwv I) + Ckl (see e.g. Drake (1972) and Hendriks et a1 (1983) for the solution for C # 0 and arbitrary times). Furthermore, the following cases can be solved exactly: Rjk = R where an arbitrary monomer production term is added to the equations (see Leyvraz and Tschudi 1980), and equilibrium solutions for Rjk =jaka if a constant monomer source is added (see White 1982). For more details on exact solutions see e.g. Hendriks et a1 (1983). It is the aim of this paper to solve exactly the following two kernels: Rjk = 1 for max( j, k) zyxwv d N =O otherwise and R,l=l; Rjk = 0 R12 = RZI = a > 0 (j + k > 3). 0305-4470/85/020321+ 06$02.25 @ 1985 The Institute of Physics 321