Research Projects

Particulate processes describe the interaction among the particles moving in random motion in a system. These interactions result in the change of shape, size, mass etc. of the particles. This changes are captured mathematically through the population balance equations (PBEs). The dynamics of a system where two particles collide to form a larger unit was first proposed and modeled by von Smoluchowki in 1916. This Smoluchowki model is mainly used to describe the time evolution of particle size distribution due to aggregation in a closed dynamical system. Later in 1972, Oort-Hulst-Safronov (OHS) have proposed a new mathematical model in which two colliding cosmic particles aggregates and thus results in the formation of large astronomical bodies like asteroids, meteors, planets etc. Note that the cosmic dust particles are in continuous random motion, therefore particle-particle interactions over millions of years result in the formation of large astronomical bodies. The literature contains enough works on the Smoluchowski’s aggregation model, however only a handful number of articles deals with the OHS aggregation model. This gap is mainly due to the complicated, highly nonlinear mathematical formulation of OHS model, and also lack of high-speed computing in the early 90’s has hindered the numerical treatment of OHS model. This project aims to address the detailed mathematical and numerical treatment of integro-partial differential equations describing OHS aggregation equation. More precisely, the OHS equation is intrinsically nonlinear evolution equation supported by some initial data. However, the highly nonlinear and complicated nature of the mathematical model has left it as less explored. In the last two decades, space-study and scientific exploration of different heavenly bodies has gained much attention from the several research institutes. More importantly, the onset of high-speed computing facility has encouraged several researchers to find an efficient numerical solution for different kinds of PBEs. Therefore, it is important to analyze the mathematical model describing the formation of large objects due to aggregation. Our study mainly proposes to analyze both the mathematical and numerical aspects of the OHS model by incorporating several physical properties like source and sink terms during particle formation. The mathematical aspects include well-posedness of the model, and proving the existence of solutions from local domain to global domain for different physics-embedded kernels. Since, PBEs are evolution equations with time, so large time analysis such models is important to study whether the solutions exhibit steady-state behavior or fails to do so. We further propose to solve the model numerically through some efficient numerical models. Many a cases, the exact solutions may not be available for the system, in that scenario solutions through MC simulations will help us to validate the efficiency of different particle properties.