Turbulence Enhancement of Coagulating Processes

We present and investigate the collision-coalescence process of particles in the presence of a fluid velocity field, examining the relationship between flow properties and enhanced coagulation. Our research focuses on two main aspects. Firstly, we propose a novel modeling approach for turbulent fluid at small scales, employing a Gaussian random field with non-trivial spatial covariance. Secondly, we derive rigorous partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) from this model, capturing the physical characteristics of particles suspended in the fluid. From an Eulerian perspective, we analyze a kinetic particle system subjected to environmental transport noise. Specifically, we rigorously study a modified version of Smoluchowski’s coagulation equation, which incorporates velocity dependence akin to the Boltzmann equation. By utilizing techniques rooted in unbounded elliptic semigroup theory and weighted Sobolev space inequalities, we establish the existence and uniqueness of classical solutions for the case of a spatially homogeneous initial distribution. Moreover, from a Lagrangian viewpoint, we employ this particle system to gain insights into the collision rate at a steady state for particles uniformly distributed within a medium. Considering a particle-fluid model, we perform two scaling limits. The first limit, involving the number of particles, yields a stochastic Smoluchowski-type system, with the turbulent velocity field still governed by a noise stochastic process. The second scaling limit pertains to the parameters of the noise, specifically targeting the direction associated with small-scale turbulence. This limit leads to a deterministic equation with eddy dissipation in the velocity variable. We conduct numerical simulations of this equation system and demonstrate the influence of turbulence on rain formation. Our qualitative findings reveal a steady increase in coagulation efficiency with escalating turbulent kinetic energy of the fluid. Additionally, we observe a power-law decay over time and in relation to the turbulence parameter. Furthermore, we recover fundamental laws governing the collision rate and relative velocity of moving particles in the high Stokes number regime.