Aggregation in non-uniform systems with advection and localized source

Abstract

We explore analytically and numerically agglomeration driven by advection and localized source. The system is inhomogeneous in one dimension, viz along the direction of advection. It is characterized by the kinetic coefficients—the advection velocity, diffusion coefficient and the reaction kernel, quantifying the aggregation rates. We analyze a simplified model with mass-independent advection velocity, diffusion coefficient, and reaction rates. We also examine a model with mass-dependent coefficients arising in the context of aggregation with sedimentation. For the quasi-stationary case and simplified model, we obtain an exact solution for the spatially dependent agglomerate densities. For the case of mass-dependent coefficients we report a new conservation law and develop a scaling theory for the densities. For the numerical efficiency we exploit the low-rank approximation technique; this dramatically increases the computational speed and allows simulations of very large systems. The numerical results are in excellent agreement with the predictions of our theory.