Motion of Particles in Complex Flows

Abstract

This thesis covers various aspects of motion of small rigid particles in complex flows. It is in two parts. Part I is concerned with motion of small spherical particles. We study extensions of two classical models for diffusion of a particle driven by random forces, namely the Ornstein-Uhlenbeck process and Chandrasekhar-Rosenbluth model. We show that both models exhibit similar scaling of the diffusion matrix, leading to the same short-time asymptotic dynamics characterized by anomalous diffusion of the momentum and ballistic diffusion of the displacement. We discuss a generalization of the Kramers model describing an overdamped particle in an external potential driven by random forces. We analyze the stationary probability density of the position in the limit when the external forcing is strong and show that the density yields a non-zero probability flux for the motion in a periodic potential with a broken reflection symmetry. We explain quantitatively an abrupt increase of the collision rate of inertial particles suspended in a flow, as the intensity of turbulence I passes a threshold. We argue that the collision rate exhibits an activated behaviour containing a factor exp(-const/I) due to the formation of fold caustics in their velocity field. Part II is concerned with patterns formed by small non-spherical, axisymmetric particles advected in a flow. Numerical simulations suggest that the direction field of the particles exhibits topological singularities of the same type as those seen in fingerprints. An exact solution of the equation of motion indicates that the direction field is non-singular, but we give a theoretical explanation arguing that the singularities are approached in an asymptotic sense. We introduce the order parameter vector characterizing the alignment of particles. We show that the order parameter field also exhibits singularities and describe their normal forms. The order parameter is related to the reflection of light by a rheoscopic fluid illuminated by three coloured light sources. We report on the results of a simple experiment supporting our theoretical findings.