Motion of Inertial Particles in Gaussian Fields Driven by an Infinite-Dimensional Fractional Brownian Motion

Motion of inertial (i.e. small heavy) particles in turbulent fluids occurs in natural phenomena as well as in technological processes and therefore has excited theoretical investigations. Examples for such processes are formation of raindrops, evolution of clouds and combusting of liquid fuel. The motion of the particle is described by Newton's second law, where the force is proportional to the difference between a background fluid velocity and the particle velocity. In this work we study the dynamics of the particles in the case where the fluid velocity satisfies a linear stochastic partial differential equation of Ornstein-Uhlenbeck type driven by an infinite-dimensional fractional Brownian motion with arbitrary Hurst parameter H in (0,1). The usefulness of such random velocity fields in simulations is that we can compute random velocity fields in an efficient way with desired statistical properties, thus generating artificial images of realistic turbulent flows. This model captures also the clustering phenomenon of preferential concentration, observed in real world and numerical experiments, i.e. particles cluster in regions of low vorticity and high strain rate. We prove almost sure existence and uniqueness of particle paths, give sufficient conditions to rewrite this system as a random dynamical system with a global random pullback attractor and derive some properties of a modified system which suggest a volume contraction in this system. Finally, we visualize the clustering of the particles depending on the parameters of the model through a numerical experiment