Theory and simulations of active and Brownian particles

When in 1827 the botanist Robert Brown was looking through a microscope he recognized particles moving through water in a chaotic way, he was not able to discern the cause of the motion of the particles. Indeed, the criticism he received when trying to publish his results was that he was actually observing `swimming' particles such as bacteria. Both particles moving due to the thermal motion of the fluid molecules and self-propelled organisms `live' in a very particular world, that is the world of Low Reynolds numbers. The physics governing this world, which encompasses both `passive' or `Brownian' particles and `active' or `alive' particles, is extremely different from the physics that governs the macroscopic world, and which we experience every day. At these length-scales, typically of the order of microns, the surface forces (i.e. friction due to the suspending fluid) dominate over the volume forces (i.e. gravity or inertial) and many interesting phenomena arise because of this. Despite being discovered more than a century ago, and being studied for decades, there are many aspects of both Brownian moving particles and of propulsion mechanisms of microorganisms that have to be unveiled yet. The aim of this thesis is to give insights, by means of theoretical analysis and numerical simulations, into two topics that have received much attention in the scientific literature over the past years, namely, the diffusion of particles under confinement and the dynamics of active micro-particles in complex fluids. This thesis is therefore naturally divided in two `macro-sections'; the first one is devoted to the study of Brownian motion of particles under confinement. In the second part of the thesis we present a well known hydrodynamical model to account for self-propulsion of micro-particles such as bacteria or other microorganisms, and we highlight the effects of a complex suspending fluid on the micro-particles dynamics and efficiency.