title: Theoretical and numerical investigation of active suspensions: Determinism, chaos and intermittency
Creators: name: Deußen, Benjamin
Publisher: name: UNSPECIFIED
Year: 2023

description: Active particles and active suspensions are a relatively new field of research. The study of such complex flows promises to yield interesting new results and possible applications range from medicine to the neutralisation of pollutants in water or soil. The term active particle refers in general to any object capable of self-driven motion. Thus, large animals such as the blue whale (Balaenoptera musculus) or technical devices such as planes are active particles, just like microscopic organisms capable of self-driven motion, e.g. Escherichia coli. Active suspensions, i.e. a mixture of active particles and a fluid, are investigated from three different angles in the present work. The focus is on microscopic particles; the Reynolds number of the resulting suspension is therefore very small. This in turn allows the assumption of a Stokes flow, i.e. the convective term of the Navier-Stokes equation can be neglected. Despite the small Reynolds number, the behaviour of an active suspension under certain conditions is called active turbulence by some researchers. This designation inspires to apply methods from turbulence research to an active suspension. The aim is to reveal the nature of the collective behaviour of an active suspension. In particular, the question is whether the behaviour is more chaotic or more deterministic, or whether both types of behaviour occur and an intermittent system is present. First, a model for active particles is developed that serves as the basis for all subsequent investigations. It is assumed that the fluid is Newtonian and described by the unsteady Stokes equation and the rigid particles are governed by the Newton-Euler equation. A special boundary condition at the particle surface is used to accelerate the particle. While one half of the particle surface is considered as passive, i.e. a no-slip condition is used, the other half is an active surface, where an active stress accelerates the surrounding fluid. Due to momentum conservation, the particle will move in the opposite direction of the active stress. The model is used to derive Lie-symmetries, which are later used to analyse simulation data. Furthermore, a statistical description of an active suspension is derived based on the Lundgren, Monin and Novikov (LMN) hierarchy used in turbulence research. Additional symmetries arise for the resulting Probability Density Function (PDF) hierarchy, which transport important information about the physical system. As already mentioned, the symmetries are used to analyse and interpret simulation data. To generate the data, a solver was developed on the basis of the eXtended Discontious Galerkin (XDG) methods implemented in the Bounded Support Spectral Solver (BoSSS) framework. The necessary extensions of the existing solver described in this paper include the implementation of a particle solver, the active boundary conditions and a collision model for the particles. The third method for the analysis of active suspensions, which is examined in the present work, is a homogenised model. In contrast to the particle-resolved approach, which was realised in the BoSSS framework, average values of the physical quantities are investigated. Similar to the Reynolds-Averaged Navier-Stokes (RANS) equations, unclosed terms arise in the model equations that describe statistical moments of higher order. These additional terms are modelled on a phenomenological basis, i.e. observations from the particle-resolved model are used to derive closure conditions. Simulation results generated with both models are linked to the theoretical results of the symmetry analysis. It becomes apparent that the behaviour of an active suspension is determined in particular by the phenomenon of intermittency, i.e. a constant alternation between deterministic and chaotic behaviour exists.