Active matter describes a class of systems that are maintained far from equilibrium by driving forces acting on the constituent particles. Here I will focus on confined active nematics, which exhibit especially rich flow behavior, ranging from structured patterns in space and time to disordered turbulent flows. To understand this behavior, I will take a deterministic dynamical systems approach, beginning with the hydrodynamic equations for the active nematic. This approach reveals that the infinite-dimensional phase space of all possible flow configurations is populated by Exact Coherent Structures (ECS), which are exact solutions of the hydrodynamic equations with distinct and regular spatiotemporal structure; examples include unstable equilibria, periodic orbits, and traveling waves. The ECS are connected by dynamical pathways called invariant manifolds. The main hypothesis in this approach is that turbulence corresponds to a trajectory meandering in the phase space, transitioning between ECS by traveling on the invariant manifolds. Similar approaches have been successful in characterizing high Reynolds number turbulence of passive fluids. Here, I will present the first systematic study of active nematic ECS and their invariant manifolds and discuss their role in characterizing the phenomenon of active turbulence.

I shall present two recent piece of work investigating how shape effects the transport of active particles in shear. Firstly we will consider the sedimentation of particles in 2D laminar flow fields of increasing complexity; and how insights from this can help explain why turbulence can enhance the sedimentation of negatively buoyant diatoms [1]. Secondly, we will consider the 3D transport of elongated active particles under the action of an aligning force (e.g. gyrotactic swimmers) in some simple flow fields; and will see how shape can influence the vertical distribution, for example changing the structure of thin layers [2]. [1] Enhanced sedimentation of elongated plankton in simple flows (2018). IMA Journal of Applied Mathematics W Clifton, RN Bearon, & MA Bees. [2] Elongation enhances migration through hydrodynamic shear (in Prep), RN Bearon & WM Durham.