In this talk, I shall review the mathematical formulation of the Born-Oppenheimer approximation and compare it to the traditional physical point of view. I shall point out some rigorous results and open questions.
We study the spatiotemporal patterns that emerge when an active nematic film is topologically constraint. These topological constraints allow to control the non-equilibrium dynamics of the active system. We consider ellipsoidal shapes for which the resulting defects are 1/2 disclinations and analyze the relation between their location and dynamics and local geometric properties of the ellipsoid. We highlight two dynamic modes: a tunable periodic state that oscillates between two defect configurations on a spherical shape and a tunable rotating state for oblate spheroids. We further demonstrate the relation between defects and high Gaussian curvature and umbilical points and point out limits for a coarse-grained description of defects as self-propelled particles.
The spontaneous emergence of collective flows is a generic property of active fluids and often leads to chaotic flow patterns characterised by swirls, jets, and topological disclinations in their orientation field. I will discuss our recent finding on the role of topological defects in regulating the morphology of growing cell colonies. I will present evidence on spontaneous formation of singularities in cellular alignment in the form of nematic topological defects, as a previously unidentified cause of cell apoptosis and extrusion, suggesting that such defects govern cell fate in epithelial tissues. Moreover, the ability to achieve structured flows and ordered disclinations is of particular importance in the design and control of active systems. By confining an active nematic fluid within a channel, we find a regular motion of disclinations, in conjunction with a well defined and dynamic flow structure. As pairs of moving disclinations travel through the channel, they continually exchange partners producing a dynamic ordered state, reminiscent of Ceilidh dancing. I will show that this state is an intermediate state governing the transition to meso-scale turbulence in living fluids and that the transition belongs to the directed percolation universality class. Finally, I will discuss means of exploiting meso-scale turbulence to produce useful mechanical power in microfluidic applications.