Curvature-controlled defect dynamics in active systems

We have studied the collective motion of polar active particles confined to ellipsoidal surfaces. The geometric constraints lead to the formation of vortices that encircle surface points of constant curvature (umbilics). We have found that collective motion patterns are particularly rich on ellipsoids, with four umbilics where vortices tend to be located near pairs of umbilical points to minimize their interaction energy. Our results provide a new perspective on the migration of living cells, which most likely use the information provided from the curved substrate geometry to guide their collective motion.Comment: Accepted manuscript. 8 pages, 7 Figures. Movies of the motion patterns can be found at https://www.youtube.com/playlist?list=PLEsE7_tnqXZ_U258VwxES8KAJTV_eO43

Topological Sound and Flocking on Curved Surfaces

Active systems on curved geometries are ubiquitous in the living world. In the presence of curvature orientationally ordered polar flocks are forced to be inhomogeneous, often requiring the presence of topological defects even in the steady state due to the constraints imposed by the topology of the underlying surface. In the presence of spontaneous flow the system additionally supports long-wavelength propagating sound modes which get gapped by the curvature of the underlying substrate. We analytically compute the steady state profile of an active polar flock on a two-sphere and a catenoid, and show that curvature and active flow together result in symmetry protected topological modes that get localized to special geodesics on the surface (the equator or the neck respectively). These modes are the analogue of edge states in electronic quantum Hall systems and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering.Comment: 15 pages, 6 figure

Hydrodynamic Coupling of Particle Inclusions Embedded in Curved Lipid Bilayer Membranes

We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent fluid, and (iii) intermonolayer slip between the two leaflets of the bilayer. We investigate relative to flat membranes how the membrane curvature and topology augment hydrodynamic responses. We show how both the translational and rotational mobility of protein inclusions are effected by the membrane curvature, ratio of intramembrane viscosity to solvent viscosity, and inter-monolayer slip. For general investigations of many-particle dynamics, we also discuss how our approaches can be used to treat the collective diffusion and hydrodynamic coupling within spherical bilayers.Comment: 32 pages, double-column format, 15 figure

Discrete exterior calculus (DEC) for the surface Navier-Stokes equation

We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on staggered grids in flat space for which we demonstrate second order convergence. We compare computational results with a vorticity-stream function approach for surfaces with genus 0 and demonstrate the interplay between topology, geometry and flow properties. Our discretization also allows to handle harmonic vector fields, which we demonstrate on a torus.Comment: 21 pages, 9 figure

Solving the incompressible surface Navier-Stokes equation by surface finite elements

We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus $g(\mathcal{S})$. The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding $\mathbb{R}^3$, penalization of the normal component, a Chorin projection method and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus (DEC) simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincar\'e-Hopf theorem on $n$-tori

Micromixing and microchannel design: Vortex shape and entropy

This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.In very recent years microdevices, due to their potency in replacing large-scale conventional laboratory instrumentation, are becoming a fast and low cost technology for the treatment of several chemical and biological processes. In particular microfluidics has been massively investigated, aiming at improving the performance of chemical reactors. This is because of the fact that reaction is often an interface phenomenon where the greater the surface to volume ratio, the higher the reaction speed, and microscale mixing increases the interfacial area (in terms of mixing-induced-by-vortices generation). However, microfluidic systems suffer from the limitation that they are characterized mostly by very low Reynolds numbers, with the consequence that (i) they cannot take advantage from the turbulence mixing support, and (ii) viscosity hampers proper vortex detection. Therefore, the proper design of micro-channels (MCs) becomes essential. In this framework, several geometries have been proposed to induce mixing vortices in MCs. However a quantitative comparison between proposed geometries in terms of their passive mixing potency can be done only after proper definition of vortex formation (topology, size) and mixing performance. The objective of this study is to test the ability of different fluid dynamic metrics in vortex detection and mixing effectiveness in micromixers. This is done numerically solving different conditions for the flow in a classic passive mixer, a ring shaped MC. We speculate that MCs design could take advantage from fluidic metrics able to rank properly flow related mixing

Hydrodynamic interactions in polar liquid crystals on evolving surfaces

We consider the derivation and numerical solution of the flow of passive and active polar liquid crystals, whose molecular orientation is subjected to a tangential anchoring on an evolving curved surface. The underlying passive model is a simplified surface Ericksen-Leslie model, which is derived as a thin-film limit of the corresponding three-dimensional equations with appropriate boundary conditions. A finite element discretization is considered and the effect of hydrodynamics on the interplay of topology, geometric properties and defect dynamics is studied for this model on various stationary and evolving surfaces. Additionally, we consider an active model. We propose a surface formulation for an active polar viscous gel and exemplarily demonstrate the effect of the underlying curvature on the location of topological defects on a torus