Membranes with rotating motors: microvortex assemblies.

We study collections of rotatory motors confined to 2-dimensional manifolds. The rotational motion induces a repulsive hydrodynamic interaction between motors leading to a non-trivial collective behavior. For high rotation speed, motors should arrange on a triangular lattice exhibiting crystalline order. At low speed, they form a disordered phase where diffusion is enhanced by velocity fluctuations. In confining geometries and under suitable boundary conditions, motor-generated flow might enhance left-right symmetry-breaking transport. All these effects should be experimentally observable for motors driven by external fields and for dipolar biological motors embedded into lipid membranes in a viscoelastic solvent.