The evolution of material lines curvature in deterministic chaotic flows

Abstract

It is by now well established that curvature plays a fundamental role in the description of the topology emerging from the partially mixed structures advected by chaotic flows. This article focuses on the dynamics of curvature in volume-preserving time-periodic flows. Previous work on the subject dealt with the evolution of curvature in the time-continuous framework. Here we derive the dynamical equations for the time-discrete dynamical system associated with the Poincaré return map of the flow. We show that this approach allows one to gain more insight into understanding the mechanisms of folding of material lines as they are passively stirred by the mixing process. By exploiting the incompressibility assumption, we analyze dependence on initial conditions (i.e. on the initial curvature and tangent vectors), and discuss under which circumstances the dependence on the initial curvature vector becomes immaterial as time increases. This analysis is closely connected with the properties of an invariant geometric structure referred to as the global unstable manifold associated with the flow system. Direct numerical simulations for physically realizable systems are used to provide concrete examples of the results that arise from theoretical considerations. The impact of this information on the prediction of the behavior of diffusing-reacting mixing processes (e.g. pattern formation and generation of lamellar structures) is also addressed.