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Context fo Chaotic Advection
Fluid flows can be described by a dynamical system with a set of ordinary differential equations that determines the evolution of a Lagrangian trajectory. These equations are called advection equations:



\frac{d x}{dt} = u(x,y,z,t) $$

\frac{d y}{dt} = v(x,y,z,t) $$

\frac{d z}{dt} = w(x,y,z,t) $$

Where $$u$$, $$v$$ and $$w$$ are the components of the velocity field, which are assumed to be known from the solution of one of the standard dynamical equations governing fluid flow, typically the Navier-Stokes equation in one of its many forms. Dynamical systems and chaos theory state that at least 3 degrees of freedom are necessary for a dynamic system to be chaotic. Indeed, in general, in flows with less than 3 degrees of freedom, Lagrangian trajectories are confined to tubes, and shear-induced mixing can only proceed within these tubes. Regarding practical application, this incomplete or ineffective mixing is not very satisfying.

That is the case for 2-D stationary flows in which there are only 2 degrees of freedom. For this type fo flow, Lagrangian trajectories of fuid particles coincide with the streamlines of the fow, as shown in the Figure. A streamline line can only cross another streamline in a stagnation point. Therefore, in the 2-d stationary flow, the topology of the streamlines corresponds to ensembles of concentric streamlines that can be separated by stagnation points. Thus, in a mixing, a blob of dye can only explore the region bounded by the most external and internal streamline on which it is lying at the initial time.

For 2-D unstationary flow which is time-dependent, instantaneous closed streamlines and Lagrangian trajectories do not coincide any more. We consider a blinking vortex flow which is piecewise stationary, and constitutes a form of time dependency. In this type of flow, two circular rods are plunged in a circular domain, and the two rods are rotated alternately in opposite senses. The rotation of the rods is alternately switched on and off implying a blinking vortex flow which is periodic. We obtain a trajectory that intersects itself which would be impossible in a stationary flow. It consists of a succession of arcs of circles matching streamlines. During the finite time of the simulation, the trajectory sweeps a large fraction of the fluid domain, like what we observed for the 3-D stationary flows, and contrary to the case of 2-D stationary flows.