User:Auroremy

Poincaré sections
In the case of chaotic advection, a particle travels in a large region, and thus encounters other particles that were initially far from it. One can then consider that a particle is mixed with particles that travel within the same region. Unfortunately, Lagrangian trajectories don't enable us to define these regions for non-stationary flows : it is thus difficult to know which particles meet each other. Therefore, we need another representation to account for these regions.

For time-periodic flows, we can solve this problem by using Poincaré sections. The Poincaré map is defined as the transformation
 * $$\begin{align}

\boldsymbol{M} \colon \boldsymbol{X}(t=0)&\to \boldsymbol{x}(T,\boldsymbol{X}). \end{align}$$

$$\boldsymbol{M}$$ transforms a point-like particle into the position of the particle after a time-interval T. Especially, for a time-periodic flow with period T, applying the map several times to a particle gives the successive positions of the particle period after period. Then, a Poincaré section is built by starting from different initial conditions and plotting the corresponding iterates. This comes down to plotting the trajectories but only keeping one point every T. Hence, by this method, we stroboscope the Lagrangian trajectory every T.



The figure above depicts the Poincaré section obtained when one applies an eight-like movement to the mixing rod. We can see that some trajectories span a large region: these are the chaotic or mixing regions, where good mixing occurs. However, there are also two "holes": in these regions, the trajectories are closed. These are called elliptic islands, as the trajectories inside are elliptic-like curves. These regions are not mixed with the remainder of the fluid. Elliptic islands have to be avoided for mixing, for two reasons :
 * Fluid particles are unable to cross the boundaries of the islands (except by slow diffusion, which is negligible), thus there is segregation.
 * The mixing in these regions is not efficient because the trajectories are closed and therefore not chaotic.

If one wants to avoid non-chaotic islands, it is useful to try and understand the physical origins of these regions. In the figure-eight flow, this can be related to the size of the stirring rods. If the rod is too small, a particle initially close to the rod doesn't see the eight-like movement of the rod. To this particle, it seems that the rod is making circles. This explains why some particles have an elliptic movement, and can't go out of their loop. If we use a larger rod, it might be possible to extract a particle from one loop to another, which allows to get rid of the elliptic islands and span the whole fluid domain, leading to a good mixing.

Poincaré sections give the possibility to analyze the mixing quality by distinguishing chaotic and elliptic regions. However, this is somehow a crude measure of the mixing process, since we cannot infer the stretching properties from this mapping method. Nevertheless, this technique is the most useful one to study the mixing of periodic flows and can be extended to a 3D-domain.