The distribution of “time of flight” in three dimensional stationary chaotic advection

Abstract

The distributions of "time of flight" (time spent by a single fluid particle between two crossings of the Poincaré section) are investigated for five different 3D stationary chaotic mixers. Above all, we study the large tails of those distributions, and show that mainly two types of behaviors are encountered. In the case of slipping walls, as expected, we obtain an exponential decay, which, however, does not scale with the Lyapunov exponent. Using a simple model, we suggest that this decay is related to the negative eigenvalues of the fixed points of the flow. When no-slip walls are considered, as predicted by the model, the behavior is radically different, with a very large tail following a power law with an exponent close to −3.

Active chaotic mixing in a channel with rotating arc-walls

“…9, suggests that the statistical distribution follows approximately a power law t −γ with an exponent γ between −2 and −2.5. Adapting the calculation proposed by Raynal and Carrière [50] for the 3-D Poiseuille flow to the 2-D case, we obtain that the exponent is equal to −3. Given that the cylinder protuberances in the mixer channel have the effect of slowing down the evacuation of the tracers downstream in comparison to the channel considered in the theoretical case, the value of the exponent obtained numerically is in agreement with the theoretical value.…”

Chaotic mixing in an acoustically driven cavity flow

“…5(a). Although much smaller in amplitude than the base flow (maximum value of about 100 compared to 1000 for the base flow), the fluctuation is highest in regions between the acoustic jets and the walls, avoiding the trapping of the fluid particles near the walls and in the corners [51][52][53][54]. The rms values of the vertical component of the velocity in the same horizontal plane is also shown in Fig.…”

Correlation Decay and Large Deviations for Mixed Systems

“…In one dimension (18) includes the case of finite-time Lyapunov exponents (15), with ψ(x) = ln |T ′ (x)|, while in higher dimensions the leading finite-time Lyapunov exponent cannot be written as a simple Birkhoff sum: nevertheless it still represents a natural indicator. Such a technique has been used in a variety of context [55]: from one dimensional maps as (4), to area preserving maps (7), and it has been also employed to corroborate universality claims [56] for correlation decay of area preserving maps with mixed phase space. More recently this method was also used in exploring mixing properties of coupled intermittent maps [57].…”

“…In what follows we will denote by weakly chaotic systems those for which a power-law decay of correlations is indeed present. We emphasise that, beyond the simple "mathematical" examples we will mention in the next section, the general observations still apply to non trivial physical settings, notably in fluid dynamics [6][7][8].…”