Relaxation with long-period oscillation in defect turbulence of planar nematic liquid crystals

Through experiments, we studied defect turbulence, a type of spatiotemporal chaos in planar systems of nematic liquid crystals, to clarify the chaotic advection of weak turbulence. In planar systems of large aspect ratio, structural relaxation, which is characterized by the dynamic structure factor, exhibits a long-period oscillation that is described well by a combination of a simple exponential relaxation and underdamped oscillation. The simple relaxation arises as a result of the roll modulation while the damped oscillation is manifest in the repetitive gliding of defect pairs in a local area. Each relaxation is derived analytically by the projection operator method that separates turbulent transport into a macroscopic contribution and fluctuations. The analysis proposes that the two relaxations are not correlated. The nonthermal fluctuations of defect turbulence are consequently separated into two independent Markov processes. Our approach sheds light on diversity and universality from a unified viewpoint for weak turbulence.

Figure 1

Images of DT at $\varepsilon =0.4$. (a) A real-space snapshot of the transmitted light intensity $I(\mathit{x},t)$. The brightness indicates the value of $I(\mathit{x},t)$. The horizontal direction corresponds to the $x$ direction. (b) Gray-scale magnitude of the spatial power spectrum $P_{\mathit{k}}$ in $\mathit{k}$ space. The brightness indicates the value of $P_{\mathit{k}}$. The solid line represents the reference axis of the phase $\theta$ and the dashed arcs $\left|\mathit{k}\right|=k_0$.

Figure 2

The real part of the modal correlation functions at $\varepsilon =0.4$; $\theta =0^{\circ }$ (red circle), $5^{\circ }$ (brown circle), ${10}^{\circ }$ (green diamond), ${15}^{\circ }$ (blue square), ${20}^{\circ }$ (purple triangle), and ${25}^{\circ }$ (gray circle), from top to bottom at the first peak ($\tau \simeq 25$ s). The solid lines are calculated from Eq. (3). The data in the plot were decimalized, whereas the values of the correlations were calculated at intervals of 0.1 s.

Figure 3

The angular dependence of the parameters in Eq. (3) with Eqs. (4) and (5). (a) The weight parameter $w$. (b) The three characteristic times ${\tau }_{\text{ch}}$: ${\tau }_{\text{E}}$ (circle), ${\tau }_{\text{D}}$ (gray square), and ${\tau }_{\text{O}}=2\pi /{\mathrm{\Omega }}_{\text{O}}$ (gray triangle).