From Equilibrium Multistability to Spatiotemporal Chaos in Channel Flows of Nematic Fluids

Abstract

We investigate channel-confined, nematic liquid crystals using the Beris-Edwards model of nematohydrodynamics. Using strong homeotropic anchoring at the walls, we find multistability i.e. multiple coexisting states where the uniform nematic state coexists with states having spatially varying scalar nematic order and director fields. When a pressure gradient is applied, flows develop, and the inherent multistability of the system organizes a variety of complex dynamics. For low pressure gradients, steady flows are established, and the director fields that emerge from the multistable states at equilibrium correspond to Bowser and Dowser configurations similar to those reported in experiments. An increasing pressure-gradient destabilizes steady Bowser and Dowser flow states sequentially, leading to unsteady periodic and chaotic regimes featuring cyclical topological transitions, pulsating flows, advecting defects and spatiotemporal chaos. These findings demonstrate that modest variations in the scalar nematic order, as captured by the Beris-Edwards model, can qualitatively modify equilibrium structures and give rise to complex nonequilibrium behaviour in confined nematics-contrasting with the Ericksen-Leslie model, which assumes a constant scalar order parameter. Our key model predictions - multistability, periodically oscillating states and advecting defect-mediated turbulence can be experimentally investigated in pressure-driven channel flows of nematic fluids.

Figures (7)

• Figure 1: Coexistence of multiple equilibrium states in channel-confined nematics with strong homeotropic anchoring. (a) Nematic order at the centre of the channel $S(0)$ as a function of the dimensionless parameter $\epsilon= \frac{K}{C L^2}$. Simulations were performed by varying $K$ for fixed $C=0.025$ and $L=20$. Red line corresponds to a uniform $\mathbf{Q}$/Bowser (B) state with a constant nematic order of $S(y)=1$. Blue line shows the variation of $S(0)$ in a state with varying $\mathbf{Q}$/Dowser (D). The varying $\mathbf{Q}$ state has a constant nematic order parameter, $S_0=1-2\pi^2\epsilon$ (dashed blue line) in the bulk of the channel for small $\epsilon$, whereas strong spatial variations emerge in $S(y)$ for larger $\epsilon$. Yellow curve shows the numerically-calculated boundary layer thickness $\delta$ defined as length from the boundary where nematic order reaches $95\%\,S_0$. Yellow dashed curve shows the fit $2\delta/L=5\sqrt{\epsilon}$. (b) Variations in the scalar nematic order $S$ and nematic orientation $\theta$ for the uniform $\mathbf{Q}$/Bowser (B) state. (c) and (d) show corresponding plots for the varying $\mathbf{Q}$/Dowser (D) states for $\epsilon=0.001$ and $\epsilon=0.03$ respectively.
• Figure 2: Nonequilibrium steady flow states for pressure-driven nematics in a channel. Panel (a) shows the nematic free energy $\mathcal{F}_{\mathrm{nem}}$ as a function of the applied pressure-gradient force $F$ for B (red dots), D$^-$ (blue dots) and D$^+$ (cyan dots) states. (b)-(d) show the evolution of (top) nematic orientation $\theta$, (middle) scalar nematic order $S$ and (bottom) velocity $u_x$ for B, D$^+$, and D$^-$ states, respectively. Bowser states originate from a uniform $\mathbf{Q}$ state at equilibrium ($F=0$) whereas Dowser states originate from varying $\mathbf{Q}$ states. The inset in panel (a) shows example nematic configuration ($S(y)$ and $\theta(y)$) and velocity field ($u_x(y)$) at $F=0.001$ for Bowser and Dowser states. Vertical dashed lines in (a) show the location of instability of the D$^-$ and B states. Other parameters were fixed to $C=0.025$, $K=0.01$, $L=20$, $\lambda=0$.
• Figure 3: Transition from steady to unsteady oscillatory states. Panels (a)-(c) show, from left to right, kymographs of (i) velocity field $u_x$, (ii) nematic orientation $\theta$, (iii) scalar nematic order $S$, and (iv) phase-space projection of the dynamics onto the $2$D space of global viscous dissipation $\mathcal{F}_{\mathrm{vis}}$ and global free energy $\mathcal{F}_{\mathrm{nem}}$ (gray curve show transients whereas black curve show the stable attractor). (a) corresponds to $F=0.0032$ where the system relaxes to the D$^+$ steady state (cyan circle), whereas (b) and (c) correspond to $F=0.00324$ and $F=0.0034$, respectively, after the onset of oscillations. Panel (d) plots the period $T$ of oscillations as a function of proximity to the transition, and from the inset a power law scaling for the divergence of the form $T\propto |F-F_c|^{-1/2}$ is evident. Other parameters of the system are fixed to $C=0.025, K=0.01$, $L=20$ and $\lambda=0$.
• Figure 4: Unsteady dynamics in pressure-driven nematic flows. (a) Bifurcation diagram showing peaks in the time series of the global free energy as a function of $F$ for unidirectional, pressure-driven flow setup. Initial flow velocity is zero, $S=1$, and different colours correspond to different initial orientation profiles with red favouring B states, blue and cyan favouring D$^-$ and D$^+$ states, respectively, and grey corresponding to random initial orientations. Panels (b)-(d) show kymographs of velocity, nematic orientation, scalar nematic order and phases-space trajectories (transients in grey and stable attractors in black) for $F=0.008, 0.012$ and $0.018$, respectively. Other parameters of the system are fixed to $C=0.025, K=0.01$, $L=20$ and $\lambda=0$.
• Figure 5: Nematohydrodynamics in $2$D pressure-driven channel flows. (a) Bifurcation diagram analogous to figure \ref{['Fig: 5']}(a) for the full $2$D equations of motion (\ref{['eq:Qevol']})--(\ref{['eq:NS']}). In $2$D flows, multistability is observed between different unsteady dynamical states. For example, at $F=0.0024$, (b,e) spatial uniform initial conditions along $x$ corresponding to a Bowser or a Dowser configuration result in periodic states where the whole channel periodically transitions between a Bowser-like and a Dowser-like state. Conversely, if the initial conditions are random, (c,f) then defects are stabilised by advection along the channel. This results in weakly chaotic flow, as the defect dynamics are not entirely periodic. (d,g) At a higher value of $F=0.006$, the turbulent flow becomes more prominent through the continuous formation, annihilation and advection of defects. The kymographs in (e)-(g) are shown for at a fixed $x=50$. Other parameter values are the same as for figure \ref{['Fig: 5']}.