Complex Spontaneous Flows and Concentration Banding in Active Polar Films

We study the dynamical properties of active polar liquid crystalline films. Like active nematic films, active polar films undergo a dynamical transition to spontaneously flowing steady states. Spontaneous flow in polar fluids is, however, always accompanied by strong concentration inhomogeneities or “banding” not seen in nematics. In addition, a spectacular property unique to polar active films is their ability to generate spontaneously oscillating and banded flows even at low activity. The oscillatory flows become increasingly complicated for strong polarity.

Figure 1

(a) Sketch of the film geometry. (b–d) Solutions of Eqs. (3a, 3b) for $\lambda =0.1$, $\xi =0.3$, $D=1$, $\tilde{C}=0.5$, $\eta \Gamma =0.5$, $w=0.13$, $\tilde{\alpha }=0.08$ and variable $\tilde{\beta }$.

Figure 2

On the left, discrete Fourier transforms of $\theta (z\ell =L/2,\tau )$ (center of the film) for $\lambda =0.1$, $\xi =0.3$, $D=1$, $\tilde{C}=0.3$, $\eta \Gamma =0.5$, $w=0.13$, $\tilde{\alpha }=0.1$ and $\tilde{\beta }=15$ (a) 18 (b) and 20 (c). On the right, space-time plots of $\varphi (z,\tau )$.

Figure 3

Phase diagram in the ($\tilde{\alpha }$, $\tilde{\beta }$) plane. The points have been obtained numerically using $\lambda =0.1$, $\xi =0.3$, $D=1$, $\tilde{C}=0.3$, $\eta \Gamma =0.5$, $w=0.13$ and $L/\ell =10$. For $\tilde{\beta }<{\tilde{\beta }}_{\mathrm{TP}}$ upon increasing $\tilde{\alpha }$ the system undergoes a transition between stationary homogeneous state (S) and inhomogeneous steady flow (SF). Above the “tricritical point” (TP) the spontaneous flow becomes oscillatory (PF).