Non-reciprocal mixtures in suspension: the role of hydrodynamic interactions

Abstract

The collective chasing dynamics of non-reciprocally coupled densities leads to stable travelling waves which can be mapped to a model for emergent flocking. In this work, we couple the non-reciprocal Cahn-Hilliard model (NRCH) to a fluid to minimally describe scalar active mixtures in a suspension, with the aim to explore the stability of the waves, i.e. the emergent flock in the presence of self-generated fluid flows. We show that the emergent polarity is linearly unstable to perturbations for a specific sign of the active stress recalling instabilities of orientational order in a fluid. Using numerical simulations, we find however that non-reciprocity stabilizes the waves against the linear instability in a large region of the phase space.

Figures (3)

• Figure 1: Two species of non-reciprocally interacting active particles [depicted in (b)] are represented by the red and blue density profiles in panel (a). Non-reciprocity gives rise to a chasing pattern, whose emergent polar order is quantified by the yellow vector field. Densities are coupled with a background fluid [streamlines in panel (a)], which bends the densities layers and the corresponding orientational order. (c) The interplay between the strengths of non-reciprocity $\alpha$ and active stress $\Gamma$ leads to enhanced stability for the polar order (travelling waves).
• Figure 2: Linear stability of the ordered state. (a) wave-number independent contribution ($q=0$) of the eigenvalue determining the evolution of the transverse fluctuations. The pattern wave number is fixed at $q_0=0.1$ and we vary the coefficient of the active stress $\Gamma$. (b) Same plot for a fixed $|\Gamma|=1$ (positive for all except the dashed line) and different initial wave-numbers $q_0$. (c) Stability diagram in the $\Gamma-q_0$ plane showing three different routes to linear instability as identified from $\lambda_0(\psi)$. (d) Streamlines of fluid velocity $\bm v$ at the onset of the h-polar instability; the colours indicate the local fluid speed. (e) Time evolution of the density $\phi_1$ undergoing the h-polar instability. The arrows on top represent the corresponding polar order parameter $\bm{J}$. The snapshots are from numerical simulations on a square domain of side $L=32 \pi$ discretized over $512^2$ grid points with $\alpha=1$, $q_0=0.5$, and $\Gamma=-1$.
• Figure 3: Order and disorder in the full nonlinear theory. (a) The snapshots show the two distinct phases we observe in the $\alpha-\Gamma$ parameter space: Travelling Waves (blue), and Disordered Phase, as shown by snapshots in the transitional state (orange) and deep in the disordered phase (green). (b) Shell-averaged structure factor $S(q)$ in different states for a fixed $\alpha$ and different $\Gamma$ marked by the same colours as the borders. For travelling bands, $S(q)$ shows a peak at the dominant wave-number. For the crossover and disordered states, we observe a large $\sim q^1$ power law, with an additional contribution at larger wave-numbers for the transitional state. In both (a) and (b), we show the data for $\alpha=4$. Inset: In the polar phase, structure factor $S(\bm q)$ is anisotropic, consistent with the travelling bands. (c) Average polar order $\bm J$ versus $h = \alpha/|\Gamma|$ shows a disorder-order transition with increasing $h$ for all system sizes. Inset : Phase diagram in the $\alpha-\Gamma$ space for $L=320\pi$.