Instabilities and turbulence in extensile swimmer suspensions

TL;DR

This work analyzes inertial, polar, extensile swimmer suspensions to map how bend and concentration-wave instabilities destabilize the ordered state. It combines a hydrodynamic continuum model with a minimal 1D amplitude framework and high-resolution 2D simulations to show that a Hopf bifurcation governs the splay-stable to splay-unstable transition, and that increasing the concentration-wave parameter $\Psi$ drives a transition from defect turbulence to concentration-wave turbulence at fixed $R$. The study develops metrics for compressibility, topological structures, and energy spectra, revealing how vortex- and aster-like features emerge and how energy is balanced across scales in different turbulent regimes. These results connect limiting cases with homogeneous concentration to more general, inertial active-fluid turbulence, providing a unified view of instability-induced turbulence in extensile swimmers with practical implications for interpreting experiments and guiding future simulations.

Abstract

We study low Reynolds number turbulence in a suspension of polar, extensile, self-propelled inertial swimmers. We review the bend and splay mechanisms that destabilize an ordered flock. The suspension is always unstable to bend perturbations. Using a minimal 1D model, we show that the splay-stable to splay-unstable transition occurs via a supercritical Hopf bifurcation. We perform high-resolution numerical simulations in 2D to study the varieties of turbulence present in this system transitioning from defect turbulence to concentration-wave turbulence depending on a single non-dimensional number, denoting the ratio of the splay-concentration wavespeed to the swimmer motility.

Figures (8)

• Figure 1: Stability diagram representing the pure bend and pure splay unstable regimes in extensile swimmer suspensions. Figure reproduced from jain2024. The non-dimensional numbers $\Psi \equiv \Gamma E c_0/v_1$ and $R=\rho v_0^2/(2 W c_0)$.
• Figure 2: Representative trajectories in the ($U,P,C$) space for (a) $\Psi=0.5$ and (b) $\Psi=5$. The initial condition is marked with an open circle, and the final state is marked with a cross. The system undergoes a supercritical Hopf bifurcation at $\Psi=f/R$. (a) For $\Psi <f/R$, the system is linearly stable and the initial perturbation spirals inward to the fixed point at the origin. (b) For $\Psi >f/R$, the system is linearly unstable and the initial perturbation settles down in a limit cycle around the fixed point. (c) shows the time evolution of the Lyapunov exponents for $\Psi=5$. As expected, for a limit cycle, the exponents converge to a near-zero ($\lambda_1 \approx -6.37 \times 10^{-7}$), and two negative values ($\lambda_2 \approx -3.91 \times 10^{-2}$, $\lambda_3 \approx -1.66$).
• Figure 3: Pseudocolor plot of the concentration field with streamlines of the polar order parameter for $R=0.1$. As $\Psi$ increases, the fluctuations in the concentration reduce and the topological structures change from spiral-asters defects at $\Psi=0.05$ to vortices at $\Psi=10$. For $\Psi=40$, the concentration-wave instability becomes dominant, giving rise to concentration-wave turbulence.
• Figure 4: (a) Variation of the compressibility ($\mathcal{K}$) of the polar order parameter with $\Psi$. The system transitions from compressible to incompressible, and again to a compressible regime when the concentration-wave instability dominates over the bend instability. Inset: Zoomed in plot showing decrease of $\mathcal{K}$ at small $\Psi$. (b) Plot of average distance between nearest neighbor defects for full $\bm{p}$ ($d_{min}$), and the compressible part $\bm{p}_c$ ($d_c$) in the concentration-wave turbulent regime. $d_{min}$ shows an opposite trend to $\mathcal{K}$: It first increases in the region where the system is compressible, then becomes nearly constant in the incompressible region, and finally, it again decreases where $\mathcal{K}$ increases. The inter-aster separation $d_c$ decreases with $\Psi$. (c) and (d) show the streamlines of the solenoidal ${\bm p}^{i}$ and the potential ${\bm p}^c$ components of ${\bm p}$ for $\psi=40$, respectively. Some of the defects are shown in (c), consisting of vortices (orange circles) and saddles (blue squares), and in (d), consisting of outward asters (orange circles) and inward asters (blue squares). Subdomains of $L=2\pi$ and $L=\pi$ are shown respectively for clarity.
• Figure 5: Schematic showing possible local structures in polar order parameter based on the eigenvalues of the $\nabla {\bm p}$ tensor in the Det-Tr plane. The parabola $\rm P\equiv {\rm Tr}^2-4~{\rm Det}=0$ separates regions with real and complex eigenvalues. The different structures are (a) $\rm P>0$: Saddles, (b) $\rm P<0$: Spirals and centers (${\rm Tr}=0$), and (c) $\rm P=0$: Star nodes or asters.