Emergence of lanes and turbulent-like motion in active spinner fluid

The study investigates how fluid inertia influences the collective dynamics of active spinner fluids composed of clockwise (CW) and counterclockwise (CCW) rotating particles. Prior work at zero Reynolds number (Stokes flow) using discrete spinner models predicted segregation into same-spin clusters and emergent patterns such as lanes, while recent experiments suggest non-negligible Reynolds numbers. The research question is whether and how increasing Reynolds number alters pattern formation (lane emergence) and induces turbulent-like motion or chirality symmetry breaking in mixtures initially containing equal fractions of CW and CCW spinners. The purpose is to provide a continuum, coarse-grained framework that captures these dynamics and yields predictions relevant to experiments on active spinner materials, particularly Quincke rotors.

Previous theoretical and experimental studies on active spinners and related active matter systems are referenced. Discrete micro-spinner models at Re = 0 showed segregation into same-spin clusters and lane or vortex formation (e.g., Yeo et al.). Experiments on colloidal spinners and magnetic rollers reported active turbulence and inverse energy cascades at quasi-2D interfaces, with non-Gaussian velocity statistics. Prior continuum models addressed active coarsening and odd-viscosity effects. Notably, discrete models observed vortex formation at higher spinner concentrations due to steric repulsion, an effect not captured in continuum approaches that neglect steric interactions. Recent colloidal experiments suggest finite inertia, motivating examination of behaviors beyond the Stokes limit.

The authors develop a continuum, coarse-grained model for a 2D suspension of spherical spinners embedded in a fluid. The model is derived from the Fokker–Planck/Smoluchowski description for the probability density Ψ(r, Ω, t) of spinner position r and angular velocity Ω, leading to coupled equations for spinner density c(r, t), spin angular velocity density ω(r, t), and fluid velocity u(r, t). Governing equations include: (1) a spin angular momentum balance for spinners with thermal noise, fluid drag torque, and externally applied torque; (2) Navier–Stokes equations for the fluid with an added body-force term driven by spinner rotations; and (3) incompressibility (∇·u = 0). Specifics of the spinner forcing are instantiated for Quincke rotation: the external torque τ = c (P × E) depends on the polarization density P and a uniform DC electric field E, with polarization dynamics given by an advection-diffusion-relaxation equation including coupling to ω. The system is nondimensionalized using the spinner radius a and the Maxwell–Wagner polarization time τ0; the key parameters are Reynolds number Re = ρ a^2/(μ τ0), inverse particle Reynolds number κ, normalized electric field γ = E/Ec, diffusion coefficients D⊥ = D∥, and area fraction α, among others summarized in Table 1. The continuum fluid equation in the low-inertia limit reduces to a forced Stokes problem with a rotlet-like source term proportional to ∇×(ω ẑ); for finite inertia, the full Navier–Stokes form is used with the same forcing. The computational setup uses a square periodic domain of side L, with typical choices: α = 0.25, γ = 1.1, κ = 0.1, D⊥ = D∥ = 0.1, L = 480 α, N = 1024 grid points, and time step dt = 0.00025 τ0. Initialization assigns random small positive or negative ω values to produce a 50:50 CW:CCW mixture. Numerical methods: pseudo-spectral discretization with FFTs and exponential time differencing for stiff terms; Stokes flow solved via stream-function formulation; finite-Re simulations advanced using a Chorin projection method. Diagnostics computed include probability distribution functions (PDFs) of fluid speed, velocity and vorticity spatial correlation functions, energy spectrum E(k) from the Fourier transform of velocity autocorrelation, and energy flux Πk using low-pass filtered velocity fields. Passive tracer transport is quantified via mean-square displacement (MSD) and mean square velocity (MSV) of the flow. The model assumes low spinner concentration (neglecting steric repulsion), two-dimensional flow with periodic boundaries, and coarse-grains the microscopic energy injection scale of individual rotors into an effective torque term.

• Flow localization and clustering: Spinners segregate into CW and CCW clusters; fluid flow is weak in interiors of same-spin regions and localized at interfaces where counter-rotating spinners pump fluid in the same direction, generating edge currents and bands of flow.
• Lane formation at Re = 0: For inertialess flow and an initial 50:50 CW:CCW random configuration, the system robustly forms steady traffic lanes of same-spin fluid. This is attributed to least-dissipation Stokes flow: minimizing interfaces between opposite-spin domains yields straight lanes under periodic boundaries.
• Quasi-stable distorted lanes at Re ≈ 0.01: Small inertia produces distorted, quasi-stable lanes with turbulent-like flow near interfaces; lanes can dissolve into transient turbulent-like states and reappear.
• Sustained turbulent-like motion for Re ≥ 0.1: At Re ≥ 0.1, the system exhibits persistent turbulent-like dynamics without large-scale structure; pre-formed lanes become unstable and dissolve.
• Chirality symmetry breaking at Re ≈ 1: The fraction of CW vs CCW spinners is no longer conserved; inertial contributions to fluid vorticity can flip spinner orientation, leading to a single-rotation-sense state after transient turbulence.
• Velocity statistics: Lane states show a bi-modal PDF of fluid velocity due to opposite flows on either side of lanes (Re = 0). Turbulent-like states show approximately Gaussian velocity PDFs (Re = 0.1); non-Gaussian tails in some simulations were linked to underresolution.
• Turbulent spectra and flux: The energy spectrum E(k) exhibits k^(-5/3) scaling at Re = 0.1. The energy flux Πk is negative only for k < k0 ≈ 0.016, indicating a limited inverse cascade range, likely due to finite domain size and coarse-graining of injection scale.
• Correlation scales: For Re = 0.1, the integral scale L1 ≈ 100 (from velocity autocorrelation) with u_rms ≈ 5, giving an integral-scale Reynolds number O(50). The Taylor microscale λ ≈ 15 (from vorticity/velocity-gradient autocorrelation).
• Transport metrics: MSD initially shows transient ballistic behavior for all Re. As Re increases, both MSD and MSV decrease; for lane-forming cases (Re ≈ 0, 0.01), MSD remains ballistic (~t^2) with roughly constant MSV. For turbulent-like cases (Re = 0.1, 1), MSD trends toward diffusive (~t), and at Re = 1, MSV eventually decreases due to chirality breaking.
• Model limitations observed: Vortex formation at higher spinner concentrations seen in discrete models is not captured, likely due to neglected steric repulsion in the continuum model.

The findings clarify the role of fluid inertia in determining the emergent organization of active spinner fluids. In the Stokes limit, lane formation minimizes interfacial length between opposite-spin regions and thus viscous dissipation, aligning with discrete-model predictions. Introducing even weak inertia disrupts this energy-minimizing configuration, producing persistent turbulent-like dynamics and, at sufficiently high Re, a chirality-breaking transition where inertially generated vorticity drives spin flips and selection of a single rotation sense. The observed k^(-5/3) energy spectrum suggests an active turbulence-like cascade, although the limited negative energy flux range indicates disparities from classical 2D inverse cascades, plausibly due to finite system size and the coarse-grained torque representation. The results connect microscale spinner forcing to mesoscale flow organization and transport, offering predictions for experiments on Quincke and magnetic spinner systems regarding thresholds for lane stability, turbulence onset, and chirality symmetry breaking.

A continuum coarse-grained model for Quincke-driven active spinner fluids reproduces and extends discrete-model and experimental observations. Key outcomes include: (i) robust lane formation at Re = 0; (ii) distorted or transient lanes at Re ≈ 0.01; (iii) sustained turbulent-like flow for Re ≥ 0.1; and (iv) chirality symmetry breaking at Re ≈ 1 leading to a single rotation sense. Quantitative characterization via PDFs, correlation functions, energy spectra/flux, MSD, and MSV elucidates transport and scaling behaviors. The study predicts how fluid inertia controls transitions between ordered (lanes) and disordered (turbulent-like) states and the emergence of chirality selection. Future research directions include incorporating steric interactions, exploring fixed and deformable boundaries and obstacle arrays, addressing non-uniform spinner densities, increasing domain sizes and resolution to probe energy fluxes, and extending the model to magnetic spinners and fully three-dimensional suspensions.

• Steric repulsion between spinners is neglected; at higher concentrations this omission likely prevents vortex structures observed in discrete models.
• The system is modeled as two-dimensional with periodic boundaries; boundary effects and confinement are not addressed here.
• Coarse-graining of microscopic energy injection into an effective torque may alter cascade properties; energy flux shows a limited inverse-cascade range, potentially due to finite domain size and coarse-graining.
• Numerical resolution can affect velocity PDF tails; underresolved simulations produce overly populated tails.
• Diffusion coefficients for turbulent-like regimes were not reliably extracted due to limited integration times.
• Parameter exploration centers on select values; broader sweeps (e.g., κ, α, γ) may reveal additional regimes.