Emergence of edge state in suspension of self-propelled particles

TL;DR

The study addresses onset of localization and bistability in bioconvection for a suspension of self-propelled particles. They analyze a two-dimensional periodic model with buoyancy $Ra$, self-propulsion $Pe$, and diffusivity-related $Pr$, performing linear stability analysis and nonlinear simulations. They find that increasing $Pe$ stabilizes the trivial state, yet localized nonlinear steady states coexist with the trivial state; an unstable edge state on the basin boundary is extracted by a bisection method and delineates the boundary between localized and trivial outcomes. The results connect to edge-state concepts in wall-bounded flows and provide a concrete procedure to identify edge states in convection models, with implications for understanding localization and bistability in bioconvection experiments.

Abstract

We numerically study a model convection system of a suspension of self-propelled particles, motivated by recent experimental findings of localized and bistable bioconvection pattern, being distinct from classical Rayleigh--Bénard convection. Linear stability analysis of the model system reveals that the trivial noncovection state is stabilized by an increase of self-propelled speed in the vertical direction. Through numerical simulations, we found a nonlinear convection state even when the nonconvection state is stable. Applying ideas and tools developed in wall-bounded flows, we numerically identified an edge state, which is an unstable solution on a basin boundary in the model dynamical systems.

Figures (4)

• Figure 1: Contour of the maximum growth rate, $\textrm{Re}(\lambda)$, obtained by the linear stability analysis for $Pr =2.5$.
• Figure 2: Density deviation $m(x,y)$ of the nonlinear stationary solutions for $Ra = 0.4$, $Pr =2.5$, and $Pe = 1$. (Left) the stable upper branch solution, $\bm{X}_{UB}$. (Right) the unstable lower branch solution, $\bm{X}_{LB}$. The unstable lower branch solution is obtained by the bisection method.
• Figure 3: Vertically-averaged density $n(x)$ for the upper and lower branch solutions.
• Figure 4: Bisection method to obtain a state on basin boundary. Time evolution of energy norm of the velocity, normalized by the energy norm of the UB solution, $E_0$, for $Ra = 0.4$, $Pr =2.5$, and $Pe = 1$. The different colors corresponds to different initial conditions, $\bm{X}_\alpha$. The red curve with stationary energy norm is the unstable lower branch (LB) solution, $\bm{X}_{LB}$, which is identified as an edge state.