Hydrodynamic bend instability of motile particles on a substrate

TL;DR

The paper demonstrates a bend instability in a two-dimensional, substrate-confined, incompressible active polar system that arises without dipolar active stress, driven solely by self-propulsion and flow alignment. Using a minimal model with polarization components $p_x,p_y$ and vorticity $oldsymbol\Omega$, coupled to a Navier–Stokes equation with substrate friction, the authors perform linear stability analysis and OpenFOAM simulations to identify a critical self-propulsion strength $\alpha_c$ at which a uniformly ordered state becomes unstable. They derive a dispersion relation $\rho \tilde{\omega}^2+\left(f+\rho\frac{K}{\gamma}q^2\right)\tilde{\omega}+\frac{K}{\gamma}fq^2=\beta\left(\alpha-\beta K q^2\right)$ and show the fastest-growing mode occurs at the smallest nonzero wave number $q=2\pi/L$, with $\alpha_c=Kq^2\left(\frac{f}{\gamma}\frac{1}{\beta}+\beta\right)$ and a crossover scale $\beta^*=\sqrt{f/\gamma}$. Depending on $(\beta,\alpha)$ there are three regimes: stable flowing, undulating bend, and disorderly flow, with a concentration-coupled analysis indicating the instability persists. Overall, the work reveals a novel mechanism for hydrodynamic bend instabilities in substrate-contacting active matter driven by self-propulsion and flow alignment, distinct from dipolar-stress–driven instabilities, with implications for cellular layers and synthetic active systems.

Abstract

The emergence of hydrodynamic bend instabilities in ordered suspensions of active particles is widely observed across diverse living and synthetic systems, and is considered to be governed by dipolar active stresses generated by the self-propelled particles. Here, using linear stability analyses and numerical simulations, we show that a hydrodynamic bend instability can emerge in the absence of any dipolar active stress and solely due to the self-propulsion force acting on polar active units suspended in an incompressible fluid confined to a substrate. Specifically, we show analytically, and confirm in simulations, that a uniformly ordered state develops bend instability above a critical self-propulsion force. Numerical simulations show that a further increase in the self-propulsion strength leads the system towards a disorderly flow state. The results offer a new route for development of hydrodynamic instabilities in two-dimensional self-propelled materials that are in contact with a substrate, with wide implications in layers of orientationally ordered cells and synthetic active particles.

Figures (5)

• Figure 1: Simulation snapshots of the three distinct states obtained from the simulations of the full model. (a) Flowing state for $\alpha=0$ and $\beta=0.5$, (b) undulating or bending state for $\alpha=2.5$ and $\beta=0.8$, and (c) disorderly flow for $\alpha=4.0$ and $\beta=0.2$. Colormap denotes the systems vorticity $\Omega$. Only small portion of the full simulation domain is shown for clarity.
• Figure 2: Stability diagram of the $\mathbf{(\beta,\alpha)}$ phase space. The blue triangles and red circles denote the unstable and stable states, respectively, determined by simulations. The theoretical prediction of the linear stability analysis, $\alpha_c$ (Eq. \ref{['eq: critical self-propulsion']}), is shown as a black solid line, with $q=\frac{2 \pi}{L}$.
• Figure 3: Dispersion relation. Growth rate $\tilde{\omega}$ as a function of the wave-vector $q$ and wavenumber $n$ for three alignment coefficients ($\beta = 0.1, 0.5, 0.9$) at fixed self-propulsion strength $\alpha = 3 \times 10^{-4}$. The inset shows an enlarged view of the stability-determining region for the fastest-growing mode.
• Figure 4: Schematic of the instability mechanism. (a) Contributions of the self-propulsion, $\alpha$ and the flow aligning term $\beta$. (b) Schematic representation of the competition between the Frank elastic energy and the flow aligning term. (c) Initial conformation and (d) conformation after the aligning contributions. Black arrows indicate the polar vector $\vec{p}$ an red dashed arrows indicate the flow velocity vector $\vec{v}$.
• Figure 5: Growth rate curves of the system coupled to concentration dynamics. Growth rate, $\tilde{\omega}(q)$ versus wave-vector $q$ plotted for the longitudinal mode (Eq. \ref{['eq: 102']}) for different $\alpha$ and $\beta$ values.