Micro-swimmer collective dynamics in Brinkman flows

TL;DR

This work addresses how viscous Brinkman flows, representing porous obstacles, modify collective micro-swimmer dynamics. The authors couple a continuum distribution $\Psi(\mathbf{x},\mathbf{p},t)$ to a Brinkman fluid with an active stress $\Sigma^p$, derive a non-dimensional system with parameters $\alpha$ and $\nu$, and perform configurational-entropy as well as linear and nonlinear analyses. They show that Brinkman resistance, quantified by $\nu$, delays and can completely suppress the long-wavelength instabilities of pushers (and stabilizes pullers), with the threshold $\nu_c \approx 0.279$ and a shortened unstable band as $\nu$ increases; translational diffusion also contributes to stabilization. Nonlinear simulations in 2D confirm the linear predictions: higher $\nu$ dampens concentration bands, reduces the global input power, and can halt the instability entirely, indicating the dominant role of bulk-fluid friction over reduced single-swimmer speed, $U_B = U\,h(\nu)$.

Abstract

Suspensions of swimming micro-organisms are known to undergo intricate collective dynamics as a result of hydrodynamic and collision interactions. Micro-swimmers, such as bacteria and micro-algae, naturally live and have evolved in complex habitats that include impurities, obstacles and interfaces. To elucidate their dynamics in a heterogeneous environment, we consider a continuum theory where the the micro-swimmers are embedded in a Brinkman wet porous medium, which models viscous flow with an additional resistance or friction due to the presence of smaller stationary obstacles. The conservation equation for the swimmer configurations includes advection and rotation by the immersing fluid, and is coupled to the viscous Brinkman fluid flow with an active stress due to the swimmers' motion in it. Resistance alters individual swimmer locomotion and the way it disturbs the surrounding fluid, and thus it alters its hydrodynamic interactions with others and and such affects collective dynamics.The entropy analysis and the linear stability analysis of the system of equations both reveal that resistance delays and hinders the onset and development of the collective swimming instabilities, and can completely suppress it if sufficiently large. Simulations of the full nonlinear system confirm these. We contrast the results with previous theoretical studies on micro-swimmers in homogeneous viscous flow, and discuss relevant experimental realizations.

Figures (10)

• Figure 1: Fluid velocity generated when the force $\mathbf{f}=(1,0,0)$ is applied at point $\mathbf{x_0}=(0,0,0)$ in a Brinkman flow with different hydrodynamic resistances: $\nu=0, 1, 4$. The vector field indicates the fluid velocity $\mathbf{u}$, whereas the field color represents $log|\mathbf{u}|$.
• Figure 2: Fluid velocity generated by two closely-applied opposite forces in Brinkman flow with different hydrodynamic resistances: $\nu=0, 1, 4$. The vector field indicates the fluid velocity $\mathbf{u}$, whereas the field color represents $log|\mathbf{u}|$.
• Figure 3: Illustration of the dumbbell swimmer.
• Figure 4: Plot of the first three terms from the asymptotic expressions Eqs. (\ref{['Hrelationexpan1_rev']}, \ref{['Hrelationexpan2_rev']}) of $\sigma_{H1}(k)$ and $\sigma_{H2}(k)$ for elongated pushers $\alpha=-1, \gamma=1$ with no translational diffusion $D=0$ for values of $\nu=0.01, 0.02, 0.03, 0.04$.
• Figure 5: Real and imaginary parts of the growth rate $\sigma_{H}(k)$ obtained from numerically solving Eq. (\ref{['Hrelation']}) for various resistance parameters $\nu= 0, 0.025, ...,0.2$, $D=0$. Whenever possible, we indicate on the $k$-axis the critical wave-numbers $k_{-}$, $k_{b}$, $k_{a}$, and $k_{+}$ with purple, green, orange and red circles.