Low-Pass Filtering of Active Turbulent Flows to Liquid Substrates

Abstract

To study the impact of active systems on their surroundings, we introduce a model that couples an active nematic fluid to an isotropic substrate fluid via friction. We numerically show that as the active layer develops turbulence, the substrate inherits the chaotic behaviour, exhibiting a novel form of turbulence driven by locally generated stochastic forcing from the active layer. In particular, the short-length-scale flow structures in the active layer are filtered out, so the system behaves as a de facto low-pass filter. We derive analytically the transfer function between the two layers and use it to predict the large-q decay of the substrate energy spectrum, and to investigate how tensorial quantities, such as the strain rate and the active stresses, are transmitted between the active layer and the substrate. Our analysis agrees with recent experiments measuring velocity-velocity correlations in mixtures of active and passive microtubules, and it may have implications for traction force microscopy measurements in cellular layers.

Figures (4)

• Figure 1: (a) Schematic of the model: an active nematic layer, with velocity $u_i^{\rm a}$ and nematic order parameter $Q_{ij}$, is coupled via friction to a substrate layer, with velocity $u_i^{\rm s}$ and stress tensor $\Pi^{\rm s}_{ij}$. The magnitude of the coupling is set by the coefficient $\xi^{\rm c}$. (b-c) Snapshots of typical flow fields in the active and the substrate layers; color represents the vorticity field $\omega^{\rm a|s} = ( \partial_y u_x^{\rm a|s} - \partial_x u_y^{\rm a|s} )/2$. Blue (red) regions correspond to counterclockwise (clockwise) rotation. The substrate inherits the turbulent flows from the active layer but filters out the short length scales of the active turbulence.
• Figure 2: (a) Root-mean-squared velocity $v_{\rm rms}^{\rm s|a}$ in the substrate and active layers as the friction coefficient ${\xi^{\rm c}}$ is varied for three different values of the activity $\zeta$. $v^{\rm s}_{\rm rms}$ increases as the friction coupling increases, but $v^{\rm a}_{\rm rms}$ decreases due to increased dissipation. For large ${\xi^{\rm c}}$, $v^{\rm s}_{\rm rms} \approx v^{\rm a}_{\rm rms}$, the effective friction experienced by the active layer vanishes, and the substrate has the same flow structure as the active layer. (b) Example of velocity-velocity correlation functions in the substrate (red) and in the active layer (blue), $C_{vv}^{\rm s}$ and $C_{vv}^{\rm a}$ respectively, for $\zeta = 0.01$ and ${\xi^{\rm c}}=0.01$. $C_{vv}^{\rm s}$ decays over a longer length scale than $C_{vv}^{\rm a}$. (c) Substrate and active length scales $\ell^{\rm s|a}$ respectively, extracted from $C_{vv}^{\rm s|a}(r)$ and (d) the ratio $\ell^{\rm s}/\ell^{\rm a}$ as ${\xi^{\rm c}}$ is varied for different activity coefficients. $\ell^{\rm s}/\ell^{\rm a} > 1$ and, as the coupling coefficient increases, decreases to unity. This curve quantifies the low-pass filtering that has occurred in the substrate turbulence, and the role of friction in this process. Correlation lengths $\ell^{\rm s|a}$ are defined by the conditions $C_{vv}^{\rm s|a}(\ell^{\rm s|a}) = 1/e$. Note that in (c) we use different scales for $\ell^{\rm s}$ and $\ell^{\rm a}$. Error bars are smaller than symbols. Similar behaviour to (c)- (d) is observed for time velocity correlations, as shown in Note S3 supp.
• Figure 3: (a) Energy spectra in the substrate layer ${\mathcal{E}}^{\rm s}(q)$ and in the active layer ${\mathcal{E}}^{\rm a}(q)$ for $\zeta =0.01$ and ${\xi^{\rm c}}=0.01$. For large $q$, ${\mathcal{E}}^{\rm s}(q)$ decays as $q^{-8}$ as opposed to ${\mathcal{E}}^{\rm a}(q)$ which decays as $q^{-4}$, consistent with the predictions of the kinetic energy transfer function, Eq. \ref{['eq:energy-spectra']}. (b) Spectral energy ratio $R \equiv {\mathcal{E}}^{\rm s}(q)/{\mathcal{E}}^{\rm a}(q)$ as the friction coupling is varied (for $\zeta = 0.01$); continuous and dashed lines are Eq. \ref{['eq:energy-spectra']} and from simulations respectively. Axes in (b) are normalized to make the comparisons easier.
• Figure 4: (a-c) Simulation snapshots (with parameters as in Fig. \ref{['Fig:fig1']}) colored according to $\varphi$ and $\chi$, defined as the angles between the director fields of the pairs $\{E_{ij}^{\rm s}, E_{ij}^{\rm a}\}$ and $\{E_{ij}^{\rm s}, Q_{ij}\}$ respectively. Regions in which $\varphi,\chi<\pi/4$ are colored blue and regions with $\varphi,\chi\ge\pi/4$ are colored red. (b-d) Probability distribution of $\varphi$ and $\chi$ as the friction coupling ${\xi^{\rm c}}$ is varied for extensile (continuous lines) and contractile (dashed lines) activity $|\zeta|=0.01$.