Bridging Elastic and Active Turbulence

TL;DR

The paper reveals a deep connection between elastic turbulence in dilute polymer solutions and active-nematic turbulence by showing a macroscopic mapping between polymer conformation dynamics and active nematics under deformable, isotropic conditions, formalized through a decomposition of the polymer conformation tensor into traceless and trace parts. Using 2D Kolmogorov-flow simulations, it demonstrates that arrowhead patterns in elastic turbulence correspond to $\pm$1/2 topological defects in the nematic director field, linked via an activity-like mechanism arising from polymer stretch. A transverse instability driven by spatial activity gradients yields secondary flows and bound defect pairs, and at high activity the system enters a flow-suppressed, jammed state analogous to spontaneous-flow transitions in channel-confined active nematics. The mapping suggests new perspectives for interpreting elastic turbulence and provides a framework to study deformable active matter, potentially enabling experimental tests with colloidal active systems and extensions to tissues and 3D dynamics.

Abstract

Remarkably, even under negligible inertia, the addition of microstructural agents can generate chaotic flow fields. Such behavior can arise in polymer solutions, leading to elastic turbulence, or from active, self-driven particles, which generate active turbulence. Here, we demonstrate a close and hitherto unrecognized connection between these two classes of turbulence. Specifically, we reveal that their continuum descriptions are analogous at the macroscopic level, such that polymeric fluids can be interpreted as a deformable analogue of contractile active matter. Moreover, our numerical results for Kolmogorov flow demonstrate that the transition into the well-known traveling arrowhead structures in elastic turbulence is marked by the emergence of $\pm 1/2$ topological defects, long recognized as a defining feature of active turbulence, in the polymer director field. Importantly, these coherent structures originate from a transverse instability driven by activity-like gradients generated by anisotropically stretched, contractile polymers. At sufficiently strong activity, the system undergoes a transition into a flow-suppressed state characterized by weak polymer stretching and ordering, a behavior that can be explained by analogy with the spontaneous-flow transition observed in channel-confined active nematics.

Figures (4)

• Figure 1: Elastic turbulence reveals $\pm1/2$ topological defects in the polymer director field. (A) Schematic of the fully periodic 2D Kolmogorov flow setup with sinusoidal forcing (shown in red) applied along the $x$-axis. The forcing wavelength is $2\pi$ and the number of wavelengths is controlled by the integer $n$. (B) Schematic illustration of half-integer topological defects, namely, a comet-like $+1/2$ defect (red marker) and a trefoil-like $-1/2$ defect (green marker). (C) Snapshot of the characteristic arrowhead structure observed for elastic turbulence over a single forcing wavelength $y=\left[0,2\pi\right]$. Top: Director field superimposed on top of the $\operatorname{tr}(\mathsfbi{C})$ field, revealing that each arrowhead corresponds to a half-integer defect pair. Bottom: Arrowheads are accompanied by strong secondary flows, as shown by the vorticity-field contours overlaid with velocity streamlines that represent the deviation from the background Kolmogorov flow. (D) Time series of the spatially averaged kinetic energy normalized by the kinetic energy of the fixed-point laminar flow, i.e., $E(t)/E_0=\langle \mathbf{u}^2\rangle/U^2$ for different levels of dimensionless activity $\zeta^*$. (E) Representative snapshots of the $\operatorname{tr}(\mathsfbi{C})$ field with the director field $\bm{n}$ superimposed for $x=[0,4\pi]$ and $y=[0,4\pi]$. (F) Number of $+1/2$ defects for different $\zeta^*$ in the statistical steady-state. The inset shows the temporally averaged flow resistance $1-\overline{E}/E_0$ for the same values of $\zeta^*$. In both plots, we observe a non-trivial sequence of transitions with increasing $\zeta^*$: I. passive Kolmogorov flow with no defects, II. a defect-rich regime accompanied by increasing flow resistance, III. a defect-free, high-resistance jammed state.
• Figure 2: Spontaneous flow instability in inextensible polymers (active nematics) with a constant contractile activity coefficient. Simulations are performed across the $Re -\zeta^*$ phase space for the spatially averaged (A) normalized streamwise velocity magnitude $\langle\lvert u_x \lvert \rangle/U$, (B) angle between the director and the $x$-axis, $\langle\lvert\theta\rvert\rangle$, and (C) order parameter magnitude $\langle S\rangle$. A rapid change in behavior occurs at $\zeta^*\approx10^0$, marking the point at which contractile active stresses begin to impact the dynamics. (D) This behavior is further illustrated in the streamwise-velocity contour plots comparing the low ($\zeta^*=0.05$) and high ($\zeta^*=160$) activity regimes. At low activity, the system recovers the passive Kolmogorov forcing laminar solution, with a well-ordered director field (white bars) aligned in the streamwise direction. As the activity is increased, the flow is strongly reduced and the director field exhibits pronounced splay. (E) This jammed state corresponds to contractile active forces (blue) that are of comparable magnitude to the imposed Kolmogorov forcing (red), but opposite in direction, resulting in a near-complete cancellation of the mean flow.
• Figure 3: Activity gradients in inextensible polymers give rise to a transverse instability and topological defects. (A) A spatially varying sinusoidal activity coefficient is defined for the Kolmogorov forcing problem (see Materials and Methods). Inextensible polymers then generate active stresses, which can create flows in the transverse direction. As a result each forcing wavelength develops a pair of coherent, counter-rotating vortices. (B) Representative snapshots obtained from numerical simulations at $\zeta_{\max}=0.0039$. Top: Contour plot of the order parameter $S$ with the director field superimposed. $+1/2$ and $-1/2$ topological defects are marked in red and green respectively. Bottom: Vorticity field with the velocity streamlines superimposed. (C) Number of $+1/2$ defects for different $\zeta_{\max}$ in the statistical steady-state. The inset shows the corresponding temporally averaged mean secondary-flow strength, $\langle\overline{u_y^2}\rangle/U^2$. Notably, the transverse instability is associated with the formation of topological defects. (D) To demonstrate the role of $\nabla\zeta$, we conduct additional simulations at $\zeta_{\max}=0.006$ and control the steepness $\phi$ (Materials and Methods) of the imposed activity profile. The transition into secondary flows is enhanced as the sinusoidal activity profile transitions into a sharper "step-like" response. The inset shows the activity profile $|\zeta(y)|$ across half a forcing wavelength for different $\phi$ values.
• Figure 4: Comparing inextensible polymers (active nematics) with activity gradients to polymers that can stretch. (A) Contour plots for (top row) the degree of alignment measured by $S$ for active nematics with spatially varying activity and (bottom row) $\operatorname{tr(\mathsfbi{C})}$ for deformable polymers where the activity gradient is space and time dependent. In both cases, the director field $\bm{n}$ is superimposed with $+1/2$ (red) and $-1/2$ (green) topological defects. Snapshots are provided for different square domain sizes $n$. (B-C) Diagrams showing the final states identified across different domain sizes and activity levels for the active nematics and polymeric fluids, respectively. In both plots, three distinct dynamical regimes emerge. At low activity and small domain size, the system exhibits passive Kolmogorov flow with no defects (yellow circles). For moderate activity and sufficiently large domains, secondary flows develop, and defect pairs appear (green squares). At high activity, the system transitions into a jammed state characterized by the absence of defects (purple triangles).