Spontaneous emergence of solitary waves in active flow networks

TL;DR

This work reveals that simple active-flow elements arranged in a ring with elastic storage can spontaneously generate solitary waves that transport localized information via coupled pressure and volume dynamics. By deriving a discrete active flow network from a continuum description and validating it with lattice Boltzmann simulations, the authors show that ASWs arise from disorder, travel at a velocity set by system parameters, and exhibit rich behaviors when non-local coupling is included. The study provides analytical scaling laws for ASW height, speed, and lifetime and demonstrates robust spontaneous emergence, collision dynamics, and the potential for information transmission, both in closed rings and open networks. These findings establish a foundation for engineering information processing in active flow systems and offer a tractable framework for exploring fluidic soliton-like phenomena in soft matter and microfluidic architectures.

Abstract

Flow networks are fundamental for understanding systems such as animal and plant vasculature or power distribution grids. These networks can encode, transmit, and transform information embodied in the spatial and temporal distribution of their flows. In this work, we focus on a minimal yet physically grounded system that allows us to isolate the fundamental mechanisms by which active flow networks generate and regulate emergent dynamics capable of supporting information transmission. The system is composed of active units that pump fluid and elastic units that store volume. From first principles, we derive a discrete model-an active flow network-that enables the simulation of large systems with many interacting units. Numerically, we show that the pressure field can develop solitary waves, resulting in the spontaneous creation and transmission of localized packets of information stored in the physical properties of the flow. We characterize how these solitary waves emerge from disordered initial conditions in a one-dimensional network, and how their size and propagation speed depend on key system parameters. Finally, when the elastic units are coupled to their neighbors, the solitary waves exhibit even richer dynamics, with diverse shapes and finite lifetimes that display power-law behaviors that we can predict analytically. Together, these results show how simple fluidic elements can collectively create, shape and transport information, laying the foundations for understanding-and ultimately engineering-information processing in active flow systems.

Figures (15)

• Figure 1: Schematic representation of the active flow network. Panel a shows a ring that is composed of active units (regions bounded by black and gray walls) and elastic units (regions enclosed by brown walls), with some elastic units being larger than others, indicating the presence of a propagating solitary wave. Panel b.1 displays the cross-sectional view of the ring at the boundary of a solitary wave, with the fluid inside represented in light blue. Arrows denote the flow through the active units, going in and out the elastic units. The ring can also be represented as a flow network model as shown in panel b.2, where active units (arrows) and elastic units (orange circles) are labeled with the index $i$. Panel c presents the characteristic behavior of one active unit (current versus pressure drop in arbitrary units). At active unit $i$ pressure drop is computed as $\Delta P_i = P_i - P_{i+1}$. Open circles correspond to results from full hydrodynamic simulations (lattice Boltzmann method), while solid lines represent a quadratic fit according to Eq. \ref{['eq:currentDeltaP']} ($\overline{Q} \approx 4.61, m \approx 0.117$, $a \approx 2\cdot 10^{-4}$ and $\Delta P_c\approx23.04$). Red and blue indicate the two possible branches in Eq. \ref{['eq:currentDeltaP']}, namely positive (clockwise, $b_i=1$) in blue and negative (counterclockwise, $b_i=-1$) in red. Panel d presents full hydrodynamic simulations (open symbols) and semi-analytical theory (closed symbols) in arbitrary units, using two different Péclet numbers ($Pe$) for the solute, and two different extents of the catalytic coating ($a_{cov}$). Panel c corresponds to the case: $Pe=5.3$ and $a_{cov}=0.35$. Please see Supplementary Material \ref{['sec:LBM']} for the details of the full hydrodynamic simulations and semi-analytical theory, as well as the definition of the arbitrary units in the plots.
• Figure 2: Spontaneous emergence of a solitary wave from a disordered configuration. The system is initialized at random $b=\pm 1$ and it evolves with time following Eq. \ref{['Eq_local_time_evolution_2']}. Snapshots at $t=0, 10, 5000$ are represented in panels (a,b), (c,d), and (e,f), respectively. Panels a, c and e present pressure (red solid line) and volume (blue dashed line), and the background color displays the branch distribution of the active pores (blue for $b=1$ and red for $b=-1$). Panels b, d and f present the net current (green solid line) at the elastic units. Each configuration is accompanied by a schematic drawing to help visualization (panels g, h, i), the elastic units are represented by orange circles and the flow in the active units by solid arrows. Simulations are carried out using the dimensionless equations, with relevant parameter values: $N = 251$, $\overline{Q} \approx 4.61, m \approx 0.117$, $a \approx 2\cdot 10^{-4}$ and $\Delta P_c\approx 23.04$.
• Figure 3: Solitary waves can be of four different types. Direction of ASW movement (yellow arrow), shown for different configurations of pressure (red line) and active pore branch domains $b_i$ (background color), blue corresponds to clockwise and red to counter-clockwise. The direction of movement is controlled by the combination of volume accumulation/depletion with the net current at the boundaries of the ASW (solid green line). We initialize the system close to each of these solutions. Simulations are carried out using the dimensionless equations, with relevant parameter values: $N = 251$, $\overline{Q} \approx 4.61, m \approx 0.117$, $a \approx 2\cdot 10^{-4}$ and $\Delta P_c\approx 23.04$.
• Figure 4: Solitary waves in the $\omega \ne 0$ case. Panel a shows the ASW height as a function of $\omega$. Dashed lines show the theoretical predictions derived in the SM. The inset in this panel illustrates the definitions of the ASW height $h$ (green arrow) and width $W$ (dark blue arrow), as measured from the ASW pressure profile (red solid line) and flow domains (background colors), for a case $\omega = 10$. Panel b shows the ASW dimensionless lifetime ($\tau$) from $\omega=1$, blue, to $\omega=10$, red, as a function of the initial width, $W_{0}$. The data displays a clear exponential behavior for large $W_0$, that we illustrate with a dashed line that follows $\tau \sim e^{\beta W_{0}}$. The inset presents the exponent of the exponential fit ($\beta$) versus $\omega$. In the inset, an analytical prediction for the scaling behavior of $\beta$ is also included as a dashed line (see \ref{['APPENDIX_sec_SM_Lifetime']}). In both panels we initialize the system with a ASW of height $P-\overline{P}=\Delta P_c$, and the system quickly evolves towards the height presented in panel a, whereas it reduces its width following the behavior shown in panel b. Simulations are carried out using the dimensionless equations, with relevant parameter values: $N = 251$, $\overline{Q} \approx 4.61, m \approx 0.117$, $a \approx 2\cdot 10^{-4}$ and $\Delta P_c\approx 23.04$. We carried out 5 simulations for each point and the error bars represent the standard deviation of the data.
• Figure 5: Diagram describing the model employed in the lattice Boltzmann simulations. The active unit has length $L_a$ and a radius profile $R(x)$ describing an hourglass with inclination angle $\theta$ (Eq. \ref{['eq:SM_hourglass']}). The black walls represent chemically inert walls, whereas grey walls are coated in catalytic material. The chemical coating is centered around the bottleneck, extending a distance of $a_{cov}L_a/2$ to each side. Such material leads to the production of a chemical species (solute) with rate $\xi$, as seen in the inset. The presence of a spatially-inhomogeneous concentration of solute leads to diffusioosmotic flows near the wall which drive motion of the bulk fluid as well (light blue arrows). A pressure drop $P_0 - P_L$ is imposed across the pore.