Nonlinear dynamics of air invasion in one-dimensional compliant fluid networks

TL;DR

The paper develops a minimal model for embolism dynamics in one-dimensional compliant networks driven by pervaporation, revealing that when the pressure-diffusion time $τ_\mathrm{diff}$ is comparable to or larger than the pervaporation time $τ_\mathrm{pv}$, nonlinear coupling between the embolism front and the internal pressure field emerges. Through discrete-network simulations and a continuum formulation, it shows how pressure diffusion delays and channel compliance produce history-dependent front propagation and transient depressurization, with a single nondimensional parameter $τ_\mathrm{diff}/τ_\mathrm{pv}$ governing the behavior. In the fast-diffusion limit, the front evolves quasi-statically with a parabolic pressure profile, while in the diffusion-limited regime, a diffusion-advection interplay creates complex dynamics including potential terminal-collapse scenarios. The work provides design principles for soft microfluidic circuits and offers a theoretical framework for interpreting embolism-like dynamics in plant vasculature, highlighting RC-like pressure diffusion as a potentially pervasive limiting factor.

Abstract

Vascular networks exhibit a remarkable diversity of architectures and transport mechanisms across biological systems. Inspired by embolism propagation in plant xylem, where air invades water-filled conduits under negative pressure, we study air penetration in compliant one-dimensional hydrodynamic networks experiencing mass loss by pervaporation. Using a theoretical framework grounded in biomimetic models, we show that embolism dynamics are shaped by the interplay between network compliance and viscous dissipation. In particular, the competition between two timescales (the pressure diffusion time, $τ_\mathrm{diff}$, and the pervaporation time, $τ_\mathrm{pv}$) governs the emergence of complex, history-dependent behaviors. When $τ_\mathrm{diff} \sim τ_\mathrm{pv}$, we uncover a nonlinear feedback between the internal pressure field and the embolism front, leading to transient depressurization and delayed interface motion. These results offer a minimal framework for understanding embolism dynamics in slow-relaxing vascular systems and provide design principles for soft microfluidic circuits with tunable, nonlinear response.

Figures (8)

• Figure 1: Left: Picture of the embolism propagation in an Adiantum leaf (scale bar = 5 mm) and embolism growth as a function of time for different veins. Courtesy of the authors of Ref. keiser_embolism_2024, where the analysis was carried out based on the experiments from Brodribb et al.brodribb_revealing_2016. For each vein, the dynamics is highly intermittent, with sudden sub-minute propagation events followed by hour-long periods of rest. Right : Embolism dynamics in an one-dimensional pervaporating biomimetic network made with PDMS (seen from the top), where analogous intermittent propagations were evidenced. Courtesy of A. Pellegrin for the picture (whose vertical dimension represents 4 mm), and of the authors of Ref. keiser_embolism_2024 for the data. The white vessels are embolized (full of air) while the grey ones are still full of water. Note that the curves representing the embolized percentage of the networks exhibit a concave shape. This is a signature of a coherent network (regardless of the number of channels $N_0$) where pressure variations are rapidly transmitted throughout the structure. The purpose of our current study is to explore configurations where pressure diffusion is slower and where networks are less coherent.
• Figure 2: Illustration of a series of compliant pervaporating channels connected by constrictions. (a) Illustration of the embolism propagation in the channel series. The width and length of the channels $w$ and $l$ and of the constriction are indicated, together with the pervaporation rate $j$ whose main contribution comes from the upper thin wall. In step $\mathsf{i}$, the embolism front is halted at the constriction between channels $i-1$ and $i$. To cross the constriction (step $\mathsf{ii)}$, pressure $p_{k+1}$ must drop below the Laplace pressure imposed by the constriction (eq. \ref{['eq:Laplace_p_c']}). In step $\mathsf{iii}$, the embolism has fully invaded channel $i$, and the interface reaches the next constriction. Step $\mathsf{iv}$ is equivalent to step $\mathsf{i}$ but with the embolism front, indexed by $k$, now located between channels $i$ and $i+1$. (b) Scheme of the deformation of the channel when loosing water volume by evaporation. In this scheme, the channel dimensions $h$, $w$, $l$ and $\delta$ are defined, as well as the section of the deformed channel $S = S_0 + \Delta S$ with $S_0=hw$ the section at rest. (c) Pressure difference between two successive channels $i$ and $i+1$ generates a flow through the connecting constriction.
• Figure 3: Hydrodynamic resistance initially delays embolism front propagation, followed by a rapid "catch-up" dynamics. Position of the front (channel index) versus time is plotted as hydrodynamic resistance $R$ is increased. For vanishing hydrodynamic resistance (dark magenta), the dynamics is that of a truncated exponential, as demonstrated by Dollet et al.dollet_drying_2019. As $R$ increases (lighter magenta), an inflection point appears in the shape of $L$ vs. $t$, a signature of the delay induced by the slow diffusion of the pressure in the resistive network. The values of the pressure diffusion timescale $\tau_\mathrm{diff} = N_0^2RC$ are expressed in the legend (see also Eq. \ref{['eq:tau_diff']}), and show that the inflection of the shapes appears as $\tau_\mathrm{diff}$ becomes comparable to the duration of the network pervaporation. (a) Example with 1D networks of $N_0=10$ channels and with $w=400\,\mu$m, $l=5\,$cm, $w_c = 15\,\mu$m and $l_c = 0.5$, $15$ or $50\,$mm. The embolism front advances intermittently, following the sequence of steps illustrated in Fig. \ref{['fig:serie']}a. (b) Example given with 1D networks of $N_0=64$ channels (as in the following figures), with $w=400\,\mu$m, $l=0.5\,$cm, $w_c = 10$, $15$ or $30\,\mu$m and $l_c = 10\,$mm. See table \ref{['tab:parameters']} for the other parameters considered in this study.
• Figure 4: Pressure dynamics in the channel series. (a) Pressure evolution in successive channels (color-coded) as a function of time, for $N_0=10$. (b) Depression in the successive channels, coded by color, as a function of time, for $N_0=64$. Only a subset of $12$ channels are shown here. (c) Zoom on the pressure minimum ($p_\mathrm{min}$ at time $t^*$) observed in (b), where the series has significant hydrodynamic resistance. (d) Pressure dynamics near a single embolism event (steps $\mathsf{i}$–$\mathsf{iv}$ in panel a of Fig. \ref{['fig:serie']}).
• Figure 5: Parametric analysis of the discrete model. Resolution of the equations (\ref{['eq:gov_eq']}), (\ref{['eq:gov_eq_k_stop']}) and (\ref{['eq:gov_eq_k_draining']}) varying the design parameters of the series. First panel line represent the temporal evolution of the embolism front. Second panel line plot the dynamics of the pressure in the channels. Last panel line represent the pressure profile $P_n$ in the series where $P_n$ is the pressure at the channel $n$ when the embolism front just finished the draining of a channel. The series considered here have $N_0 = 64$, $l_c = 1\,$cm, $l = 5\,$mm and $h = 50\,\mu$m.