Supplementary material for "Nonlinear dynamics of air invasion in one-dimensional compliant fluid networks"

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, $\tau_\mathrm{diff}$, and the pervaporation time, $\tau_\mathrm{pv}$) governs the emergence of complex, history-dependent behaviors. When $\tau_\mathrm{diff} \sim \tau_\mathrm{pv}$, we uncover previously unreported 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.