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Description
Capillary-driven transport through narrow conduits is central to microfluidic systems, where surface forces, wall geometry, and externally imposed fluxes jointly determine flow dynamics. In this work, we study water uptake in plants by modelling the xylem as a naturally occurring microfluidic capillary, in which sap rises under capillary action and transpiration-induced suction. Based on this biologically motivated system, we develop a modified Bosanquet-type model that integrates key physical effects relevant to microscale transport [1].
Specifically, we incorporate an effective friction term arising from wall protrusions and corrugation, analogous to roughness in microfabricated channels, and include corrections to surface tension due to dissolved ions and curvature-dependent effects through a Tolman-type correction. An externally imposed transpiration flux is introduced via boundary-driven diffusion and evaporation at the leaf level, enabling controlled forcing of the capillary flow. The resulting system provides a minimal yet rich framework to study the interplay between wetting, confinement, and imposed flux in narrow channels.
We identify a dimensionless tuning parameter, ξ, representing the relative strength of capillary and hydrostatic forces, which governs qualitative changes in system dynamics. In the absence of imposed flux, the model exhibits transitions between oscillatory and non-oscillatory capillary rise depending on ξ, while the rate of rise is controlled primarily by the effective friction parameter. When transpiration is included, the system stabilizes around a nonlinear center, while significantly increasing the maximum height attained by the fluid column.
Using dynamical systems analysis, we obtain scaling laws for (i) the time required for the column to reach its maximal height as a function of wall friction, and (ii) the characteristic decay time of oscillations as a function of ξ. We further analyze the competing effects of imposed flux and wall corrugation, highlighting regimes in which increased roughness can suppress sustained oscillations, maintaining the flow regime in the vicinity of the equilibrium point.
Our results demonstrate how biologically inspired capillary systems can illuminate general principles of microscale flow, wetting, and transport under confinement. By bridging capillary physics with nonlinear dynamics, this work offers insights relevant to both natural and engineered microfluidic systems operating under strong surface and boundary-driven effects.
[1] Mahalanabis, R., Ashok, B. Effect of conduit friction and presence of charged species on rise of xylem sap. Eur. Phys. J. E 49, 6 (2026). https://doi.org/10.1140/epje/s10189-025-00543-x