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The flow within porous microtubes is a key issue in many industrial contexts, particularly for hollow fiber membranes often made of organic materials (such as cellulose) with amphiphilic properties. While flow in microtubes with impermeable walls has been extensively studied, little attention has been given to the case of porous walls. Our recent work focused on wettability effects by investigating microtubes coated with a thin film in the inertialess regime [1, 2]. We identified specific flow regimes related to partial wetting, including traveling waves of droplet trains where droplets cluster without coalescing.
This article aims to model and simulate the dynamics of liquid film flow both inside and outside a narrow porous-walled tube, driven by a longitudinal force such as gravity. Due to the amphiphilic nature of the hollow fiber, Darcy’s model is no longer applicable, and a free energy-based model must be considered for the wall [3, 4]. Furthermore, as shown in [5], flows are not only hydrodynamically but also thermodynamically coupled. We propose to combine the hydrodynamic model from [1] for free-surface flows over the fiber with the approach developed in [5] for water exchange between thin films and the porous wall. Each medium (thin liquid films and porous medium) is associated with a free energy functional depending on an order parameter. The evolution equations are formulated as gradient dynamics for non-conserved order fields [6].
Simulations of axisymmetric thin film flows reveal complex interactions between internal and external flows, primarily due to the amphiphilic properties of the porous wall (see Figure). For instance, spontaneous spatial variations in water content within the fiber may lead to intermittent flow. This rich behavior is analyzed using time integration, path-following methods, and numerical bifurcation analysis.
[1] P. Beltrame, Partial and complete wetting in a microtube, EPL 121:64002 (2018).
[2] P. Beltrame, Drop train flow in a microtube, Eur. Phys. J. Spec. Top., 232(4), 435–442 (2023).
[3] P Beltrame and F. Cajot, Model of hydrophobic porous media applied to stratified media: Water trapping, intermittent flow and fingering instability, EPL 138:53004 (2022).
[4] F. Cajot, C. Doussan, P. Beltrame, A free energy based model for water transfer in amphiphilic soils, Advances in Water Resources 198 (2025).
[5] F. Cajot, C. Doussan, S. Hartmann and P. Beltrame, Model of drop infiltration into a thin amphiphilic porous medium, J. Colloid Interface Sci. 684, 35–46 (2025).
[6] O. Kap et al., Nonequilibrium configurations of swelling polymer brush layers induced by spreading drops of weakly volatile oil, The Journal of Chemical Physics 158: 174903 (2023).
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