Capillary flow in semicircular open channel

Capillary flow in open microchannels is governed by a subtle balance between surface tension, viscosity, and inertia, and reliable prediction remains challenging. We propose an energy-based model for capillary penetration in semicircular open channels that rigorously preserves constant-mean-curvature of the liquid–air interface while avoiding step-wise resolution of the complete meniscus by introducing a “flat-front + concave-tip” description. Validation against volume-of-fluid simulations and independent experimental data confirms that the new formulation produces significant smaller errors than modified Lucas–Washburn and momentum-balance models throughout the first few centimeters of imbibition. A systematic parametric analysis on channel radius demonstrates that smaller radii trigger higher peak entrance velocities but lose that advantage rapidly to viscous dissipation, whereas larger radii sustain faster flows over longer distances. In the limit of long flow distance, we further derive a closed-form solution that reproduces the full numerical model outside the initial acceleration stage yet mitigates the complex numerical integration. Finally, generalization to elliptical cross sections shows that increasing the channel aspect ratio enhances early-time penetration, whereas increasing depth enhances long-distance flow rate. These findings provide quantitative design guidelines for capillary-driven microfluidics and thermal management systems and represent, to our knowledge, the first constant-mean-curvature energy model that yields a closed-form long-distance solution for open channels of arbitrary aspect ratio.