Scaling analysis of capillary channel flow

Capillary channel flow (CCF) is important in microgravity and microscale open-channel systems, where operation is limited by a critical flow rate above which the free surface collapses and gas is ingested, causing choking. Although one-dimensional models and three-dimensional computations can determine this limit numerically, a compact explicit scaling expression for different geometries remains desirable. In this work, steady CCF is analyzed using a balance among capillary pressure, convective acceleration, and viscous pressure loss. The resulting formulation predicts the critical flow rate from the critical velocity scale and channel cross-sectional area. Comparison with International Space Station experimental data shows that the model captures the measured length dependence for parallel-plate (PP), groove (GR), and wedge (WE) channels. The fitted coefficient is s=0.71 for both the PP and groove channels and s=0.45 for the wedge channel, indicating that neglected geometric and numerical factors remain of order unity. A parametric analysis based on the Ohnesorge number and visco-geometric parameter quantifies the transition between convection-dominated and viscous-loss-dominated regimes. In the convection-dominated and viscous-loss-dominated limits, the dimensionless critical velocity approaches V=s(1+2/Ω)1/2 and V=2s/(Kfl̃), respectively. These asymptotic expressions clarify how inertia, viscous loss, channel length, and free-surface curvature control choking onset. The fitted coefficient is further interpreted using a cross section-based Weber number as an a posteriori measure of the inertial–capillary balance. Once calibrated by one representative experiment or high-fidelity computation, the scaling analysis can estimate critical flow rate trends for geometrically similar open capillary channels under comparable boundary conditions.