This article presents the results of a study on an axisymmetric gravity-driven slender free-surface flow down a cone by deriving depth-averaged conservation equations on a cone-adapted coordinate system and obtaining a backwater-type differential equation for steady, axisymmetric films with prescribed apex discharge. Analysis of this equation reveals a location-dependent critical condition separating supercritical and subcritical regimes and shows that a classical constant normal depth does not exist; instead, the flow approaches an equilibrium between gravity and resistance forces as it develops downstream. Asymptotic expansions for the flow and critical depths recover previously established results for the laminar leading-order and first-order corrections under consistent velocity shape coefficients, confirming that capillarity affects only first-order terms. The framework predicts a critical length beyond which the flow must be subcritical, Reynolds number decays inversely with the distance, leading to inevitable relaminarization on sufficiently long cones, and the potential need for hydraulic jumps to compatibilize supercritical and subcritical flow regimes, paralleling open-channel hydraulics on mild slopes. Numerical solutions of the backwater equation agree with existing measurements where the slender-film assumptions hold, providing a practical basis to compute flow depth and regime transitions on conical surfaces.
González et al. (Tue,) studied this question.