Low-order models for a circular hydraulic jump on a spherical cap

In this work, we analyze the flow of a thin layer of viscous liquid over the outer surface of a sphere due to inertia and gravity. We use the classical problem of a circular hydraulic jump as a starting point and observe the changes in the flow structure as the gravity component along the surface becomes significant. We assume that the flow is stationary and axisymmetric, the curvature of the spherical surface is small, and the capillary forces are negligible. The depth-averaged thin-layer equations describe the flow. We perform a qualitative analysis using a one-parametric representation of the longitudinal velocity distribution and find the necessary conditions for the hydraulic jump existence. The intensity of the jump monotonically decreases, and its radius grows to a certain finite value. The jump vanishes at a finite distance from the axis of symmetry. Using a two-parametric representation, we locate zones of recirculating flow and find the condition of their existence. We find the optimal strategy of averaging by comparing the results of our calculations with the data obtained experimentally and by using simulations in the framework of the full Navier–Stokes equations.