Harmonic solutions for polygonal hydraulic jumps in thin fluid films

Abstract

This article contains numerical and theoretical results on the circular and polygonal hydraulic jumps in the framework of inertial lubrication theory. The free surface and velocity fields are computed along with cross-sections of the vorticity and pressure, in agreement with experimental data. The forces that drive and resist the instability are identified with the radial shear force, the azimuthal surface tension and the hydrostatic azimuthal force, in addition to a nonlinear term in the radial coordinate. Periodic solutions are obtained from the first orders of a perturbation theory by considering azimuthal symmetries. The thresholds of the instability are defined at closed jumps for discontinuous solutions and at one-sided hydraulic jumps for continuous curves that conserve fluid mass density.