Surface tension and instability in the hydrodynamic white hole of a circular hydraulic jump

TL;DR

This paper models the circular hydraulic jump as a hydrodynamic white hole by analyzing surface tension effects and flow instability through a wave equation derived from linearized perturbations.

Contribution

It introduces a hydrodynamic metric framework to study surface tension-induced instability and derives scaling relations for the hydraulic jump radius.

Findings

Surface tension causes flow instability and wave blocking.

Derived scaling laws for hydraulic jump radius involving viscosity, gravity, and surface tension.

Identified the hydraulic jump as a hydrodynamic white hole due to wave blocking.

Abstract

We impose a linearized Eulerian perturbation on a steady, shallow, radial outflow of a liquid (water), whose local pressure function includes both the hydrostatic and the Laplace pressure terms. The resulting wave equation bears the form of a hydrodynamic metric. A dispersion relation, extracted from the wave equation, gives an instability due to surface tension and the cylindrical flow symmetry. Using the dispersion relation, we also derive three known relations that scale the radius of the circular hydraulic jump in the outflow. The first two relations are scaled by viscosity and gravity, with a capillarity-dependent crossover to the third relation, which is scaled by viscosity and surface tension. The perturbation as a high-frequency travelling wave, propagating radially inward against the bulk outflow, is blocked just outside the circular hydraulic jump. The amplitude of the wave also diverges here because of a singularity. The blocking is associated with surface tension, which renders the circular hydraulic jump a hydrodynamic white hole.