The influence of surface tension in thin-film hydrodynamics: gravity free planar hydraulic jumps

TL;DR

This paper develops a theoretical framework for understanding surface tension-driven hydraulic jumps in thin films, challenging the gravity-centric view and providing analytical predictions for jump characteristics in zero-gravity conditions.

Contribution

It introduces a new theoretical model for surface tension-driven hydraulic jumps in thin films, emphasizing the dominant role of surface tension and deriving parameter-free governing equations.

Findings

Identifies a singularity at the Weber number of one as the control criterion.

Provides a similarity solution for the velocity profile in surface tension-driven jumps.

Predicts jump location and structure analytically.

Abstract

Hydraulic jumps in thin films are traditionally explained through gravity-driven shallow-water theory, with surface tension assumed to play only a secondary role via Laplace pressure. Recent experiments, however, suggest that surface tension can be the primary mechanism. In this work we develop a theoretical framework for surface tension driven hydraulic jumps in planar thin-film flows. Starting from the full interfacial stress conditions, we show that the deviatoric component of the normal stress enters at leading order and fundamentally alters the balance. A dominant-balance analysis in the zero-gravity limit yields parameter-free governing equations, which admit a similarity solution for the velocity profile. Depth-averaged momentum conservation then reveals a singularity at unit Weber number, interpreted as the criterion for hydraulic control. This singularity is regularised by a non-trivial pressure gradient at the jump. This work establishes the theoretical basis for surface-tension-driven hydraulic jumps, providing analytical predictions for the jump location and structure.