The influence of surface tension in thin-film hydrodynamics: gravity free planar hydraulic jumps
Rajesh Kumar Bhagat
TL;DR
The paper demonstrates that surface tension can act as the primary control mechanism for hydraulic jumps in planar thin-film flows by performing a dominant-balance analysis in the zero-gravity limit, which yields a parameter-free leading-order system and a similarity solution for the velocity profile. It reveals a critical Weber-number singularity at $\mathrm{We}=1$ in the depth-averaged momentum balance, which is regularised by the interfacial pressure, providing a robust mechanism for jump formation and location. The work connects the jump to a zero-mode standing wave of the capillary–gravity dispersion relation and shows consistency with standing-wave criteria when comparing to a capillary-dominated limit, yielding a practical criterion for hydraulic control. Overall, it establishes surface-tension-driven hydraulic jumps as a theoretically solid and practically relevant phenomenon for thin-film flows, with implications for microfluidics and coating processes where capillarity governs flow transitions.
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.