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

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.

💡 Research Summary

This paper presents a theoretical framework that fundamentally reinterprets the mechanism behind hydraulic jumps in thin-film flows, demonstrating that surface tension can serve as the primary controlling agent, even in the absence of gravity.

The study challenges the classical gravity-centric shallow-water theory, which traditionally relegates surface tension to a secondary role via Laplace pressure. Motivated by recent experimental observations, the authors investigate steady, planar, thin-film channel flow under zero-gravity conditions to isolate the effects of surface tension. The core innovation lies in a detailed asymptotic analysis that reveals the leading-order contribution of the deviatoric part of the normal stress at the free surface, a component often neglected in conventional scaling.

The theoretical journey begins by re-framing the hydraulic jump as a “zero-mode” standing wave. Analyzing the gravity-capillary dispersion relation shows that criticality—where the flow velocity matches the wave propagation speed—can be governed by either gravity or surface tension depending on the film thickness. For thin films (low Bond number), the criterion shifts to a Weber number condition, suggesting capillary-wave control.

The authors then systematically derive the governing equations and interfacial boundary conditions. A key step is the exact resolution of the stress conditions at the free surface, showing that the viscous normal stress enters the balance at the same order as capillary pressure in the relevant scaling regime. A dominant-balance analysis for the critical, surface-tension-controlled regime yields unique characteristic scales, resulting in a parameter-free system of equations where the Reynolds and Weber numbers are order-one constants.

This simplified system admits a similarity solution for the streamwise velocity profile. Substituting this solution into the depth-integrated momentum conservation equation unveils a mathematical singularity when the local effective Weber number equals unity. This singularity is interpreted as the criterion for hydraulic control, marking the jump location. The paper argues that this singularity is regularized by a non-trivial pressure gradient within the jump structure itself, allowing for a physical transition.

In summary, this work establishes a rigorous theoretical basis for surface-tension-driven hydraulic jumps. It provides analytical tools to predict jump location and structure in thin films, moving beyond the classical gravity-dependent paradigm. The findings have significant implications for understanding fluid dynamics in contexts where gravity is negligible or subdominant, such as in microscale flows, coating processes, and space fluid mechanics.