Surface-Tension-Driven Hydraulic Jumps

• Surface-tension-driven hydraulic jumps are abrupt thickness changes in thin liquid films where capillary forces, rather than gravity, control the flow.
• Experiments and theoretical models reveal that in micro-scale regimes, the jump’s location scales with flow rate and fluid properties in a capillary–viscosity regime.
• The insights inform applications in microfluidics and jet rebound design, emphasizing the importance of capillarity in controlling thin-film dynamics.

Surface-tension-driven hydraulic jumps (CHJs) are abrupt discontinuities in the thickness of a thin liquid film, where the primary force arresting the spreading flow is surface tension, rather than gravity. These phenomena are realized most clearly in small-scale (sub-millimetric to millimetric) regimes, such as the rebound of water jets on hydrophobic surfaces or circular/planner jumps observed at kitchen sink scales. In these systems, the classical mechanism attributed to hydrostatic pressure is supplanted by capillary forces, fundamentally altering the scaling, geometric structure, and control parameters that govern the formation and location of the jump. This entry synthesizes key experimental findings, theoretical models, and debates within the current literature.

1. Governing Mechanisms: From Hydrodynamic Control to Capillarity

In classical hydraulic jumps, inertia is balanced by viscous and hydrostatic (gravity) contributions, culminating in a thickness discontinuity near the location where the incoming flow speed matches the gravity-wave speed (Froude number $Fr=1$). In contrast, in the CHJ regime, the film dynamically transitions from supercritical to subcritical flow when the local radial velocity matches the capillary-wave speed, that is, when the Weber number $We = \rho\,u^2\,h/\gamma$ reaches unity. Surface tension enters through the Laplace pressure at the free interface and, for thin films, fundamentally governs both the jump's position and its structure (Bhagat et al., 2018, Bhagat et al., 2017, Bhagat et al., 2020, Bhagat, 16 Dec 2025).

Mass and momentum conservation in the axisymmetric, thin-film limit yield a set of ordinary differential equations, with the jump radius $R_j$ identified by a singularity in curvature or the denominator of an energy/momentum balance, both at $We \sim 1$ (Bhagat et al., 2018). This geometric control plays a central role in both circular and planar CHJs (Bhagat, 16 Dec 2025).

2. Experimental Evidence and Gravity Independence

Microgravity experiments provide compelling evidence for the dominant role of capillarity in setting the jump scale. Investigations by Avedisian et al. and Painter et al. measured the jump radius under terrestrial and $2\%\,g_E$ (microgravity) conditions, showing no perceptible change in $R_j$, while the downstream profile evolved due to lower hydrostatic pressure (Bhagat et al., 2020). Systematic variation of gravity orientation (horizontal, vertical, inverted) similarly leaves the jump position unchanged (Bhagat et al., 2018, Bhagat et al., 2017).

Comprehensive experimental datasets for water, aqueous surfactants, glycerol mixtures, and silicone oils consistently show $$R_j \propto Q^{3/4} \rho^{1/4} \nu^{-1/4} \gamma^{-1/4}$$ over several orders of magnitude in $Q$, $\nu$, and $\gamma$, provided $Q \ll Q_C \sim \gamma^2/(\nu\,\rho^2\,g)$ so that gravity is negligible (Bhagat et al., 2020).

3. Scaling Laws: Regime Classification and Dimensional Analysis

CHJ systems exhibit two principal scaling regimes, distinguished by which force (gravity or capillarity) limits the expansion of the film:

Regime	Radius Scaling	Dominant Physics
Gravity–viscosity (Bohr type)	$R_j \propto Q^{5/8}\nu^{-3/8}g^{-1/8}$	Hydrostatic plus viscous drag
Capillarity–viscosity (Bhagat type)	$$R_j \propto Q^{3/4}\rho^{1/4}\nu^{-1/4}\gamma^{-1/4}$$	Surface tension versus viscous dissipation

The surface-tension-dominated scaling (capillary–viscosity regime) is observed for flows with $Q \ll Q_C$. At higher $Q$, or for thicker films, gravity becomes significant and the classical scaling (gravity–viscosity) is recovered (Bhagat et al., 2018, Duchesne et al., 2021, Bhattacharjee et al., 2020).

The critical Weber–Froude condition $We^{-1} + Fr^{-2} = 1$ encapsulates the crossover between the two physical regimes, with $We \approx 1$ in the capillary limit and $Fr \approx 1$ in the gravity-dominated limit (Bhagat et al., 2020, Bhagat et al., 2017).

4. Surface-Tension-Driven Jet Rebound and Microfluidic Implications

Celestini et al. experimentally demonstrated that for sub-millimetric water jets impinging on hydrophobic or superhydrophobic substrates, capillarity can dominate to the extent that the jet, after decelerating via a capillary hydraulic jump, rebounds off the surface without destabilizing into drops. The transition from supported ('landing') jet to stable rebound to destabilization is controlled by the incident velocity and substrate wettability (quantified via the equilibrium contact angle $\theta_e$). The scaling $R/r_0 \approx We/2$ applies to the jump radius in terms of the jet radius $r_0$, and the reflection angle obeys a modified specular law tied to restitution coefficients and substrate hydrophobicity (Celestini et al., 2010).

This regime map is central to emerging 'jet micro-fluidics,' where fine control of flow paths, splitting, and rebound on patterned surfaces is achieved purely by manipulating capillarity via substrate design and flow rates.

5. Instabilities, Polygonal Jumps, and Wave Blocking

Surface tension not only determines the steady-state jump but also underpins the symmetry-breaking and dynamic stability of the system:

Instabilities and Polygonal Hydraulic Jumps

In the 'roller' region of a type II jump, azimuthal instabilities—governed by the Bond number and surface-tension-driven Laplace pressure—can destabilize the jump into $N$-sided polygons. The most unstable wavelength scales with the local roller width, reminiscent of the Rayleigh-Plateau instability mechanism. The linear stability criterion predicts instability bands in terms of gravity, capillarity, and geometry (Martens et al., 2011).

Analogue Gravity and Wave Blocking

The jump front can be interpreted as a white-hole horizon for surface waves: inward-propagating (capillary-gravity) disturbances are blocked and amplified at the jump radius. Dispersive corrections due to capillarity induce instabilities of capillary ripples and smoothed wave horizons in circular jumps (Bhattacharjee et al., 2020). The phase velocity for capillary waves replaces the gravity-wave speed in setting the hydraulic control, and the amplitude of blocked upstream waves formally diverges at the jump.

6. Theoretical Controversies and Model Limitations

There is an ongoing debate concerning the appropriate energetic and thermodynamic accounting of surface tension in the CHJ context. Bhagat et al. argued for the dominance of a surface-energy transport term in the energy balance, leading to their scaling law. Subsequent critique demonstrated that, under rigorous derivation—including both convective and surface terms—surface tension's effect in thin films reduces strictly to Laplace pressure via curvature, entering the energy and normal-momentum balances but not generating new distributed “$\sigma\kappa$” terms in the film interior (Duchesne et al., 2019, Duchesne et al., 2021). This clarifies the conditions under which capillarity sets the jump, reaffirming that, except near the rim where the free surface is highly curved, distributed capillary terms are subleading in the governing balances.

Additionally, in planar geometries and under zero-gravity scaling, the full interfacial stress conditions and the role of the deviatoric normal stress show a unique singularity at $We=1$, interpreted as the criterion for hydraulic control. This singularity is regularized by a pressure gradient across the jump, providing a closed analytical prediction for the jump location (Bhagat, 16 Dec 2025).

7. Summary and Perspectives

Surface-tension-driven hydraulic jumps reveal a distinct regime of thin-film hydrodynamics where capillarity, rather than gravity, is the primary force determining jump location, structure, and stability. The critical features are:

• A sharp, capillarity-controlled radius scaling of the form $$R_j \propto Q^{3/4} \rho^{1/4} \nu^{-1/4} \gamma^{-1/4}$$ in circular and planar geometries.
• Independence from gravity and orientation for sufficiently thin films and low flow rates.
• Rich phenomenology including jet rebound, patterning via wettability, capillary instabilities, and analogy to horizons in black-hole hydrodynamics.
• The decisive role of jump rim and interface geometry, as opposed to distributed capillarity in the interior, in determining the hydraulic control.
• Ongoing debates about the correct energy- and momentum-balance representations, especially regarding the inclusion (or exclusion) of convective surface energy transport (Duchesne et al., 2019, Duchesne et al., 2021).

A plausible implication is that further refinement of jump models, including detailed treatment of contact lines, time dependence, and three-dimensional flows, will clarify quantitative discrepancies while reinforcing the foundational role of surface tension in thin film hydraulic phenomena.