Capillary currents and viscous droplet spreading

We present the results of a combined experimental and theoretical study of the spreading of viscous droplets over rigid substrates. First, we experimentally investigate the wetting of a roughened glass surface by a viscous droplet of silicone oil, wide and shallow relative to the capillary length $\ell_c$. The horizontal radius of the droplet grows according to an $R_\mathrm{drop}\sim t^{1/8}$ scaling reminiscent of viscous gravity currents (Lopez et al. 1976). The droplet is preceded by a mesoscopic fluid film that percolates through the rough substrate, its radius increasing according to $R_\mathrm{film}\sim t^{3/8}/(\log t)^{1/2}$. To rationalize these observed scalings, we develop a new ‘capillary current’ model for the spreading of shallow droplets with arbitrary radius on rough surfaces. Furthermore, on the basis of established similarities between droplet spreading over wetted rough and smooth substrates (Cazabat & Cohen Stuart 1986), we argue its relevance to a broader class of spreading problems. We propose that, throughout their evolution, shallow droplets maintain a quasi-equilibrium balance between hydrostatic and curvature pressure, perturbed only by unbalanced contact line forces arising along the droplet’s edge. For drops with horizontal radii small with respect to $\ell_c$, our model converges to the original description of Hervet & de Gennes (1984) and thereby recovers the classic spreading laws of Hoffman (1975), Voinov (1976), and Tanner (1979). For drops wide with respect to $\ell_c$, it rationalizes why millimetric, surface-tension-driven capillary currents exhibit the same spreading behavior as relatively large-scale viscous gravity currents.

💡 Research Summary

The authors present a combined experimental‑theoretical investigation of viscous droplet spreading on rigid substrates, focusing on the interplay between capillary forces and gravity. In the laboratory, a silicone oil droplet (high viscosity, total wetting) is placed on a roughened glass surface. The droplet spreads shallowly and laterally, and its bulk radius follows the classic gravity‑current scaling (R_{\text{drop}}\sim t^{1/8}). Ahead of the visible front a thin “Darcy precursor film” percolates through the roughness; its radius grows as (R_{\text{film}}\sim t^{3/8}/(\log t)^{1/2}).

To rationalize these observations, the paper introduces a “capillary‑current” model that treats the droplet interior as being in quasi‑equilibrium between hydrostatic pressure ((\rho g h)) and curvature pressure ((\sigma \nabla^2 h)). This balance holds throughout the evolution, while unbalanced surface‑tension forces at the contact line provide the driving work. Energy dissipation is assumed to occur primarily in a narrow edge region, leading to a logarithmic correction factor (\ell_D=\ln(L/a)) that recovers the classic Hoffman‑Voinov‑Tanner law for small droplets ((R\ll \ell_c)).

When the droplet radius exceeds the capillary length ((R\gg \ell_c)), the hydrostatic pressure becomes essentially uniform and the curvature pressure is confined to a boundary layer of thickness (\sim \ell_c). The resulting gravity‑viscous balance yields the (t^{1/8}) law, explaining why millimetric capillary currents behave like large‑scale viscous gravity currents.

The precursor film is modeled as Darcy flow through the rough substrate, with pressure gradient set by the same hydrostatic term. Solving the Darcy equation gives the (t^{3/8}) growth together with a ((\log t)^{-1/2}) correction, matching the measurements. Because the film separates the liquid from the solid, the classic contact‑line singularity is avoided, making the framework applicable to both rough (Darcy) and smooth (microscopic) substrates.

The model also extends to partial wetting ((S<0)), reproducing results from de Ruijter (1999) and Durian (2022) for early‑time spreading. Comparisons with prior data (Dorbolo 2021, Cazabat 1986, Ehrhard 1993) show good agreement, supporting the claim that the same underlying physics governs a broad class of spreading problems.

Future work suggested includes incorporating non‑Newtonian rheology, anisotropic permeability of the substrate, and external fields (electric, magnetic) to broaden the applicability of the capillary‑current paradigm.