Universality of capillary rising in corners

We study the dynamics of capillary rising in corners. Using Onsager principle, we derive a partial differential equation that describes the time evolution of meniscus profile. We obtain both numerical solutions and self-similar solutions to this partial differential equation. Our results show that the advance of the meniscus front follows a time-scaling of $t^{1/3}$, in agreement with the experimental results and theoretical conjecture of Ponomarenko et al.

💡 Research Summary

In this paper the authors investigate the dynamics of viscous capillary rise in a corner formed by two curved walls described by the power‑law function y = c xⁿ (n > 1). Using the Onsager variational principle, they construct a Rayleighian that combines the rate of change of free energy (gravity and interfacial contributions) with the viscous dissipation calculated under the lubrication approximation. By minimizing the Rayleighian with respect to the volume flux Q(z,t) and invoking the conservation of fluid volume, they derive a nonlinear partial differential equation governing the evolution of the meniscus profile G(z,t).

Specializing to the power‑law geometry, they obtain explicit expressions for the geometric functions B(G), A′(G) and the derivative of the curvature term, leading to a compact evolution equation (3.6). After introducing characteristic length and time scales based on the capillary length a_c = √(γ/ρg) and the viscous time η a_c/γ, the equation is rendered dimensionless (3.8)–(3.9).

The authors then seek self‑similar solutions of the form G̃(ẑ,t̃)=F(χ) t̃^{α}, χ=ẑ t̃^{β}. Matching powers of t̃ yields α = −1/(3n) and β = −1/3, which implies that the tip position Z_m grows as t^{1/3} irrespective of the exponent n. The ordinary differential equation for the similarity profile F(χ) (3.15) is solved numerically with boundary conditions that enforce the known equilibrium shape at small χ and a finite tip at χ = χ₀. The resulting χ₀ depends weakly on n; for n ranging from 1 (linear corner) to 5 the prefactor C = χ₀ (n² cos²θ/3)^{1/3} varies by less than 10 %.

To validate the theory, the authors compute numerical solutions for two representative cases: (i) a linear corner (n = 1) corresponding to two nearly parallel plates, and (ii) a quadratic corner (n = 2) where the walls follow y ∝ x². In both cases the full time‑dependent meniscus profiles are obtained by solving the dimensionless PDEs (4.2) and (4.10). The tip position follows the predicted t^{1/3} law, and the extracted prefactors (χ₀≈1.8098 for n = 1 and χ₀≈1.0646 for n = 2) agree with experimental measurements reported by Higuera et al. (2008) and Ponomarenko et al. (2011).

The key contribution of the work is the demonstration that capillary rise in corners exhibits a universal scaling behavior that is independent of the detailed geometry of the corner. By employing the Onsager principle, the authors provide a systematic variational framework that can be extended to more complex geometries, non‑Newtonian fluids, or dynamic contact angles. The finding that the prefactor C is nearly geometry‑independent explains the experimental observation of data collapse across different corner shapes and supports the broader notion that gravity‑dominated capillary flows in confined geometries are governed by a simple t^{1/3} law. This insight has practical implications for microfluidic device design, enhanced oil recovery, and the interpretation of natural capillary phenomena where corners or wedges are prevalent.