Capillary Rise in Sharp Corners: Not Quite Universal

We study the capillary rise of viscous liquids into sharp corners formed by two surfaces whose geometry is described by power laws, h_i(x) = c_i xⁿ, i = 1,2, where c₂ > c₁ for n ≥ 1. The lubrication approximation is used to derive a partial differential equation for the evolution of the liquid column that rises into the corner. It is found that the dynamics are unaffected by potential asymmetry of the corner and that the shape of the liquid column is independent of c_i when described using the interface radius. Furthermore, despite the lack of geometric similarity of the liquid column cross-section for n>1, there exists a scaling and a similarity transformation that are independent of corner geometry. Consequently, the t^1/3 power-law for capillary rise applies regardless of the corner geometry. However, the prefactor, which corresponds to the tip altitude of the self-similar solution, is a function of n, and it is shown to be bounded and monotonically decreasing as n → ∞. Theoretical results are compared against experimental measurements of the time evolution of the tip altitude and of profiles of the interface radius as a function of altitude. The generalization to asymmetrical corners (c₁ ≠ -c₂) enables comparison of analytical and experimental results for corners with n>1.