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_ixⁿ, i = 1, 2, where c₂ > c₁ for n ≥ 1. Prior investigations of capillary rise in sharp corners have shown that the meniscus altitude increases with time as t^1/3, a result that is universal, i.e. applies to all corner geometries. The universality of the phenomenon of capillary rise in sharp corners is revisited in this work through the analysis of a partial differential equation for the evolution of a liquid column rising into power-law-shaped corners, which is derived using lubrication theory. Despite the lack of geometric similarity of the liquid column cross-section for n > 1, there exist a scaling and a similarity transformation that are independent of c_i and n, which gives rise to the universal t^1/3 power law for capillary rise. 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 → ∞. Accordingly, the profile of the interface radius as a function of altitude is also independent of c_i and exhibits slight variations with 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.