Capillary rise in sharp corners: not quite universal

Abstract

We study the capillary rise of viscous liquids into sharp corners formed by two surfaces whose geometry is described by power laws $h_i(x) = c_i x^n$ , $i = 1,2$ , where $c_2 > c_1$ for $n \geq 1$ . 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 $n\to \infty$ . 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.