Capillary rising in a tube with corners

TL;DR

This paper investigates the dynamics of fluid rising in cornered capillary tubes, revealing distinct early and late-stage behaviors and the influence of finger formations on the overall rise height.

Contribution

It introduces coupled evolution equations for bulk and finger parts, showing how corners alter fluid rise dynamics compared to circular tubes.

Findings

Early stage dynamics match circular tube behavior.

Late stage finger height scales as t^{1/3}.

Coupling reduces the equilibrium bulk height below Jurin's height.

Abstract

We study the dynamics of a fluid rising in a capillary tube with corners. In the cornered tube, unlike the circular tube, fluid rises with two parts, the bulk part where the entire cross-section is occupied by the fluid, and the finger part where the cross-section is only partially filled. Using Onsager principle, we derive coupled time-evolution equations for the two parts. We show that (a) at the early stage of rising, the dynamics is dominated by the bulk part and the fluid height $h_0(t)$ shows the same behavior as that in the circular tube, and (b) at the late stage, the bulk part stops rising, but the finger part keeps rising following the scaling law of $h_1(t) \sim t^{1/3}$. We also show that due to the coupling between the two parts, the equilibrium bulk height is smaller than the Jurin's height which ignores the effect of the finger part.