Capillary Filling Dynamics in Polygonal Tubes

TL;DR

This paper investigates the dynamics of capillary filling in polygonal tubes, deriving coupled equations to describe flow and saturation, revealing a $t^{1/2}$ scaling and the influence of polygon shape on flow behavior.

Contribution

It introduces a coupled modeling approach for capillary flow in polygonal tubes, highlighting the shape-dependent effects on flow dynamics and deviations from classical predictions.

Findings

Flow follows $t^{1/2}$ scaling in polygonal tubes.

The shape influences the flow rate prefactor, approaching Lucas-Washburn as the number of sides increases.

Coupled equations accurately describe the saturation and flow profiles.

Abstract

We study the dynamics of capillary filling in tubes of regular polygon cross-section. Using Onsager variational principle, we derive a coupled ordinary differential equation and partial differential equation, which respectively describe time evolution of the bulk flow and the saturation profile of the finger flow. We obtain both numerical solution and self-similar solution to the coupled equations, and the results indicate that the bulk flow and the finger flow both follow the $t^{1/2}$ time-scaling. We show that due to the coupling effect of the finger flow, the prefactor for the bulk flow is smaller than that of the Lucas-Washburn prediction. The reduction effect is more pronounced when the side number $n$ of the regular-polygon is small, while as $n$ increases, the prefactor approaches Lucas-Washburn prediction.