Capillary filling dynamics in polygonal tubes

We study the dynamics of capillary filling in tubes of regular polygon cross section. Using the Onsager variational principle, we derive a coupled ordinary differential equation and a 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 t1/2 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.