Capillary rising in a tube with corners

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.