Capillary imbibition in a square tube

When a square tube is brought in contact with bulk liquid, the liquid wets the corners of the tube, and creates finger-like wetted region. The wetting of the liquid then takes place with the growth of two parts, the bulk part where the cross section is entirely filled with the liquid and the finger part where the cross section of the tube is partially filled. In the previous works, the growth of these two parts has been discussed separately. Here we conduct the analysis by explicitly accounting for the coupling of the two parts. We propose coupled equations for the liquid imbibition in both parts and show that (a) the length of each part, $h_0$ and $h_1$, both increases in time $t$ following the Lucas-Washburn's law, $h_0 \sim t^{1/2}$ and $h_1 \sim t^{1/2}$, but that (b) the coefficients are different from those obtained in the previous analysis which ignored the coupling.