  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.
