Wetting equilibrium in a rectangular channel

When a capillary channel with corners is wetted by fluid, there are regions where the fluid fills whole cross-section and only filled fluid. The fraction of partially-filled region, $s^*$, an important quantity related to pressure. We calculate value $s^*$ for channels slightly deviated from rectangle: height larger in center than those on two short sides. find that small change geometry leads huge $s^*$. This result consistent experimental observations.