Equilibrium bridge solution from a sessile drop partially covered by another fluid

We study the equilibrium solutions of a fluid bridge that connects a horizontal solid substrate with a fluid phase (typically, air) while surrounded by another immiscible liquid. The shape of each interface is considered as a single curve with axial symmetry and constant curvature. The interface between the bridge and the air phase is a spherical cap, while the one between the external liquid and the air has zero curvature. Finally, the liquid-liquid interface corresponds to portions of onduloids or nodoids. The equilibrium solution must satisfy the boundary conditions given by Neumann's at the three fluids junction (triple point) and Young's law at the two fluids-solid contact line (at the substrate). For given fluids, we find that the solution control parameter is the $\gamma$ angle, which is the slope of the surrounding interface respect to the horizontal at the triple point. Using this procedure, we are able to find the $\gamma$ regions where different kinds of solutions are possible. In order to assess which solution is more likely to be found in nature, we compare their surface energies looking for that corresponding to the lowest one. Finally, we find numerical solutions that consider the presence of a vessel wall containing the fluids, by using the analytical results to provide guess values needed to initiate the required iterative process.