The shape of two-dimensional liquid bridges

We have studied a single vertical, two-dimensional liquid bridge spanning the gap between two flat, horizontal solid substrates of given wettabilities. For this simple geometry, the Young-Laplace equation can be solved (quasi-)analytically to yield the equilibrium bridge shape under gravity. We establish the range of gap widths (as described by a Bond number [Formula: see text]) for which the liquid bridge can exist, for given contact angles at the top and bottom substrates ([Formula: see text] and [Formula: see text], respectively). In particular, we find that the absolute maximum span of a liquid bridge is four capillary lengths, for [Formula: see text] and [Formula: see text]; whereas for [Formula: see text] and [Formula: see text] no bridge can form, for any substrate separation. We also obtain the minimum value of the cross-sectional area of such a liquid bridge, as well as the conditions for the existence and positions of any necks or bulges and inflection points on its surface. This generalises our earlier work in which the gap was assumed to be spanned by a liquid film of zero thickness connecting two menisci at the bottom and top substrates.