Consider a (unperturbed) symmetric configuration as sketched below. The effect of gravity is neglected. The cross-section of the liquid free surface is an arc of circle of radius r, of curvature and half-angle . The liquid-solid contact angle is a prescribed material property. The area of the liquid cross-sectional domain is A, and the liquid capillary pressure is (assuming the pressure of vapor phase to be zero). We call the curvature of the wall at the contact line (where the three phases meet).
We have found that, when the inequality
is satisfied between the contact line mean curvatures and of the liquid and solid interfaces, the resulting liquid ridge is stable with respect to all transverse sinuous perturbations.
Otherwise, we find that each meniscus is characterized by a critical wavelength above which all such transverse perturbations are destabilizing, and in this case, the liquid ridge may break into separate droplets, and dewet some portion of the substrate. We also showed that this stability criterion is equivalent to
where the meniscus pressure p is viewed as a function of its cross-sectional area A, keeping all other physical parameters (contact angle, substrate geometrical parameters) constant.
Consider for example a substrate in the form of a wedge of half-angle , . Two possible configurations of symmetric liquid ridges are possible. For a negative-pressure meniscus, the liquid/vapor interface is convex as long as the contact angle satisfies , and the liquid ridge is stable to all perturbations. Conversely, for , a concave interface, that is, a positive-pressure meniscus, results to an unstable liquid configuration. Note that menisci in wedges are always stable in the limit of zero contact angle.
For more detail, consult On the Stability of Liquid Ridges, R.V. Roy & L.W. Schwartz, to appear in Journal of Fluid Mechanics.