On the stability of liquid ridges

We consider the stability of a rectilinear liquid region whose boundary is composed of a solid cylindrical substrate of arbitrary shape and a free sur face whose cross-section, in the absence of gravity, is a circular are. The liquid-solid contact angle is a prescribed material property. A variationa l technique, using an energy functional, is developed that predicts the min imum wavelength for transverse instability under the action of capillarity. Conversely, certain configurations are absolutely stable and a simple stab ility criterion is derived. Stability is guaranteed if, for given substrate geometry and given contact angle, the unperturbed meniscus pressure is an increasing function of the liquid cross-sectional area. The analysis is app lied to a variety of liquid/substrate configurations including (i) a liquid ridge with contact lines pinned to the sharp edges of a slot or groove, (i i) liquid ridges with free contact lines on flat and wedge-shaped substrate s as well as substrates of circular or elliptical cross-section. Results ar e consistent with special cases previously treated including those that emp loy a slope-small-slope approximation.