Influence of gravity on the stability of capillary liquid ridges
Roy RV, Schwartz LW
Journal of Fluid Mechanics, (in review).

Abstract

The stability problem of liquid ridges partially wetting solid, homogeneous surfaces of arbitrary shape in a constant gravity field is examined. More specifically, both the equilibrium capillary surface and the substrate boundary are assumed cylindrical, with generatrices parallel to some y-axis perpendicular to the direction of gravitational acceleration. The liquid-solid contact angle gamma is a prescribed material property. For a given substrate geometry, a liquid ridge is characterized by its cross-sectional area A measured in some plane y= constant, and may be ``sessile'' or ``pendant'' depending on the direction of gravity. Restricting our analysis to symmetrical solid/liquid configurations, we find that the relevant modes of perturbation are either planar modes eta_0 (s), function only of the arclength s measured along the equilibrium line, or transverse sinuous modes eta_1 (s) cos ( 2 pi y / lambda) of wavelength lambda ranging from zero to infinity. By generating numerically the one-parameter ``master family'' of two-dimensional menisci, we find the equilibria neutrally stable to both modes of perturbation, and we build for a given substrate geometry, the stability boundaries in the parameter plane gamma -A. When a given liquid ridge is unstable to transverse modes, we determine the minimum (or ``critical'') wavelength of instability below which destabilization cannot occur under the action of capillarity. We apply a numerical procedure to a number of fundamental geometries, some of which having been studied in the past. We start with the case of rectangular channels for both pendant and sessile ridges and extend the stability diagram first studied by Concus (1963). Next, we give a complete stability picture of pendant ridges with pinned contact lines first studied by Majumbar and Michael (1976). We then examine the case of flat, wedge-shaped, and right-circular cylindrical substrates. We find small-slope approximations of the critical wavelength of instability in the case of both ridges with pinned contact line and ridges on flat substrates; we recover the result of Sekimoto et al in the latter case; these approximations are quite reliable even for large contact angles. We also extend a result which we first proved in the case of negligible gravity (Roy and Schwartz, 1999): liquid ridges remain stable to transverse sinuous perturbations of all wavelengths if they satisfy the condition dp/dA > 0, where the capillary pressure p, measured at some common horizontal level within the fluid, is viewed as a function of A only, keeping all other physical parameters constant. A meniscus satisfying dp/dA =0 is neutrally stable to sinuous perturbations of infinite wavelength. The role of planar perturbations in the case of pendant ridges is to provide a maximum allowable family of equilibria in the gamma -A plane; this family is characterized by solutions of maximum cross-sectional area A. These perturbations play no role in a zero-gravity environment.