Roy RV, Schwartz LW

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.