Minimum energy shapes of one-side-pinned static drops on inclined surfaces

The shape that a liquid drop will assume when resting statically on a solid surface inclined to the horizontal is studied here in two dimensions. Earlier experimental and numerical studies yield multiple solutions primarily because of inherent differences in surface characteristics. On a solid surface capable of sustaining any amount of hysteresis, we obtain the global, and hence unique, minimum energy shape as a function of equilibrium contact angle, drop volume, and plate inclination. It is shown, in the energy minimization procedure, how the potential energy of this system is dependent on the basis chosen to measure it from, and two realistic bases, front-pinned and back-pinned, are chosen for consideration. This is at variance with previous numerical investigations where both ends of the contact line are pinned. It is found that the free end always assumes Young's equilibrium angle. Using this, simple equations that describe the angles and the maximum volume are then derived. The range of parameters where static drops are possible is presented. We introduce a detailed force balance for this problem and study the role of the wall in supporting the drop. We show that a portion of the wall reaction can oppose gravity while the other portion aids it. This determines the maximum drop volume that can be supported at a given plate inclination. This maximum volume is the least for a vertical wall, and is higher for all other wall inclinations. This study can be extended to three-dimensional drops in a straightforward manner and, even without this, lends itself to experimental verification of several of its predictions.