  An analytical model is proposed for the Young-Laplace equation of
two-dimensional (2D) drops under gravity. Inspired by the pioneering work of
Landau & Lifshitz (1987), we derive analytical expressions of the profile of
drops on flat surfaces, for arbitrary contact angles and drop volume. We then
extend our theory for drops on inclined surfaces and reveal that the contact
line plays a key role on the wetting state of the drops: (1) when the contact
line is completely pinning, the advancing and receding contact angles and the
shape of the drop can be uniquely determined by the predefined droplet volume,
sliding angle and contact area, which does not rely on the Young contact angle;
(2) when the drop has a movable contact line, it would achieve a wetting state
with a minimum free energy resulting from the competition between the surface
tension and gravity. Our theory is in excellent agreement with numerical
results.

Fluid Dynamics;Soft Condensed Matter