Q:

Height of water layer on flat surface

Let's consider some flat surface which extends to infinity in all directions. 
Suppose we slowly spill some water near some point of the surface. Due to surface tension, the water doesn't spread out indefinitely. If done properly, the water will spread essentially uniformly over a section of the surface, and within this section there won't be any "holes" where the surface isn't covered by water.
If we add some more water, the water will redistrubte itself to maintain a nearly constant height $h$. Furthermore experimentally it seems that $h$ doesn't depend on the amount of water spilled.
Is this true, and how might we calculate the height of the water layer?

A:

I believe this is a standard question in undergraduate physics. The height of the puddle on a flat smooth hydrophobic surface will be given by the following equation:
$h= \sqrt{\frac{2 \gamma (1- cos( \theta)}{g \rho}}$, where $\rho$ is the density of the liquid, $\gamma$ is the surface tension of the liquid, and $\theta$ is the contact angle, that ranges from $\frac{\pi}{2}$ to $\pi$. 
For a formal derivation see :

Pierre-Gilles de Gennes; Françoise Brochard-Wyart; David Quéré (2002). Capillarity and Wetting Phenomena—Drops, Bubbles, Pearls, Waves. Alex Reisinger. Springer. ISBN 978-0-387-00592-8.

EDIT: As I defined above, $\theta$ is within that range for hydrophobic surface. If the surface is hydrophillic, $\theta$ will be less than $\frac{\pi}{2}$. You can take these angles as the definition of hydrophobic/hydrophillic surfaces.

