# 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?

• What are your experimental results for h? Or where is the source? Also, if the surface has dust particles on it then the water may not behave as you expect.
– user207455
Jun 3, 2019 at 17:23

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 :

1. 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.

• Does that change if the surface is hydrophyllic or hydrophobic? Water behaves differently on a car roof depending on whether it has been waxed or not...
– user207455
Jun 3, 2019 at 17:55
• @SolarMike, that's a good point, just edited my answer. Jun 3, 2019 at 17:58
• I was going to ask the OP soon anyway... well done though plus 1 from me.
– user207455
Jun 3, 2019 at 17:59
• Based on the comments, I'm curious now; would this equation already account for how hydrophobic the surface is with the contact angle; or do you need to assume hydrophobic surface to arrive at that relationship?
– JMac
Jun 3, 2019 at 18:07