Water, water everywhere! (and drops to spare!)

Water is your immutable best friend. It amounts for 70% of the Earth’s mass, is essential for biological processes (as well as industrial processes), and is currently coating Boston in a very pretty display of snow:

2016-02-05 11.58.41
View of the snow in Boston

But why, you might ask? Why is water so important? Is it the 104.5° bond angle caused by double lone-pair repulsion? Is it because ice is less dense than liquid water due to the crystal structure and hydrogen bonding? Is it because it’s an incredibly good solvent from the aforementioned hydrogen bonding capability? Perhaps a combination of all of the above. One can try to characterize many different aspects of water (as we have done in previous posts), but today we will discuss the concept of water droplets.

Droplet movement is probably one of the coolest events to observe, evidenced below:

What’s going on here? Let’s construct a naive force diagram to understand the forces acting on a droplet when it hits the surface:

force_diagram
Force diagram of a droplet landing on a liquid surface.

Gravity drags the droplet downwards, but there exists a surface tension between the surface of the droplet and the surface of the liquid that it lands upon. The key is that the surface tension exists parallel to the surface of the liquid in question- so to minimize this force, the droplet wraps itself into a sphere. How large of a sphere is formed? This is governed by the Young-Laplace Equation, which states:

Δp = 2ϒ/R

where Δp is the interfacial pressure drop, ϒ is the surface tension, and R is the radius of the sphere formed. So the pressure difference inside the liquid bubble and the outside air drives the sphere formation. And water has a high surface tension due to the strength of the O-H hydrogen bonds. This important property allows for things like this or this. Or maybe best visualized by this picture:

surface_tension_01
Paper clip floating on water due to surface tension. Source: Wikimedia

Understanding tiny droplets is cool and all, but how can we relate this to a more realistic example? For example, what happens in a water faucet, when water seemingly spills in a column? Check out this video to get inspired about the idea!

Let’s break it down. When water flows quickly out of a small faucet/nozzle, it is in the turbulent flow regime. Here water is flowing so fast that it doesn’t have ‘time’ to form droplets. But what happens if we try to interrupt the flow higher up in the stream? You can actually try this out!

Go to a nearby sink and let the tap run at a slow rate, i.e. a thin stream. Place your finger at the bottom of the water column- the flow will probably be uninterrupted. But if you move your finger closer to the nozzle, you will notice that the water will actually separate into different droplets! My 10.301 friends will remember this as one of the questions on PSet #1 :). Why does this happen? Check out the below diagram:

bond_capillary_numbers
Description of breaking up water flow into tiny droplets. Bo, or the Bond number, represents the ratio of gravity to surface tension forces. Likewise Ca, or the capillary number, represents the ratio of viscous to surface tension forces.

The above diagram attempts to explain why this phenomenon occurs. The Bond number, which is a ratio of the gravity to surface tension, is notably dependent on the length of the column flowing, L, and the surface tension, ϒ of the water. This ratio of forces indicates when one force dominates the other; if Bo << 1, then gravitational forces become unimportant. Similarly, the capillary number is a ratio of the viscous to surface tension forces. Viscous forces are important when the flow is laminar (for more information, see this), i.e. when you slow the water flow out of the faucet. Let’s just say that I did the calculations as below:

martian-donald-glover
Like Rich Purnell from The Martian, my calculations are correct. Source: ReBrn

and that Bo << 1 (gravity negligible) but Ca ~ 1 (viscous forces compete with surface tension to form water drops). So at the end of the day, we found that when the surface tension force dominates the viscous forces acting on the water, then droplets form. And there, my friends, is the first lecture of Fluid Mechanics en route to becoming a Chemical Engineer. Remember, it’s never too late to switch majors 😉

Links

Young-Laplace Equation

Thickness of surface tension ‘skin’

How to create the perfect skipping stone

Jesus walks on water

Coalescence cascade- water droplet falling on water

YouTube video of coalescence cascade

Reynolds Number: Inertial to viscous forces

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