A Churning Ring of Water

[Title with apologies to Johnny Cash.]

The sink in our new home has an interesting setting that I was curious about:

It sprays a thin film outward, but the water curves back to meet in the middle again. When the water leaves the sprayer, there are only two forces acting on it: gravity pulling it down, and surface tension pulling the droplets together. I mentioned surface tension long ago, but I've never dug into the mechanics of it.

Surface tension is a force that acts to decrease the surface area of a fluid. For a given volume, a sphere has the smallest surface area, which is why water forms drops, and why shot towers can make round bullets. The magnitude of the force is given as

where γ is a constant that depends on the two materials being considered (air and water in this case) and L is the length of the edge that F will act to reduce. The sink is spraying out a ring of water, so if we take a cross-section, L is the inner plus the outer circumference of the ring. We can rewrite the force as

where m is the total mass of water, a is the acceleration, ρ is the density, A is the area of the ring, and Δh is the small vertical slice we're considering. Now this a refers to the radial acceleration of each water molecule, but we want to relate it to back to L. To do that, we can write two equations expressing L and A in terms of the inner and outer radii of the ring:

Since the ring is thin, we can take r1 approximately equal to r2, and after a bunch of algebra write

Since the flow of water is constant, A must be constant, so we can use the above equation to get a timeseries for L, then find r1 and r2.

In order to integrate this, we need initial values for L and Ldot. We can approximate the opening on the faucet to get r1 and r2, and find the initial L and the constant A. Then we can use A with the typical flow rate of 2.2 gallons/minute to find Ldot. Something didn't quite work out with my estimations, since the scale is way off in the following plots, but the shape matches great. Here's a side view of the spray:

and an animation descending through cross-sections:

So far I haven't found much use for this setting when cleaning dishes, but it did give me something interesting to think about!