From Water Jet to Little Droplets

A: Hey B, you accidentally left the water faucet not fully closed!

B: Really??? Cause I don’t hear anything even though the faucet is just behind this door.

A: Well I didn’t hear it, I saw it!

B can’t say a word after that..

So is there anything wrong with B’s hearing? Perhaps no, we can’t blame her/him; sometimes a small water stream can make a lot of noise, while some other times it is really quiet. The quiet stream is identical with a cylindrical column of water giving a continuous disturbance to the sink. However it may happen that the column of water reaches a critical length and it breaks up into tiny droplets and falls in disjoined cluster. If the water jet breaks up into droplets before hitting the sink, the disturbance will be more concentrated, hence it will give rise to louder and more annoying sounds. Let’s call the point where the water jet breaks up into droplets point x. I observed this point x is also moving up and down in somewhat jerky manner or probably the motion is actually periodic but it is too fast that our eyes won’t be able to catch up.

Okay let’s play with it experimentally! Take a plastic cup and make a small hole at its bottom, I used a hot nail to melt a hole on it. Then put some water on it, and observe the breaking point. It is hard to observe where exactly the point x is, so I illuminated the water with a flashlight. Since the water column won’t scatter many photons, this way we can clearly distinguish it with water droplets which can scatter many photons. This is how the set up looks like

But just a moment, there is an unwanted effect, which is the water vortex. If the water is spinning before exiting the hole, the resulting jet won’t be cylindrical and it can breaks into droplets much easily and it will be crazy and complicated. Here is one simple way to minimize this effect:

Yupee it works! Ok good, now the experiment can be more peaceful. The length of the water jet depends of the height of water in the container at that instance. Thus I measured the length of water jet as function of the height of water in the container and I found that they are proportional. The higher the height of water column, the longer the jet length will be. And further observation shows that the length of the jet is longer for a larger orifice diameter. Then I think for a while, go to pee, realized that the breaking up occurs independent on the direction of gravity, and decided to stop the measurement. Later I confirmed it using holed water bottle to make a parabolic water jet, and it really turned out to be independent of the direction of gravity.

The first plausible explanation that came to my mind is:

Suppose we drop one ball per second from the roof of a building, initially the distance between two consecutive balls is $g(1)^2/2$. We know that at any time, a ball is always faster than the ball above it, and is always slower than the other ball below it, it means that the distance between two consecutive balls will increase as the balls go down. Similarly this effect may also happens in water stream, as it goes down, it tends to move apart.
The independent of gravity direction argument obviously crushes this explanation.. At least this effect might not be dominant.

Let’s start over and contemplate at this problem more carefully. First it would be awesome to get rid of viscosity, the problem will be much simpler without it. Let’s compare it with surface tension.

$\frac{F_{viscosity}}{F_{surface tension}}=\frac{\eta v}{\gamma}=0.0172$

Where $\eta$ is the viscosity of water, $\gamma$ is the surface tension of water, and $v$ is the characteristic velocity between adjacent layers of water, I assumed it to be equals to the jet velocity which is about $v=1.25 m/s$(I am a bit too generous here). At $20^0C$, $\eta=1.002\times10^{-3} P_a s$, and $\gamma=7.28\times 10^{-2} N/m$.

So the viscosity is no match for surface tension. The surface tension has no alibi! Let’s get him!

A liquid desires to be in a minimal energy state, any movement that will lead to reduction of surface area is favorable by surface tension.  On a level of quasi-static motion it would thus be desirable to collect all fluid into one sphere, corresponding to the smallest surface area. Evidently it does not happen. The surface tension has to work against inertia, which opposes fluid motion over long distances.

The cross section of water jet with circular shape has the lowest energy, in other words the cross section of the jet tends to be circular. In general, the faucet’s hole is not circular, thus it will induce a cross sectional oscillation. Initially the cross section of the water jet is not circular and it tends to be circular. When it reach the circular form, the water still has some kinetic energy (inertia), therefore it overshoots and the cross section continues to evolve further until all the kinetic energy goes into the surface tension potential energy after that it tends to be circular again, and so on. I suspect that this disturbance is amplified as it is transmitted down the jet until it breaks up into drops at a point. Then I take a fork and using its tip I give a small perturbation on the jet, I could see the perturbation propagates upwards and downwards like a wave.  The perturbation that goes upward decays and eventually vanished, but the perturbation that goes downward grows in size, and it moves the point x higher or the water jet shorter. Then I did another experiment, If I put the disturbance higher the point x moves down, if I put the disturbance lower the point x moves up. Which proves my hypothesis that the disturbance amplifies as it goes down. The longer the disturbance travels down the jet the stronger it will be. Thus the measurement of the length of water jet as function of the height of water in the container is pointless because it depends on how uncircular the orifice is. It just proves that for higher water height in the container, the velocity of jet is higher thus it allows the jet travels more distance before the disturbance goes sufficiently large to breaks into drops. And it also proves that for smaller orifices radius, the uncircularness becomes more effective.

The last thing that I tried: Dimensional analysis. I wonder that the skill of using dimensional analysis might be useful for me in the future. The argument that sounds like “there’s only one way to construct <insert a dimension here> by combining the relevant parameters alone” might be useful, it may give us some physical insights, the meaning of that unique dimensional combinations. Some physics problems can be solved easily using this method, e.g. a hoop of rope problem; to find the acceleration of the end point of the rope if the rope is allowed to slide down through a hole. Now back to the case, The only time scale that can be formed from fluid parameters alone is

$t=\sqrt{\frac{r^3 \rho}{\gamma}}$

But what does it means? I don’t know, it looks like some inertial terms divided by the surface tension term. The get some idea, let’s try to plug in the numerical values

$r=2mm$

$\rho=1 kg/m^3$

$\gamma=7.28\times 10^{-2} N/m$

We get

$t=0.331 ms$

This is sooo small, it is not possible that this is the time it takes from the orifice to breaking point. I think it might be the time scale of the point x’ periodic motion, probably it is the time it takes to form a single droplet.

Look at this, it is really so fast