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

Physics of Walking While Holding a Plastic Bag

When we are walking in a straight line, while holding a plastic bag on our hand, with an acceleration much smaller than g. There are two possible cases that may happen….

Sometimes the plastic bag would swing so peacefully that it makes almost no difference whether it is there or not….

But sometimes it may swing with a large amplitude, annoying enough to force us to make a conscious attempt to stop it.

Then how should we walk in order to avoid the latter case?

Let’s take a look at the plastic bag:

Yes, that is how it looks like in the physics world!

We can neglect the mass of the plastic bag relative to the total mass of the stuffs inside the bag. Due to centrifugal force, the stuffs inside will distribute themselves as far as possible from the pivot(finger holding the bag), so it is quite likely that they will end up having moment of inertia close to ml^2 relative to the pivot. And since the plastic bag has a large cross sectional area, we cannot underestimate the power of air drag. Let’s say it is proportional to the velocity of the bag, so we can write the torque due to the drag force as \tau_{drag}=-k l^2 \frac{d\theta}{dt}. Further I will assume that the mass of the plastic and the stuffs inside is small enough that we can neglect the transient effect. Yet another assumption, we may say that while walking human’s center of mass translates in approximately sinusoidal motion so that we can write the acceleration of center of mass as a=a_0sin\omega t. Since we are working in the hand’s frame, actually the drag force will change the equilibrium position so that it is not vertical anymore, but we will assume that this effect is small.

Okay now let’s write down the equation of motion, here is the torque about pivot equation:

ml^2\frac{d^2\theta}{dt^2}+kl^2\frac{d\theta}{dt}+mgl\theta=ma_0 lsin\omega t

Forget about the homogeneous solution, the steady state solution for this differential equation is:

The amplitude of oscillation

\theta_0= \frac{a_0}{l\sqrt{(\gamma^2\omega^2+((g/l)^2-\omega^2)^2}}

where \gamma=k/2m

So the high amplitude motion will occurs at resonance, when the \omega of walking pace close to \sqrt{\frac{g}{l}}

But it is not that simple.. I measured my own standard walking pace using my ipod, the average time it takes for me to do ten steps is 4.91 seconds, hence the average time between steps is 0.491 seconds or the frequency of pace is 2.04 Hz. And the natural frequency of oscillation of a random plastic bag filled with snacks and drinks can be calculated using f=\frac{1}{2\pi}\sqrt{\frac{g}{l}} , and the result is 0.760 Hz for l=43cm. Thus resonance won’t occur at around the standard walking pace.

But we may pass through the resonance frequency during the process of building up speed. During the process, the frequency pace is increased from zero to 2.04 Hz, at some point it will hit 0.760 Hz. Now the thing is during the process how long will it stay inside the “near resonance region” which is roughly \omega_r -\gamma<\omega<\omega_r+\gamma? Where the amplitude of the oscillation is at least half the maximum amplitude. The longer it stay in that region the more effective the resonance will be. We know that the characteristic time required for the oscillations to die out is 1/\gamma. Therefore our walking pace must pass through this “near resonance region” in time more than 1/\gamma to get significant excitation of the resonance. The value of gamma can be calculated by comparing the natural frequency of oscillation with the experimental frequency of oscillation which turns out to be f_1=0.735Hz. We get \gamma=1.45 Hz

The average walking velocity is related to the pace frequency as follows

v=d\frac{\omega}{2\pi}

Where d is the pace length. It can be measured by wetting my sandals before walking on the dry road, and then I came back and measure the distance between two  footsteps using a ruler.

The average distance turns out to be d=110cm . Assuming constant pace length, the change in velocity corresponds to the change in \omega as follows:

\delta v=\frac{d}{2\pi}\delta\omega

Thus the time required to pass through the resonance width \delta\omega=2\gamma is

\delta t=\frac{d\gamma}{a\pi}

Which must be larger than the damping time 1/\gamma in order to excite an effective resonance. At the end we get

a<\frac{d\gamma^2}{\pi}=0.736 m/s^2

Therefore to avoid an annoying high amplitude swinging plastic bag, we must accelerate faster than 0.736 m/s^2, at least for this plastic bag

Childhood Question Finally Solved?

When you look at a light source you will notice something awesome, don’t you?

The bright point of the light will appear to be surrounded by a spray of radial light rays!

This thing is so cool that one may not be able to sleep because he or she can’t stop playing with it.

But what are those rays???

It is one of the biggest questions that I have since I was a child…

You may say, “Maybe it is due to scattering by the air”

OK now look at the light source again, and rotate you head, what do the rays do?

They rotate with you!

Then rotate the light source if you can.

Notice that the rays do not rotate!

So it is something to do with our eyes? Perhaps something related to our eyelids, eye liquid, brain perception or such things?

Not sure yet? Ok now choose an opaque object around you, it could be your hand, your Gundam toy, a book or etc..

Hold that thing between your eyes and the light source.

Try to block the rays without blocking the light source.

Notice that even if you hold the thing very close to your eyes the rays are always magically appears between the

the thing and your eyes. And also the rays will vanish if you cannot see the light bulb.

So it is a “biological” effect? But my camera does not agree, she says that she can capture that effect too!

What is Going On?

Wait a second… Look at that innocent little bottle of chili sauce over there!

Did you see the “white line” on the bottle?

Now why? Each “blob” on the bottle will produce one copy of mini light bulb, as a result we got a line of mini light bulbs. If the blobs are much smaller, our eyes won’t be able to distinguish the gaps between the mini light bulbs! I think water ripples also do the same thing:

Now look at this:

Even more epic one:

Thanks to the chili sauce bottle.. Now I have a special power to control light!! Maybe I am the chosen one!

I must protect the world from the dark forces!

Okay okay enough bullshitting.. What did I do??

When I rub the camera using my finger horizontally the light line becomes vertical however when I rub it vertically the line becomes horizontal. The light line is always perpendicular to the direction in which I rub the camera. And if I twist my finger while touching the camera, I got lines in every direction or you can say there is no line at all.

I think when I rub the camera in a certain directions, I will leave trace amounts of oil that I get on my fingers maybe from rubbing my forehead or nose. The trace is in a form of series of parallel lines, as in the chili sauce bottle case and water ripples case it will reflect light perpendicular to itself .

But most photos of such light rays that I saw have very neat six dominant lines. like this one

I think it is a completely different phenomena. Maybe it is because the camera’s light receptor is not isotropic, usually it has hexagonal symmetry.

Let’s get back to our eyes. I would assume that there are scratches on our eye lens and they are more or less randomly oriented. This scratches will act as reflectors as in above cases. The problem is if the scratches are randomly placed why are the rays radial? Only a few selected scratches glint light towards our retina.  The basic principle is a scratch will reflect light perpendicular to itself, thus only scratches perpendicular to our line of sight or “tangential” to the light source will reflect light to our eyes. Therefore it would be very unlikely to have a non radial line, the scratches must be very specially ordered. So the scratches produce lines of light radiating in all directions around the point source. We can also play with the lines by goggling and narrowing our eyes. That way we can change the shape of our eye lens, it’s like stretching the scratches vertically and thus the lines. When we goggle the lines are more “vertical”, when we narrow our eyes the lines are more “horizontal”.

There is another optical effect that we can observe. At first you can look at a light bulb without squinting. Then by squinting the eyes you can see the rays being extended. And when we look above or below the light bulb the the rays are extended further upward or downward. I think it is because when we move our eyelids some eye liquid is moved and covered the cornea , it will change the liquid’s thickness profile at some part. To prove this hypothesis, I tried to yawn several times to produce some tear before looking at the light source. And what did I get? I can see similar long line even without squinting!

Yet another interesting stuff that I discovered while playing with light:

If we look at a bright light source for a long enough  time and then we look the other way, the part where the bright light was will become dark. If we put the source at the center of our vision, it would be awesome. After the light source is removed, the dark spot will show us where we are looking at. For example I tried to read something while the dark spot still there, I found that upon reading my eyes are focused at left edge, middle, and right edge of the text repeatedly. Moreover you can also know where your eyes are focusing while staring at optical illusion picture!