Mathematical model for water droplet shape -Award

Neglecting air resistance and initial distortion from the release,
it will be a sphere held in shape by surface tension.
As it's under free fall, there is no distorting effect from the accelleration.

If you do want to consider the effects of air resistance and
reaction to release, .............thats too much darn work.

Yes, it is a problem of fluid dynamics involving air, liquid, and gravity acceleration at the same time on internal forces on the water drop governed by its internal forces of adhesion. They change the shape of the droplet as it moves. A mathematical model should predict all that in time. I am also planning a high speed recording of the event at 1 million frames per second to see if model is fitting the actuals.

I am also interested in knowing how inkjet droplets hit the paper in accelerated mode, their shape and their splash on impact to be recorded.

It is wonderful field to predict the shape of the event. Think of predicting splash of the dropped egg on the floor.

I have just played with a glass of cooking oil . Dripping water onto the surface eventually forms a water droplet large enough to break surface tension and fall slowly. I don't know if this has any relevance to you , but it was interesting to watch .The form of the water droplet changes nice and slow , the only complication being the breaking free from surface tension.

Hi Kris

Yes that is nice to observe water on oil surface. Most fascinating thing is oil surface on water forming a thin film and giving rise to colors and a simple way of thin film light interference observation. This can be seen on the road in rainy season wherever your car spills some oil.

Rain after a long dry spell is a major road hazard . The rain forces oil that has soaked into the road material back to the surface . This can be worse than the tyres aquaplaning on water alone - after a brief shower inexperienced drivers do not readily see this hazard (made worse because oil is often found at road bends ) if lighting prevents you seeing the colours off the oil.

Your point is very realistic. Oil also causes environmental hazard and worst when it spills in the sea due to ship fire etc.

Some fifty years ago a mathematics professor (Dr. Wray) at Georgia Tech, USA was interested in this problem and once said he did not know the solution, but that he did know lots of non-solutions -- good luck!

Agreed. One such great problem was tear drop model which is glass capsule filled with water, which breaks with lots of energy. I am very much interested in knowing the shape parameters and impact parameters as I am one in supper high speed imaging technology.

You must be make a kidding! use a high speed camera up to 1M frames to shoot the water drop?

Anyone but the you siad. Im afraid.

We use the camera only for high speed object reaserch, like bullet fly out of the gun instantaneously. even at this situation, the speed of 30,000f/s is enough to show clearly the instantaneou state. and can be enlarge to 300,000f/s is very clearly showing its action of in /out etc.

at the station, the noise is very clear as the film behavior itself. the high speed the large partical.

even in track field, we shoot 100M race, we use to take 1/1000 S shutter, sometimes we use 5000f/s speed to record their "spurt" at end line by hs vedio camera.

The 1mf/s camera is very expensive, maybe 300,000usd depend on the record time and revolution etc.

wow you must be very rich or get support of goverment or do something with them.

btw, could youi show us the camera youi are using now?

I saw only one time three years ago and havent operate it at such high speed shoot yet.

Its true that water droplets jetting out are very fascinating for high speed imaging. I agree that 1M frame per second may sound too much but I have designed one for 100 million frames per second to record explosions. It is low resolution camera of 25x25 pixel frame and each pixel has separate sensor attached to fiber, detector, amplifier and data recorder. You can say it consists of 256 transient recorders sampling at 100M samples per second using flash ADCs. It was developed in 1988 and was presented in an International IEEE conference.

These two areas are different research and have the tool so I can use it for any purpose. The new one I am integrating has lower frame rate to 1 million frame per second but has pixel density much higher to 256x256 to record a movie at that rate. There is a line of 256 fiber pixels and I need to go on adding as many lines possible. For moving objects, one line is also OK but picture is formed by object crossing the line like you get picture from an scanner. It is some what like progressive scan image.

I am not afraid of any kind of electronics designs. I need money to make things which at times is limited to my cup of tea, which gets preference over all developments.

The picture you posted here is not good, Im afraid, at least it should be a elliptoid bellow the cone before it drop out of a taper (suppose). although we can image the surface tensile force make the drop to sphere (minimum surface area) shape. but the stream concentrate non-uniform and get mimumum potential energy, its made into ohter curve than sphere.

We like to paint to beautiful water drop when we were in middle school and used to observe it when we were called out to country to re-educated by peasants (haha, its a very interesting, every other one except live in our country will not make clear what it is) so I know its not sphere strickly speaking.

after it take off the taper or other subjject, the shape will be change according to tensile force principle, the cone will be changed and ellipsoid tend to sphere shpage.

rain drop is another thing differnt from drop from taper. air resistance is a veriable force while ti dropping and when it approch ground it will be uniform speed of about 10m/s( I forget the number)

I cannt write its precision formula, but I thnk i can draw its shape by nurbs sometimes. in fact Im sure you would find as much as you can find from the net. maybe. and it animation, and other more like impact ground and spread out etc.

we like to paint the , not to. beautiful shape.

Yes, it is not a real picture. It is a model generated by paintbrush software to give an idea. Actual images at high speed are like balls of all sizes moving together and some are attached to each other forming an oil tanker like image. As the velocity goes up they go almost in straight line so do not change shape very much but at mid level speed where gravity takes over and droplets fall, they get into different shapes.

If you have plunix 1000 fps camera then you can take those pictures of free fall droplets easily but accelerated water droplets need some what faster imaging system. To capture water hitting event, requires almost 1us per event recorder.

Water cutting jet speeds are much higher. I am not sure if you know the speed at which ink jet throws the droplet of ink out. I am working on charged water droplets in accelerated electric field and getting them to know is my research. They are like aerosol of micron size to visible droplets of mm size.

I wonder what you are talking about? " water droplet shape from initial zero velocity to free fall under gravity field ."

or water power cutter tools? They are very different concept! The last works at powerful pressure and get fast speed at outlet. The action of gravity can be ignored simply. the different behave either at scale or direction. I dont know very much about the ctter tools but I know it was produced in chinese half century ago and widely use for many fields like mining etc.

The front speed you can simple calculate by newton law. you can shoot it by only 200-=500f/s enough. if you are in Mumbai, you can go to your movies city t o visit photograhper who shoot special effect ever,

your fim city is famous, it seems called Baolywood?, I forget how to spel.l.

The charge liquid drop move in electric field is another situation either. thier america germany and japan has strong tech at it.

my ink jet printer something wrong, you can give some suggestion how to fix it.

who will offer you such high speed camera, if you lack money?

could you show us more details about the 100M speed camera? Im interesting in it as well.

accelerated model reduce distance and time. Idea is only for free fall modelling of the water droplets. Did you work out something on it?

wow, are you still waiting for answer? the water drop is a sphere shape when it fall free to ground, if you dont take account of the air resistance.

The reason is surface tension, as I speaked above.

The force trend the small drop or droplet to the minimum suface area. tahts why its sphere shape.

this is drawed easily.

The speed it approaches to ground can be calculated by the law of universal gravitation, the equation is:

the force does work will equal to its end kinetick engergy .and intigratial path is from infinite to earth radii.

you can look for the paramter, I cannt remember them. its not difficult.

I try to draw it initial position shape and last shape with 3d softwre. if I cannt forget my painting art.

A big water drip in 3D.

Nice picture cnpower.

Shyam - you may want to allow for the fact that local g value varies around the globe . Most people automatically enter 9.18 , but it can lead to error.

Nice picture cnpower.

Shyam - you may want to allow for the fact that local g value varies around the globe . Most people automatically enter 9.18 , but it can lead to error.

Hello CBPower

This drop in the picture I can see that it is flat at the bottom and there is shape cusp joining the leaf and what you see as sphere is not the complete sphere at all.

Back to the main question--->

Water drop as it comes out from tap hole and elongates due to surface tension bound to the metal and then on its release, it changes shape and forms an oscillator on its way down journey under gravity and its shapes undergoes such oscillations and these oscillations get damped due to air friction.

On motion the tip of the ball faces first impact like a jet and there is air drag around reducing pressure at the back of the drop and it elongates behind compared to the front which has greater pressure on it acceleration path under gravity.

What is its real shape is the question? and this question remain there even now unanswered.

Hello Shyam , I have enlarged the picture and it appears to me as perfectly spherical ! I have not yet superimposed a known circle to check this , but the picture is unexpectedly round. I do not want to divert from the intent of the thread , but an explanation to the picture must have some relevance to the issue .

Surface tension is proportional to 1/R where R is the drop radius, which means greater inner force in smaller drops than in bigger drops. Water also does not stick to leaf surface due to other factors and dew is often seen as round drops. These may even be hanging from the tip of the leaf.

when entire image is out of focus, you can not have only a drop of water in focus in a picture.

Hi, Kris,

We meet again at the thread.

I try to draw a 3D picture, but I suddenly find I forget almost operation. Its a chance from me to review my lession, although its leisure avocation.

would you pls correct my wrong sentence every time you read my thread. Im thankful.

I've sent a reply to your mailbox , Kris

Thank you Kris. I read it. its wonderful.

Hi, Sham,

The picture I posted is a less compression jpg file. you can enlarge it at your computer to see clearly.

The big water ball is real sphare shape, equaltion is x^2 + y^2 + z^2 = a^2. no any problem. if you cannt see circle shape at your compter, its problem of configuration.or distortion of screen,

as to color, you may see different from I see from my computer is normal, because we have different white balance setting. mine color temperature is stting about 6000 degree. between 3600 and 9600.

The background picture is shot from nature, is a photo. the water drop on it is real. no any distortion and simulation and any modification at all.

It is not a real picture but an added image. Those teath like structure are from cut image. Where did you get this picture?

Those color mixing marks tell that this picture was added and edited.

The water drop on the leave is what it si the world. I shall make an animation pic for the thread next time.

your question is complex and your award is tiny.inappreciable.

I can only reply you with my high school knowledge.

your thinking is not quit right. differnt madium has different behavior, this is famous soackage of liquid. I forget the formula now. and Im laze to read book again.

you can see it from different objects when water in/on it.

the drop of water will drop out of the iron tap, ( I type a wrog word of taper in the past)

will different from drop from lotus surface. because one is soakage and another is not.

the drop can be fall out becaue of the surface tension is less than gravity of itself.

after tear be the gravity, they will tend to come back to shapere shape.

Thats why I said your top post pic is quit wrong.

The path of the small water flux crawl along the shortest path on the drop suface before the drop fall out. and accumluate at bottom.

The shortest path can be calculated by a n intigratial equation. ( I can write the equation, but I cannt solve ti)

if youi take accountof oscillation. that will be more cpmplex question. we can ingore it here. especially at state of symmetry. or uniform condition.

Have you every shot the shpe with your high speed camera?

Im interested in it aswell.

look at left pic, A is differ from B. because of the different material osf the cup.

the right pics, drop fall from beginning to end,

3 steps simplify.

different liquid diffent shape. it relates to material, liquid tension, viscosity, density etc. its out of my depth.

Im waiting for your high speed image. and good answer from another specialist.

I look forward to a high speed image as well. I can imagine that a drop reaches a momentary point where it is spherical (having done so after the initial tear shape ) but my feeling is that the drop would progress to a more complex form after (due to surface tension pulling it toward a sphere , but air resistance opposing this). I have suggested that others view this thread because the photograph is great , and the whole topic is interesting. I do not understand the physics here , but hopefully more people who look can add some clarification.

To:28#; 29#

Hi, Sham,

it seems you research the pic carefully. and mark it . But you are wrong. quit wrong. Its really picture drawed by me yeterday morning.

Its 3D picture. I hve to rend it, so that we can see it in form of water feeling.

I add some reflection and refraction, so that you can see it more activity and lovely.

none like its original model. I thnk.

Its a n easy work. if you understand art. you'll get use to it.

I like this one ;