Checkpoint List

Drop Pinch-Off (Winter 24)

The formation and pinch-off of a drop is governed by a myriad of fluid physics phenomena. With the help of high-speed photography, we can study probe different stages in the evolution of a drop and gain insight into the length and time scales over which different forces dominate. This experiment uses dimensional analysis to provide potential models for the narrowing radius of a drop's neck nearing pinch-off, and tests these models against data collected in different time regimes close to drop separation.

Before we get into the nuts and bolts of the experiment, spend some time watching the following videos and looking at the images of fluids undergoing drop pinchoff. Forget about math and physics for a moment, just take a step back and appreciate the beauty of nature in action. Click on the videos and images to view the full screen, change the playback speed of the videos and watch them in super slow motion and sped up. Step back and observe the behavior of the fluids as a single body of fluid, falling under the influence of gravity stretches out, noticing how its shape changes as it elongates. Pay particular attention to the transitions which occur between different shapes and behaviors. Too often we get so caught up in the rigorous mathematical formalism that we lost sight of the beauty of nature. Whether you aspire to be a theorist or an experimentalist, the best insights come from paying attention to the world around us. Even something as simple as water dripping from a faucet can provide scientific inspiration. So just observe… explore… play…

  1. This Nature Communications paper attempts to generalize similar modeling methods for complex systems. The work this lab is based on is a citation for one such system.
  2. Capillary pinch-off dynamics help explain how cats and dogs drink water.
  3. Similar power-law scaling and video analysis techniques are used to investigate how adhesives fail.

Teaching Points

In this lab you will not be testing a model which has been derived rigorously from first principles, or comparing a final result to some well known literature value as you are used to doing. Instead, this lab is about investigating a complex phenomena for the purpose of gaining insight into what physical processes are likely to be most relevant. The goal is to learn how to study a phenomena for which you do not have the knowledge to analyze mathematically. Doing so requires you to learn how to observe a phenomena, how to look for and notice patterns of behavior, and to make connections between these behaviors and the physics concepts which you do know.

The phenomena you will study is the pinch-off process which occurs when a drop of water separates from the main body. This is a fluid dynamics problem which will introduce you to the concepts of power laws and scaling relationships.

  • Learning to gain insight into a complex phenomena through exploration.
  • Gain familiarity with some important concepts in fluid dynamics.
    • How forces manifest in fluids on the macroscopic scale; surface tension, viscosity, density.
    • Power law relationships and scaling phenomena.
  • Using high speed video techniques.

A Note About Uncertainties

In this lab you will not be obtaining numerical results which are then compared to some expectation. Therefor careful analysis of uncertainties is not a focus. But it is nonetheless important to always consider how well you know the value of any measured quantity. You still need to ask yourself “what is limiting how well I know this number I am writing down”. We will expect that you identify and record the dominant uncertainty in all of your measurements.

Before coming to lab...

This is a Fluid Dynamics experiment. You may not have had a course in fluid dynamics, but that is ok. It is enough to familiarize yourself with the following terms:

  • Surface Tension
  • Viscosity
  • Capillary Length

We will use the technique of Dimensional Analysis as well as Scaling to help us construct plausible models to test. The subject of dimensional analysis can take up an entire course. (In fact, the Department of Physics sometimes offers courses in dimensional analysis and fluid dynamics.) Therefore, it is not our intent to teach you these subjects in this lab, but you should be familiar with what it entails – in particular as regards fluid dynamics problems. A Google search will suffice.

The book Dimensional Analysis: Examples of the Use of Symmetry by Hans G. Hornung is a very good introduction to the subject if you find the subject particularly interesting.

Also read through the following section titled Imagery, watch the videos and spend time studying the images.


Unless otherwise specified all images and movies of drops are courtesy of Mark Chantell, University of Chicago.

Video clips

Two different video clips of the drop pinch off process recorded with a high speed camera running at several thousand frames per second. When watching through them keep in mind that the whole process took less than a second. Play through them at normal speed, then go back and lower the playback speed (controls are accessed via the three dots in the lower right corner of the frame) and slowly watch the process. The first video is of a water drop. The second is pure glycerin. Although both are fluids, the details of how they breakup provide insights into the differences in their properties.

Water droplet Glycerine droplet

Images captured one at a time using a digital camera

Successive stages of drop pinch off process in water. Notice how the contour of the drops change as they progress further from the nozzle, or thought of another way as they progress closer to the moment of pinch off. This sequence of images does not follow the same drop from start to finish, rather each image is of a different drop at a different part of the process. It is striking however how repeatable the process is. Multiple images of the same drop and a sequence of images of different drops will look the same because the underlying physical processes are unchanged from one drop to the next.
A more magnified sequence of images of a water drop pinching off. Again these images were taken with a still camera and show different drops at different stages of the process. Do you notice a subtle difference in the curvature of the neck where it meets top of the drop between the first two images? This is indicative that the underlying physical processes have changed in some manner. The third and forth images show the fluid which was in the neck region of the drop after pinch off has occurred. Notice the symmetry in the shapes and patterns. The shapes you see are not random, clearly there are underlying physical processes occurring between the molecules which make up the fluid that are producing these intricate shapes. These shapes are repeatable too. If you photograph multiple drops of the same fluid, under the same conditions, at the same time relative to the moment of pinch-off, you will see the exact same shapes every time!
These last two images don't show anything new, they are just two of my favorites from a purely artistic point of view. - Mark Chantell

Now that you have hopefully spent some time observing the phenomena lets ask a few questions to get the scientific process started. Did you notice the following in the videos and images:

  • The neck connecting the drop to the faucet exhibits at least two different characteristic shapes. Inward curving like a dumbbell in the early stages, and a much thinner, longer cylindrical shape closer to pinchoff. Strong curvature is indicative of surface tension effects while long and straight strands are more likely dominated by viscosity.
  • There are different time scales apparent in the videos. The early part of the development is slower than what happens at pinch off. And what about the transitions between the viscosity and surface tension shapes, do these happen quickly or slowly.
  • After pinch-off the remaining strand of fluid from the long thin neck forms back into a drop. Did you notice that the shape of this drop oscillates with a characteristic frequency.
  • The behavior of the water drop and the glycerin drop are markedly different in terms of how the neck develops. Viscosity and density are two of the primary characteristics of these fluids which are very different from one another.

Fluids are incredibly complicated systems to study in rigorous mathematical detail. Fluids are made up of loosely interacting molecules. But you are not going to get very far if you attempt to model the system using quantum mechanics at the molecular scale. You could instead approach the problem using a macroscopic description of the forces and momenta involved which works well with solids such as bricks sliding down inclined planes. However these are fluids and their boundaries are constantly changing which means that quantities such as internal forces, velocities, etc. are not only position dependent within the fluid but are also changing in time in a position dependent manner. This makes it nearly impossible to find closed form solutions to the equations of motion describing the system as a whole.

How then do we study such phenomena if we cannot write down precise equations? One way is to approach the problem computationally, which is a valuable tool. But it is time consuming and tends to focus on very specific conditions. We will use a more general approach which can be highly suitable for initial investigations in to complex phenomena as a way to gain insight into the dominant physics at work and how it changes through out the process.

By watching how the drop pinch-off process evolves using high speed photography and video techniques, and using our physics intuition we have identified that the physical properties of surface tension, viscosity and density are likely to be involved. We also see reasons to expect that the influence of these factors change as a function of time relative to the moment pinch-off occurs.


1. Play with the high-speed camera (5 points)

Yes, you read that right. The first task for which you get credit is to play with a 30,000 frame per second camera!

On the practical side you will need to be proficient in operating the camera, exporting and analyzing the video. When using the camera to study the pinch-off of a fluid, you will spend a fair bit of time fine tuning your setup to capture very small and very fast processes. It helps to already be familiar with the basic operation of the camera and analysis tools.

However there is another aspect to consider. Doing experimental physics usually involves working with some really cool toys (I mean scientific apparatus ;-)), and it should be fun. For this particular lab you have the opportunity to play with a high speed camera and take some really cool and fascinating videos of things which you are not usually able to observe. As a first exercise we want you to do just that.

Some important concepts which this exercise will introduce you to are:

  • Aiming and focusing at high magnification which creates a narrow depth of focus.
  • Lighting and aperture. Recording at 1000 frames per second (fps) or higher means each frame in the video receives one thousandth of a second exposure time or less. At higher fps lighting becomes increasingly challenging.
  • Using the camera controls to set the frame rates.
  • Saving selected portions of the video stored in the cameras memory to export to the computer for analysis.
  • Converting the cameras video format from mp4 to aiv on the computer.
  • Opening the avi version of the video on the computer in ImageJ for analysis.
  • Making basic measurements using ImageJ.

As for what to record, try to think of something which happens in less than one second. Some examples include:

  • The vibrations of a plucked string on a Ukulele. (There should be a Ukulele in one of the two drop pinch-off rooms).
  • The vibrations of the tines of a tuning fork. (There should be a couple of these in the two drop pinch-off rooms).
  • A drop of water splashing in a cup.

Use your imagination, but please consult with the lab staff before proceeding.

To receive credit for this exercise you need to

  • Record a video at 1000fps or higher.
  • Select a portion of the video to save and export the save file to the computer.
  • Convert the video from an MP4 file format to an AVI file format so that it can be read into the image analysis software.
  • Read the video into ImageJ and use its measurement tools to measure how long it took for something to happen. e.g. one full cycle of motion of a vibrating string.

Keep in mind that we are not after a precise physics measurement here, we just want you to demonstrate that you can use all of the tools to record and make measurements before proceeding.

Some tutorial videos and general notes on using the high-speed camera can be found here.

Once you have recorded a video and are ready to open it in the image analysis program you will first have to convert it from an MP4 format to an AVI format. Instructions for how to do this can be found here.

2. Record and analyze the pinch-off of a water drop (5 points)

Figure 1: The drop pinch-off apparatus
  • Use the apparatus illustrated above to record a video of the full drop pinch-off process for water.
  • Select the frames corresponding to the pinch-off and export them to the computer.
  • Use ImageJ to measure the radii of the narrowest part of the neck through out the pinch-off.
  • Make a log-log plot of the minimum neck radius vs time from pinch-off.
  • Investigate the impact of the uncertainty in the time of pinch off on the power exponents.

3. Record and analyze the end of the pinch-off of a water drop (5 points)

Repeat what you did for the previous task, except this time focus on the very last moment before pinch-off occurs. One of the things which you hopefully noticed when looking at your video and analyzing the minimum neck radius vs time from pinch-off is that the process of the narrowing of the neck started off slow and accelerated. As the drop gets closer to the moment of pinch-off things are happening both fast and on a very small length scale. It is possible that in the interval between the frames just before and just after pinch-off that something interesting may be happening. The only way to know is to go back and record another video of just the bottom of neck where it attaches to the drop which is about to separate. Since things are happening very fast you should attempt to achieve a frame rate of at least 15,000fps, and 30,000fps would not be too much. Doing this will require you to be much more precise in positioning the drop in the field of view of the camera and both lighting and focusing will be more of a challenge.

Some points to consider:

  • The neck of the drop is going to be very narrow. You will need to get it as close as possible to the camera in order to magnify it enough to make measurements. You may need to make use of a higher magnification lens which we have available (ask the staff) or what are called extension tubes (again, ask the staff).
  • The way you get higher frame rates on the camera is to use only a small portion of the full area of the imaging sensor. The primary limitation in frame rate is how long it takes to read out the pixel data from the sensor, and restricting yourself to fewer pixels allows for faster read out times. Don't be surprised if you have to reduce the imaging resolution to an area of a hundred pixels or so on a side.
  • Lighting will be more challenging at higher frame rates. It is ok to remove the diffuser that you used for the previous measurement and shine the light straight into the camera. Doing so will not harm the camera.
  • With limited resolution on the sensor and higher magnification it will be challenging to evaluate focus on the back screen of the camera. You might want to record several videos, making slight changes to the focus for each one. Then when you view the videos on the larger computer screen you can compare them to decide which had the best focus and go with that one for your analysis.

Receiving credit for this task is essentially the same as for the previous one.

Record a video and process the data all the way through to the log-log plot.

What you are investigating is:

  • Does anything about the drop pinch-off process change when you look more closely at the final stage?
  • Does the shape of the neck change?
    • Is there a power law scaling behavior and if so is the coefficient different than it was for the early part of the process?

4. Record and analyze the drop pinch-off for a 50/50 mixture of glycerin and water (5 points)

You probably notice the pattern now, rinse and repeat. Repeat the drop pinch-off measurement, this time for a 50/50 mix of Glycerin and water. This will be a much more viscous solution than pure water. Since we expect the pinch-off process to be governed by properties of the fluid (i.e. surface tension, viscosity, density…) we might expect to observe different behavior as viscosity increases.

Go through the same procedure as before with a zoomed out view that allows you to capture the entire process with the highest frame rate you can get. When you process the data and produce the log-log plot look for evidence that something has changed. Do you see clear power law behavior? Is there more than one power law involved? What are the power law(s) you observe and how to they compare to the case for pure water?

Receiving credit for this exercise is the same as for the previous two exercises.

5. Record and analyze the end of the drop pinch-off for a 50/50 mixture of glycerin and water (5 points)

For the final exercise record and analyze a video of the very end of the pinch-off process. Compare and contrast with the previous three exercises.

Receiving credit for this exercise is the same as for the previous two exercises.

Further investigations

By this point you should be comfortable with the operation of the camera and its use. Investigate each one of these subjects.

Power law behavior

The power law relationship between $r_{min}$ and $\tau$ in water glycerin mixtures is known to be highly dependent on the density, viscosity and surface tension of the fluid. There are both theoretically and experimental reasons to expect certain behaviors at different points in the pinch-off process as a function of viscosity.

The paper quoted here reports on measurements showing that the drop pinch-off for liquids in air (air is considered a fluid in this context) undergoes a transition from an inertial regime to a viscous regime as the viscosity of the viscosity of the fluid changes. Papers in this field are filled with jargon and terminology that can make them challenging to follow. Rather than attempt to give you a crash course in fluid dynamics, we will instead frame this portion of the lab as follows.

Among other things, this paper reports on measurements of their observed power laws for the cases of both a low and a high viscosity solution of glycerine and water. The relevant part of the linked paper is highlighted on page 350. To summarize they find that:

  • At low viscosity the development of the neck of the drop follows an exponent of $\frac{2}{3}$, which they refer to as being consistent with an inertial regime.
  • At higher viscosity they observe an exponent of $\frac{2}{3}$ in the early stage of drop formation, followed by an exponent of 1 in later stages. The exponent of 1 corresponds to a viscous regime.

Using the pure water and the 50/50 glycerin/water solutions, do you see similar behavior.

Top pinch-off

Through out the exercises you have focused on investigating the behavior of the bottom of the neck where it pinches off from the top of the droplet which then falls away from the nozzle. However you may have noticed that after the pinch-off, the fluid remaining in the neck also pinches off at its top and forms into a small secondary drop which then follows the main drop out of the field of view of the camera. For this to happen the top of the neck must also undergo a pinch-off. Since there is no reason to believe that the physics is any different at the top of the neck, one would expect to see the exact same pinch-off profile as occurs at the bottom of the neck.

Collect data on the pinch-off at the top of the neck. On one log-log plot of $r_{min}$ vs $\tau$ show the data for the pinch-off at both the top and bottom of the neck. Is the process the same at both ends of the neck.


The ideas of scale invariance and self-similarity are important in fields like fluid dynamics. Self-similarity can be seen in many aspects of nature as well as in mathematics and even music. These concepts are fascinating, but they are somewhat abstract and it is not always apparent how they actually manifest in the natural world around us. Self-similarity comes in different forms, but loosely stated it describes situations where the same pattern appears within a system at different length scales. Wikipedia has a good description of self-similarity.

The drop pinch-off process is one phenomena which exhibits self-similarity, you may already have noticed it in some of your videos, which is most apparent at high viscosities. Investigate the 80/20 glycerin to water solution looking for behavior in the evolution of the neck through out the pinch-off which repeats itself at different length scales. Among other things, you will need to focus your attention on what happens to the fluid in the neck after the pinch-off occurs.

Random or repeatable

At this point you have been watching videos of the pinch-off process for water & glycerin solutions under different conditions. You should have noticed that after the pinch-off at the bottom of the drop, the fluid which remains in the neck can exhibit a wide range of behaviors. Sometimes the neck pulls itself into a new drop which falls out of view, sometimes the neck continues to stretch and pull apart. For the case of pure water you should have observed that the neck reforms into a drop but in doing so there is a lot of fascinating structure as shown in the image below. The curious mind might ask whether or not this behavior is the same for every pinch-off event when conditions effecting the system are unchanged. It may seem at first as if there is a lot of randomness going on immediately after the pinch-off occurs, but if the physics is unchanged then the process should be the same. To what degree would you expect the behavior of the neck to be repeatable from one drop to the next? If the behavior is the same, to what degree of precision is it repeatable?

Collect video clips showing the behavior of the fluid in the neck after pinch-off for the water sample. Carefully examine what happens for 3 or 4 such events, try to find some characteristic feature of the process which you can use to determine as precisely as possible just how repeatable is this process. Check with an instructor to make sure that your plan is acceptable.

We want you to use your own creativity in deciding how to establish repeatability. Whatever you choose to focus on you should find some way to quantitatively establish whether or not the behavior you choose to study repeats from one drop to the next. Minimally this should involve:

  • Identify a specific feature such as the oscillation frequency of the rebounding fluid in the neck after pinch-off, or the distance between the bottom of the rebounding neck and the top of the free falling drop, the diameter of some feature of the neck as a function of time from the moment of pinch off…
  • Record a video of several successive drops so that you can compare data from one drop to the next.
  • Make some series of measurements of the chosen behavior which you can use for direct comparison from one drop to the next. You could also create a movie or sequence of images that show the behavior of multiple drops side by side and synched to the time from pinch-off.

Use your creativity and have fun with this part, but we do expect you to produce some sort of scientifically motivated analysis.

Final data analysis

You have one week to perform a full and complete analysis of the data you collected in-lab and submit the following assignments. You can score up to a maximum of 50 points total on these assignments, but you should notice that the pool of points available is actually 60 points. This means that you don't need to do everything perfectly to get full credit (though you should still try to do everything as well as possible, because we will not consider make-up or extra credit). If you score more than the maximum of 50 points… “hooray”! (But you will still only get a score of 50; you don't get extra credit.)

All plots should be appropriately labeled and of publication quality.

With this experiment including representative images taken from your videos, appropriately annotated, can really help to make a lot of your discussion much more clear.

Viscosity (25 points)

Do the following:

  • Provide your raw data, measurements of neck radii, including units and uncertainties. Note that in this context, where you may have a lot of numbers recorded from your videos, that it is acceptable to provide plots of the data, properly labeled and with error bars which reflect the measurement uncertainties.
  • Provide enough description for the reader to understand how you measured the neck radii and how you estimated the uncertainties. Including a representative image of one such measurement with annotations can save you a lot of words.
  • Include a log-log plot of your measured neck radii vs time from pinch off for both the low viscosity (water) and medium viscosity (50/50 mixture) samples. Indicate any regions where you identified distinct power law behavior and show your fits to these regions.
  • Summarize the power laws you found in your data for the two different samples.
  • Comment on whether or not your data confirm the results presented in the paper.

Symmetry (25 points)

Do the following:

  • Provide your raw data, measurements of neck radii, including units and uncertainties.
  • Provide enough description for the reader to understand how you measured the neck radii and how you estimated the uncertainties. Including a representative image of one such measurement with annotations can save you a lot of words.
  • Include a log-log plot of your measured neck radii vs time from pinch-off for both the top and bottom pinch-off events. Indicate regions where you identified distinct power law behavior and show your fits to these regions.
  • Comment on the degree to which the pinch-off process is observed to be the same at both ends of the neck.

Repeatability (25 points)

You should do the following:

  • Describe the behavior which you are investigating and how you are testing for repeatability.
  • Use images and measurements to show that behavior of the neck after pinch-off is repeatable from one drop to the next. Remember you need to show some sort of quantitative assessment for this part.
  • Briefly describe what appear to be some of the more important physics concepts involved in the behavior of the phenomena you have chosen to investigate. There is no “correct” answer here, the intent is for you to some reasonable degree of insight into what you observe in the videos.