Table of Contents
Drop Pinch-Off
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 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.
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…
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 might be 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 each detail mathematically. Doing so requires you to learn how to observe a phenomena, 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.
The teaching points for this lab are as follows:
- Learning to gain insight into a complex phenomena through exploration.
- Gaining familiarity with some important concepts in fluid dynamics, including the following:
- how forces manifest in fluids on the macroscopic scale (e.g. surface tension, viscosity, density); and
- power law relationships and scaling phenomena.
- Using high speed video techniques.
Before you come to lab...
In order to prepare for the lab, read the following and complete the prelab exercises.
Overview
This lab is different than others in this course in the sense that it is an observational experiment. Most of the physics problems you are used to solving are ones which you approach from well defined first principles which you use to develop a rigorous mathematical model. Not all research questions come about this way, however. Many, if not most, come from observations of something happening in the natural world around us and we want to understand why it behaves the way it does.
But what do you do when the phenomenon you want to study is so complex and new that there is no basis for starting from first principles to build up a theoretical understanding?
This lab is intended to give you experience approaching such a problem – and such problems start with observing, noticing patterns and trends that are indicative of physical processes, and experimenting to uncover basic functional relationships among relevant physical processes. The idea is to use experiment to provide insight into what might be going on, and which might inform you as to how to go about modeling the observed behavior in more detail. Historically, new developments in theory typically arise from experiment – not the other way around.
Some terms
This is a fluid dynamics experiment. You may not have had a course in fluid dynamics, but that's okay. It will be enough to familiarize yourself with the following terms (either through an internet search or text book):
- surface tension;
- viscosity; and
- capillary length.
We will use the technique of dimensional analysis as well as the concept of scaling to help us construct plausible models to test. These subjects can take up an entire course. (In fact, the Department of Physics sometimes offers courses in dimensional analysis and fluid dynamics.) It is not our intent to teach you these subjects in this lab, but rather to give you some exposure to them. 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.
Videos
Spend some time watching the following videos and looking at the images of fluids undergoing drop pinch-off. 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.
Observe the behavior of the fluids as a single body of fluid – falling under the influence of gravity and stretching out – and notice 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.
| Water droplet | Glycerine droplet |
| Here are 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 playback speed of the video is arbitrary. (For example, the video on the right represents less than one second of real time.) 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. | |
1. Prelab exercises (4 points)
As you watch the above videos, try to notice patterns and behaviors which might indicate something interesting going on in terms of the underlying physical processes that are driving the system. More specifically, we are interested here in how the minimum radius $r_{min}$ of the neck changes as a function of time relative to the moment of pinch-off $t_{0}$. (The moment of pinch-off is when the drop separates from the neck. The minimum radius is simply the narrowest part of the drop profile at any given moment in time.)
Keep in mind that the shape of the drop and how it evolves over time is governed by the interaction of different forces acting on the system. Consider the following:
- Surface tension is a force which tries to minimize the surface area of the drop.
- Viscosity is the internal resistance of the fluid (akin to friction in kinematics).
- The mass density of the material leads to inertial forces (keeping the drop in motion or resisting change).
- Gravity is pulling down on the fluid.
- The external air pressure is pushing inwards against the internal fluid pressure pushing outwards.
As you study the behavior of these two drops, consider the following questions and try to make some qualitative sense of what you see. Note that these are guides as to the “type” of things you should be looking for, not specific points you need to address.
- Does the overall shape (profile) of the drop give you any insight into what the dominant physical process might be?
- Do you observe that the profile of the drop stays the same throughout the process? Does it change? If it changes, might this be an indication that something about the underlying physics has changed? If so, what might have changed?
- How does $r_{min}$ evolve over time? Is it always in the middle vertically? Does its rate of change remain constant? Is it linear in time? If you observe some change in the behavior of $r_{min}$, what might this indicate?
- How does the overall the profile of the drop change with time? As you get closer to the moment of pinch-off, $t_{0}$, does the drop appear to behave differently?
Consider what is or might be going on with respect to the physical processes driving the pinch-off for each movie separately. Also, look for differences in behavior of the two drops. (The fluid in the video on the left has lower viscosity than the one on the right. How might this affect your interpretation of the differences between the two.)
To receive credit, write-up a summary of your observations and your initial insights.
The above are not intended to be a bullet point list of questions for you to answer. They are just ideas to help you grasp how to “observe” the behavior of the system in a scientific way. We are not looking for any particular or correct answers. We want you to articulate some things which you observed and what you think they might plausibly be telling you about the physics that is going on.
It is not required, but you can also browse through this collection of images and see different behaviors that come out of the drop pinch-off process.
Exercises
Before you begin to collect the bulk of the data, you will complete a number of specific tasks, each of which is focused on a skill or technique which you need to understand in order to complete the experiment.
Completing these exercises will likely take most of the first one or two days of lab. Go slowly, and make sure you understand each step!
Playing with the high-speed camera
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 files, 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.
2. Making a high-speed video (7 points)
You will make a high-speed video of whatever you like.
Some important concepts which this exercise will introduce you to are the following:
- Aiming and focusing at high magnification (which creates a narrow depth of focus).
- Learning how to adjust 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 following:
- 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.
Do the following:
- Record a video at 1000 fps 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. (For example, the time it takes for 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.
Studying the pinch-off of a water drop
Use the apparatus illustrated below to record a video of the full drop pinch-off process for water.
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| Figure 1: The drop pinch-off apparatus |
In the camera, select the frames corresponding to the pinch-off for a single drop, export them to the computer, and convert the video to .avi. Use ImageJ to measure the radii of the narrowest part of the neck throughout pinch-off.
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3. Look for power law behavior (7 points)
We want all of the students in a group to learn how to use the tools to make measurements whenever possible. For this exercise, each student should measure the neck radii for the drop throughout the process. This will give you multiple data sets that will allow you to assess the degree of repeatability of your neck radius measurement technique.
Do the following:
- Produce a professional quality log-log plot of the minimum neck radius vs. time from pinch-off using the data which you collected from the video.
- Identify regions of the data which appear to follow distinct power law behavior and fit them to determine the exponent of the power law.
Work together in the lab to produce the power law plot. A point of emphasis in this lab is for you to gain experience in determining “what to do next”. You have just collected data that covers the full drop pinch-off process. We want you to use this data to inform your what you need to do next in your investigation. When you have produced the log-log plot of minimum neck radius vs time from pinch-off spend some time studying it. In particular look for the following:
- Are there regions of your data which appear to exhibit clear power law behavior? If so you will want to fit these regions to the form of a power law $y=Ax^{\gamma}$ where $\gamma$ is the power of interest.
- You are likely to observe that there are more than one regions of the data which exhibit power law like behavior. When you look at the parts of your video which correspond to these regions do you see anything in the time evolution of the drop which is consistent over that region. Do regions where the power differs appear to behave differently in the video? You are trying to “interesting” behavior which might be indicitave of what is driving the process at different stages of the drop pinch-off process.
- Are there regions where your data might be showing a change in the behavior of the drop, but it is unclear because your error bars are too large, or you do not have enough data points? If so this would indicate that perhaps you need to collect more data with higher magnification or higher frame rates in order to tell if something is indeed going on in that region. This is most likely to happen closer to the moment of pinch-off.
The point of this exercise is to use your first video to learn what the next step in studying this process should be.
If you spend enough time WATCHING your video and examining the log-log plot of the data, you should notice two different factors which are limiting your ability to understand what might be happening just before pinchoff occurs. One of these factors is temporal and the other is spatial.
To finish this checkpoint use your plot of the data, and frames from the video to articulate what is limiting your ability to observe what is happening just before pinch off occurs, and what you can do experimentally to better study this part of the phenomena. To receive credit for this checkpoint you need to write a single page that clearly describes:
- What are the two factors limiting your data just before the moment of pinch-off. Precisely what is it you see in the data and video that motivates this insight.
- What you need to do to improve your data in this region. Be specific. If you need to go to higher frame rates, state how much higher and provide a data driven justification. If you need more magnification, say something quantitative about how much more magnification you need and what you need to do to obtain it.
The goal is to have a plan for the next check point.
Note that each student in the group needs to produce the plot and do the fits, and write up the plan for the next checkpoint in order to get credit. But you should collaborate on this in the lab, and once an instructor is satisfied with your results proceed on to the next checkpoint. In other words, you do not have to both produce the final plots and write up the next step before proceeding.
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4. Record and analyze the end of the pinch-off of a water drop (7 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 here, you should attempt to achieve a frame rate of at least 15,000 fps (or up to even 30,000 fps). 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 okay 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. As before, each student in the group should collect a separate set of measurements of the neck radii.
What you are investigating is the following:
- 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?
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 theoretical and experimental reasons to expect certain behaviors at different points in the pinch-off process as a function of viscosity.
The paper 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 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 glycerin and water. The relevant part of the linked paper is highlighted on page 350. To summarize, they find the following:
- 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 more viscous glycerin/water solutions, do you see similar behavior?
Top pinch-off
Throughout 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 a single 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?
Self-similarity
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?
In the videos you have collected so far, 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, and 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?
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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.
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 the following:
- Identify a specific feature such as…
- …the oscillation frequency of the rebounding fluid in the neck after pinch-off,
- …the distance between the bottom of the rebounding neck and the top of the free falling drop, or
- …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 synced 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 collect in lab and submit the following assignments. You can score up to a maximum of 75 points total on these assignments.
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. It may be useful to include a representative image of one such measurement with annotations.
- Provide a log-log plot of your measured neck radii vs. time from pinch-off for both the low viscosity (water) and medium viscosity (Glycerol/H2O 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 are consistent with 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. It may be useful to include a representative image of one such measurement with annotations.
- 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)
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 that 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.
Related applications
- 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.
- Capillary pinch-off dynamics help explain how cats and dogs drink water.
- Similar power-law scaling and video analysis techniques are used to investigate how adhesives fail.
- Studying drop formation can help optimize the design of inkjet printers.