Autumn 2025

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…

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:

In order to prepare for the lab, read the following and complete the prelab exercises.

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.

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 it particularly interesting.

Your tasks for day one of this lab are:

• Learn how to use the high speed camera and use the analysis software to make measurements from the saved video files.
• Use the camera to record a video of the pinch off process for water, measure the minimum radius of the drop as a function of time from the moment at which pinch off occurs, then plot and analyze this data to look for power law behavior.
• Use what you learned from the water pinch off video to motivate, plan and execute further investigation of the process.

Using the high speed camera to record a video and then exporting that video to the computer where it can be analyzed is not complicated. Once you have gone through the process a couple times you will find it is pretty easy to obtain a well focused video with the frame rate and resolution needed.

Still, it is easier to learn how to work with the camera without also worrying about getting “good” data. To facilitate learning how to use the camera your first task will be to record a video of something unimportant, which you can then export to the computer and practice using the image analysis software. This way you can focus your attention on things like;

• focusing the camera,
• setting the frame rate and resolution,
• adjusting your illumination,
• how to save a portion of your video and export it to the computer,
• converting the saved video into a format that can be read by the image analysis software,
• and using the image analysis software to make measurements.

Using your imagination, record a video of something that happens quickly in time. The ignition of a match, popping a balloon, a falling coin…

For this task you need to do the following:

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 necessary to record and make measurements before proceeding.

You are now familiar with how to use the high speed camera, export the video and open it in imagej to measure the sizes. Now you will apply this technique to investigating the pinch off process for ordinary water. You will use the apparatus illustrated below.

Figure 1: The drop pinch-off apparatus

Be thoughtful in how you setup the apparatus and configure the camera. You want to obtain the best quality data you can. What defines best? One way to think about this is to consider what are the quantities that you will be measuring? In this case you will be measuring the size of the neck of the drop at its narrowest point as a function of the time from the moment of pinch off. Your measured quantities will be a length in pixels using the tools in imagej, and a time which will be related to the frame rate of the camera. It is reasonable then to conclude that two important considerations are:

• The size of the image of the drop in the field of view of the camera. If you are too zoomed out the entire pinch off process will take place over only a small portion of the pixels in the image sensor, limiting your spatial resolution. Ideally you would like the size of the image of the drop to fill as much of the image sensor as possible while still keeping the moment when the drop separates from the neck in frame.
• The frame rate will limit how well you know the time at which your measured neck sizes occurred. A higher frame rate will provide better time resolution. Keep in mind that all times need to be measured relative to the moment at which pinch off occurred.

With this in mind do the following.

There are at least two different regimes in the pinch off process for water which you should be able to observe given the capabilities of the apparatus you are working with. The different regimes are related to different length scales for the minimum neck radius. As the neck radius becomes smaller it is reasonable to expect that physical factors that dominate at larger radii become less important at smaller radii and vice versa.

In studying the first video you should have noticed that in one part of the process where your data may be indicating that something is changing in the behavior of the drop, but the data is not “good enough” to tell for sure. It should also be clear how you can modify your setup and camera settings to obtain “better” data in this region to see more clearly if some new behavior is emerging.

Using what you learned about the pinch off process from your first video, decide what changes you will make to the apparatus setup and the configuration of the camera to investigate this region of your data where it appears something might be happening. Before you start making changes to the setup however think carefully about what changes you want to make and why. Why means what did you observe in your first data set that suggests making these changes. You will be expected to be able to clearly articulate the What and the Why as part of your assignment. So check with an instructor to make sure you have a good grasp on this before proceeding. Then go ahead and perform this new set of measurements.

• Modify your setup and camera settings as needed and record another video of just the portion of the pinch off process that you want to follow up on.
• Analyze this video, make a log-log plot and fit to the power law models as you did before.
• Examine this results of this analysis along with the previous analysis, in particular try to answer the following questions:
  • Does the data from this second drop match up with the data from the first one over the regions you expect the two to overlap?
  • Where the data from the two videos does overlap do they both agree with the same power law model?
  • Does the data in the second video show agreement with one of the power law models in the region where the first video gave ambiguous results?

Go over your results with an instructor before you leave the lab.

When we used dimensional analysis to come up with our three power law models we made an assumption that the size of the nozzle from which the drop appears can be neglected because it is “much” larger than the size of the drops neck.

It may seem arbitrary how we chose what factors to include in our dimensional analysis, and which factors we neglected. To some degree this is correct, we were guessing as to what the relevant parameters might be. But we can test the assumption that the size of the nozzle can be neglected, and that is the focus of todays lab activity.

One easy way to test this assumption is to ask the question, what happens when $R_[o]$ is comparable to the size of the nozzle? You can the data and video you already took on Day 1 of this lab.

In the lab there should be a collection of nozzles in three different sizes.

Note that there is no separate out of lab analysis for todays assignment. You will be asked to assess the validity of the assumption as part of the analysis you do after day 3 in lab. So take careful notes so that you can clearly and precisely articulate at what point, if any, does the assumption break down.

At this point you have experience working with the high speed camera, analyzing video data, and thinking about fluid dynamics as having power law behavior where the value of the exponent is related to the underlying physics which are driving the system for water. You have also investigated an underlying assumption that the nozzle size can be neglected in the power law models which we constructed.

Today you will make use of what you have learned to investigate the drop pinch off for a solution of water and glycerin which is much more viscous that pure water. The fact that the higher viscosity glycerin solution behaves differently from water should be evident as soon as you begin observing it.

After going over the results of your initial investigation with an instructor, you should have a clear plan for what to investigate next. Adjust your apparatus as necessary and collect additional video.

Make sure that you analyze the video, plot the measured neck radii vs. time from pinch off, and fit this data to the three power law models while you are in the lab. Go over your findings and interpretation of the results with an instructor to ensure that you have the data you need for your analysis. For your analysis you will be expected to understand the relationship between the features and power laws which are present in your data as well as how they relate to the behavior that you observe in the video.

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.
4. Studying drop formation can help optimize the design of inkjet printers.