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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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?

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.

Do the following:

Do the following:

You should do the following: