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

Before your first day in lab, you and your lab partner should do some background research. 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, so it is not our intent to teach you these subjects fully in this lab; a thorough 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 it 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 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 glycerine. Although both are fluids, the details of how they breakup provide insights into the difference 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 dumbell 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 visosity.
• 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.

In dimensional analysis one begins by identifying relevant physical parameters which would be expected to play a role in the phenomena under consideration.  So what physical parameters might one expect would play a role in determining how a liquid dripping from a faucet stretches, thins and eventually breaks in two.  Obviously gravity is pulling on the drop.  The size of a water molecule might matter.  The radius of the faucet might be expected to matter.  The shape and flow of fluids are dependent on properties such as viscosity, surface tension and density. 

Each of these parameters is defined by a set of dimensions, mass (M), length (L), time (T), etc.  The two parameters we are interested in studying are the radius of the neck at its narrowest point $r$ (dimension L) and the time $\tau$ (dimension T) which we will measure relative to the moment at which the drop breaks in two.  Some of these parameters should be less important due to scaling considerations.  For example in this experiment we will be using a high speed camera to record the pinchoff process.  As you will see for yourself when you begin the experiment the radius of the neck that we will be able to observe with the camera will range from a scale of mm to tens of μm and the pinchoff process will occur on a time scale of thousandths of a second. 

The size of a molecule is many orders of magnitude smaller than the observable neck size so it can be neglected.  The same consideration can be used to neglect the radius of the faucet which is ten times larger than the neck radii we will be measuring.  This leaves us with Surface Tension, Density and Viscosity as a set of parameters which might reasonably be expected to influence the behavior of the fluid during the pinchoff process.

In terms of dimensions our list of potential parameters looks like:

Now that we have identified a set of parameters which might be involved in the pinch-off process all that is left is to find combinations of $\tau$, $\rho$, $\gamma$ and $\eta$ for which the units work out.  We are looking for functions of the form: $r_{min} = F(\rho, \gamma, \eta, \tau)$

You should be able to find three different combinations of parameters for which the units on the right hand side of the equation reduce to L which matches the left hand side.  You can multiply and divide different combination of parameters as well as raise individual parameters to different powers in order to make the units work out.

As a lab in development, the experimental procedure for this setup is not yet formalized.  As such, it is even more important than usual for you to record what you're doing (and why you're doing it) so that you can interpret your findings when it is time to write the report.

The apparatus is illustrated in Fig. 1.

Figure 1: The drop pinch-off apparatus

The TA and laboratory staff will show you how to configure and use the high-speed camera. Below are some things you will need to take into consideration when setting up. To assist you here are some short videos showing how to use the camera.

In order to collect enough data at the moment when the pinch-off occurs, you need to run the camera at a frame rate of at least 10,000 frames per second (fps). (However, 20,000 fps is even better.)

When setup to use the full resolution of the sensor, the maximum frame rate is about 1000 fps. As you reduce the resolution (thereby using fewer pixels) the maximum frame rate increases. To get an adequate frame rate you will need to make the resolution as small as possible. The question you need to answer is “How small can you go?”

Your goal is to record a high speed movie of the moment the drop pinches off. You will be making measurements of the narrowest part of the neck of the drop, so the active area of the camera sensor has to be wide enough to see the full width of the neck throughout the whole pinch-off process. You also need enough vertical pixels to record the drop as it falls, right up to the point where it pinches off. Therefore, you will need to play with different camera resolution settings as well as the height of the syringe until you are satisfied that you are recording all of the features you need to see at a sufficiently high frame rate.

A consequence of increasing the frame rate is that each frame is exposed for a shorter period of time. This results in dimmer images as frame rate increases. For this reason you need to get as much light into the camera as possible. There are two factors under your control that affect image brightness.

First is the brightness of the lamp which illuminates the drop. Note that we are back-illuminating the drop so that we actually record its shadow. To provide more uniform illumination of the drop, it is best to place a diffusing material (such as wax paper) in front of the light source. Experiment with the placement of both the diffuser and the lamp to obtain the brightest and most uniform illumination of the drop.

The second factor under you control is the aperture of the camera. Wikipedia has a nice explanation of how changing the aperture affects the image. In short, decreasing the aperture (higher f number) reduces the amount of light reaching the sensor but increases depth of focus, while increasing the aperture (smaller f number) increases the amount of light at the sensor at a cost of decreasing the depth of focus. The aperture is controlled by rotating the blue ring on the camera lens.

Filling the video frame without wasting space (thereby utilizing the full resolution of the camera) may require adjusting the position of the camera relative to the drop.

These instructions are specific to installation on the Windows 7 computers used in the lab.

1. Go HERE and download the appropriate version for your operating system.
2. Unzip the downloaded file and rename the resulting folder “FFmpeg”.
3. Move the folder “FFmpeg” to C:\Program Files.
4. I could not get the system path updated to recognize the location of the binary, so invoke ffmpeg using its full path. 
5. Open a command terminal by typing cmd into the search box in the Start bar.
6. Test the installation by executing the following command in the cmd prompt window:

C:\Program Files\FFmpeg\bin\ffmpeg -version

1. Create a folder where you will store your movie files.
2. Open a command terminal by typing cmd into the search box in the Start bar.
3. Use the dos “cd” and “dir” commands to navigate to the folder containing your movie files.
4. Enter the following command in the cmd window.  “fname” is the name of the input and out put file.

“C:\Program Files\FFmpeg\bin\ffmpeg” -i fname.mp4 -f avi -vcodec mjpeg -qscale 0 fname.avi

Your written analysis that you submit to be graded should be built around your final conclusions. Everything in your analysis should support your final result and conclusions. For this experiment your final result(s) may end up being a discussion of what physical processes dominate the shape of the pinchoff process at different length scales. Your conclusions would be your evaluation of how well your measured values did or did not match a theoretical prediction and a discussion of anything you may have discovered about how the results depend on any factors encountered in the lab.

You need to make clear things you did, decisions you made in the lab which are important to understanding how you arrived at your results and conclusions. This might include:

• Details on sample preparation if you made your own samples.
• A description of the different power law relationships you were able to come up with and an interpretation of why they seem like reasonable descriptions of the pinch off process.
• How you assessed, quantified and propagated uncertainties.
• Plots of the data showing different power law regimes.
• How you determined minimum neck radii.

The above list is not intended to be complete, nor should it be treated as a checklist of what should go into your written analysis. Your analysis needs to make clear to the reader what your results and conclusions are, show how your data support those conclusions, demonstrate how you processed the data, etc.

For this quarter we are focusing on developing your skills in data analysis and drawing appropriate conclusions from your data. Your analysis should focus on these things. You should not include sections on the apparatus, background theory, historical significance, and things like this. This is not to say that these things are unimportant, they are just not part of a report on your analysis and results.

You may also use the Python notebook online Here. Be aware that it may take several minutes to start up, and it will not save your data when you leave the page.