# Viscous flow between parallel plates – lubrication and interfacial instabilities

Another guest post this week. Morris Flynn from the ept. of Mech. Eng., at U. Alberta writes about a couple of viscous flow demonstrations. This demonstration serves to highlight two important features of (immiscible) viscous flows, namely (i) their capacity for supporting large compressive loads, and, (ii) the Taylor-Saffman instability that develops when a less viscous fluid pushes on a more viscous fluid.

Equipment

1. Two pieces of acrylic plastic. Each piece should be at least 5 mm thick and have a cross sectional area about the size of two business cards.
2. Dish soap. Lubricating oil can also be used, but requires a more careful cleanup.
3. (Optional) An overhead projector so that you can project the demonstration onto an overhead screen.

Demonstration

1. Place one of the pieces of acrylic flat on the overhead projector stage.
2. Add several drops of dish soap to the top surface of the acrylic piece.
3. Gently place the second acrylic piece on top of the first. The soap drops should expand into a pancake shape.
4. Gently pull on the upper acrylic piece while holding the other fixed to the stage. Notice the interesting instability pattern that appears.

Analysis, part i (squeezing the acrylic pieces together)

Reynolds’ lubrication equation describes flow that is primarily or exclusively in one particular direction, say the horizontal or x-direction. Regarding the x-component of the Navier-Stokes equation, the inertial terms are smaller than the viscous terms by a factor of (L/h)/Re. Here, L and h represent, respectively, the characteristic length scales in the horizontal and vertical directions. Also Re=ρUh/μ is the Reynolds number where ρ is the fluid density, μ is the fluid dynamic viscosity and U is a characteristic flow speed. For a lubricating flow where Re is less than 10 say, (L/h)/Re is expected to be small so that fluid inertia is negligible.

If the upper acrylic piece is pushed downwards at a constant speed, Reynolds’ lubrication equation can be written as

As the gap width h becomes indefinitely small, the only way this equation can be satisfied is if the pressure gradient and therefore p, the fluid pressure, becomes indefinitely large. Thus solid-solid contact is inhibited because an infinite force is necessary to squeeze out all of the interstitial fluid. This is the reason why lubricating oils play such a critical role in reducing wear in engines and related mechanical equipment.

Analysis, part ii (pulling the acrylic pieces apart)

By analogy with the Darcy’s Law equations, which consider flow in a porous medium, flow in the horizontal direction is described by

U now represents the (constant) speed of the interface between the soap, having a dynamic viscosity μ1, and the intruding air, having a dynamic viscosity μ2. Also k, the analogue of the porous medium permeability, is related to the gap width between the pieces of acrylic.

For whatever reason, let’s suppose that the otherwise smooth interface develops a “kink.” We can apply the above equation to understand whether this perturbation is likely to grow or decay in time. The force on the soap displaced by the air due to the kink is proportional to p2p1 where

Here r1 and r2 denote, respectively, the radial position of the interface in the unperturbed and perturbed states. If the force in question is positive, there is a positive feedback leading to an amplification of the perturbation or, in other words, a further distortion of the interface. From the last equation, p2>p1 whenever μ12. The interface is therefore unstable when the displacing fluid (air in this case) has a smaller dynamic viscosity than the displaced fluid (dish soap in this case). This result has long been the bane of petroleum engineers, who frequently inject water into hydrocarbon formations to maintain reservoir pressure and to sweep oil towards a producing well. The fact that the interface between the injected water and the oil is very often unstable leads to phenomena such as fingering and water breakthrough. These cause a commensurate drop in the rate of oil production.

Reference:

Homsy, G. M., 1987: Viscous fingering in porous media. Annu. Rev. Fluid Mech., 19, 271—311.

Description compiled by Morris R. Flynn, Dept. of Mech. Eng., U. Alberta (mrflynn”at”ualberta”dot”ca, 780-492-5593)

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# Video of “Atmospheric pressure – plunger tug-o-war”

Here are some video of the “Atmospheric pressure – plunger tug-o-war” demonstration. The full videos are linked from the GIF titles

Setting up the plungers

Pulling them apart

Thanks to Ali and Tanjina for helping with the videos.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# Atmospheric pressure – plunger tug-o-war

One of the difficulties some students have with any engineering course is to have a feel for the scale of particular forces or the scale of different parameters. This can be hard to communicate. Here is an easy demonstration to illustrate the scale of atmospheric pressure.

Equipment

1. Two suction cup plungers
2. A chap stick (or some petroleum jelly)
3. Two student volunteers

Demonstration

1. Rub the chap stick or petroleum jelly along the rim of the two plungers
2. Align the two rims and push them together pushing out as much of the air between them as possible
3. Have the two student volunteers try and pull the plungers apart. This will be quite difficult if the seal is good enough and the bulk of the air is removed from between the plungers.

Analysis

It is not possible to easily calculate or measure the force required to pull the plungers apart. The answer mainly depends on the air pressure and volume in the gap between the plungers before you start to pull. The easiest way to get a rough (over) estimate of the force is to assume that there is a vacuum between the plungers and that the chap stick and the plungers do not provide any significant mechanical resistance. In that case one can draw a free body diagram of one plunger and sum forces along the direction of the axis of symmetry.

∑Fx=0=Patm πD2/4 – Fstudent

Substituting in values for atmospheric pressure (14.7 psi) and the plunger diameter (5½” in my case) then each student will need to provide approximately 350 lb of force to pull the plungers apart. In reality the plungers are unlikely to be perfectly aligned and the seal will also be less than perfect so this will be an upper estimate of the required force. Even if the force actually required is a bit less than this the students will still get a sense that atmospheric pressure is quite large.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# videos of Turbulence in stratified and unstratified environments

Here are some videos of the “Turbulence in stratified and unstratified environments” demonstration. The full videos are linked to from the GIF titles.

I had not tried this before I made these videos and I do not think I got as sharp a stratification as is possible. That said, the demonstration still worked quite well.

Placing the upper layer of fresh water. The basic idea is that you pour the water into the boat and have it slowly overflow and spread out over the dense water below. I used salt rather than sugar, and it worked fine.

Adding the food coloring and getting the 2D flow. Again, I messed this up a bit and added too much dye with too much momentum. However, I did end up with a single vortex. The demonstration is quite forgiving.

3D demonstration. Here you just inject the dye into a jar of fresh water and watch the fine scale structures develop.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# Turbulence in stratified and unstratified environments

Welcome to our first guest post! It is a simple experiment that was developed by Bruce R. Sutherland for a high school newsletter. However, it could also be used to demonstrate a range of different phenomena including that 2D turbulence tends toward larger scales over time whereas 3D turbulence decays to smaller scales. I plan on using this in my environmental fluid mechanics class and may use is as a quick demonstration in my undergraduate fluids class as an interesting phenomena (if I can find a spare 15 minutes in the schedule). Bruce Sutherland is a Professor in the Department of Mathematical and Statistical Sciences at the University of Alberta. His e-mail address is (Bruce dot Sutherland at ualberta.ca). His website is http://taylor.math.ualberta.ca/~bruce. The write up also includes general reflections on the subject of fluid mechanics, its applications, and why he decided to study the subject. I decided to leave the full text of the write up with Bruce’s reflections rather than try to edit it down to just the demonstration since it flows nicely as is.

The study of fluids is pervasive in such scientific disciplines as mathematics, physics, chemistry, engineering and medicine. To name but a few examples, Fluid dynamics researchers might examine methods for extracting oil from Alberta’s tar sands and transporting it through pipes, or they might study how medicine is distributed though the body’s cardiovascular system, or they may try to predict how the cold waters in the Equatorial Pacific will affect Canada’s weather during a La Nina event. Although the equations describing the motion of fluids were derived two centuries ago, exact solutions have been found for only a few special cases. The challenge of finding exact or approximate solutions have continually pushed the frontiers of mathematics, most recently through devising efficient and reliable computer codes, and through the development of new fields of mathematics, including chaos theory and pattern formation, about which many popular science books have recently been written.

Much of my work examines mixing and waves in fluids with varying density. Such fluids are said to be “stratified” because they act as if they are composed of slabs of fluid layered one on top of the other. Oceans, lakes and the atmosphere are stratified fluids. (Indeed, the stratosphere gets its name because its density decreases relatively rapidly with altitude.) The air in the room where you are sitting is a stratified fluid: hot, less dense air floats near the ceiling and cooler air is closer to the floor.

I have been drawn to study fluid dynamics not only because its applications are of such practical importance, but because of the intuition and breadth of knowledge required by the discipline. Furthermore, because of the remarkable growth in the speed and memory size of computers, many fundamental problems in fluid dynamics that were previously unsolvable can now be modelled numerically and studied in laboratory experiments using lasers and digital image processing. The following describes an experiment that can be done in the kitchen and which demonstrates some of the beauty and surprising complexity of stratified fluid motion. You will need the following:

• 9″ x 11″ glass baking dish (or similarly large glass dish)
• a 4 cup measurer or bowl of at least this volume
• 8″ x 8″ piece aluminum foil
• ½ cup sugar
• food coloring (two colors)
• water

Figure 1: What you need to make a stratified fluid.

Figure 2: After filling the bottom half of the pan with sugar water, pour tap water, dyed red, into the aluminum boat. This will inhibit mixing and most of the red water will end up floating on top of the sugar water solution.

Figure 3: Put in a few drops of blue food coloring and watch the patch evolve into spirals.

Figure 4: Dragging a knife through the patch makes more complex patterns of vortices.

To make a stratified fluid, add 4 cups of water and 1=2 cup of sugar to the glass dish. (You may wish to put the dish on a white cloth to observe the fluid motions more easily later on.) Mix these together to form a strong sugar solution (the density of the solution should be about 1.1 gcm-3, compared with fresh water which has a density of about 1:0 gcm-3). Now measure out another four cups of water and add a light colored dye to it (four drops of red food coloring should be enough). We want to layer this dyed fresh water on top of the dense sugar water. The following is a crude but effective way to do this. Make a “boat” from the aluminum foil with a flat bottom and sides as high as the sugar water in the baking dish (about 1 cm). Float the boat on the sugar water and slowly pour in the dyed water. The boat will lower into the sugar water and the dyed water will eventually overflow spilling over the sugar water. You will notice that as the dyed water spills out, it floats over the sugar water. Continue to pour all four cups of the dyed water into the overflowing boat, pouring at a rate so that it takes about a minute to do this. When you are done, you should still be able to see some clear, undyed sugar water at the bottom of the dish.

Congratulations! You have made a stratified fluid!

At this point it is easy to see an astounding property of stratified fluids: add a single drop of dark colored dye (such as green or blue) to the center of the dish and observe what happens. If you do this within a few minutes of making the stratified fluid, it is likely you will see the dye stretched out into a spiral-like vortex looking not unlike a nebula in various Star Trek movies. A coherent slowly swirling vortex such as this typically does not occur in unstratified, homogeneous fluids. To see this, just add a drop of dye to fresh water in a bowl or another baking dish. You will likely find that the dye in this case gets pulled into filaments of ever finer structure in a motion that is typically chaotic and progressively less predictable.

To emphasize further the difference between the behaviors of stratified and unstratified fluids, take a knife and slowly drag it through both fluids creating a wake about 1 mm wide. In both cases, you will see small-scale turbulence in the knife’s wake. But what happens over time as the turbulence decays? In the unstratified fluid, mixing occurs near the wake and the resulting motion dies down after a minute or so. In the stratified fluid, from the small-scale mixing emerge large, slow-moving vortices which grow in size as they combine with other vortices and which continue to evolve for many minutes. The collapse and decay of turbulence in a stratified fluid involves many complicated processes that are the subject of active research today. How might a mathematician approach this problem? The first step is to write the exact equations of fluid motion appropriate to this problem. Although it requires an understanding of calculus to make any sense, the equations describing the motion of sugar water are given below (with an English translation of their meaning in parentheses below them):

Figure 5: But if you leave the fluid undisturbed for a long time, it will form one large spiral. This spiral took 5 minutes to form.

Here ρ is the density,  the velocity, and p the pressure, all three of which are functions of space and time. The constant g is the acceleration due to gravity and ν is the kinematic viscosity (which is a measure of friction within a fluid). The symbols D/Dt , , and ∇2  are convenient notations involving derivatives which are used to describe infinitesimal changes in time and space. Similar equations also exist describing how the density changes in time. The equations describing the motion of unstratified, fresh water are the same as those for stratified water but with the ρgz term removed. Although, with experience, it is a relatively simple matter to write down the equations, at present they cannot be solved to describe the turbulent motions in the above experiment. Indeed, they may never be solved; one would be hard pressed to think of a function that could encompass such complexity of evolution in time and space. Nonetheless, mathematical progress has been made. From experiments it was realized that unstratified (homogeneous) turbulence exhibits a special kind of symmetry, which today we describe as being fractal: a close up view of turbulence looks almost identical to turbulence seen from farther away.

For example, the turbulent plume formed by pouring cream in your coffee is similar in many respects to the turbulent plume from a chimney or an exploding volcano. Using scaling theory, scientists have been able to estimate how quickly energy is dissipated and how fast pollutants are mixed in turbulence. Stratified turbulence is much more difficult to model in this way, however. Mathematically, this is due to the presence of the ρgz term in the equations of motion for stratified fluids, which represents buoyancy forces acting vertically to carry relatively heavy fluid downward and light fluid upward. Physically it means that fluid, loosely speaking, “prefers” to move horizontally when it is stratified. You can see this in the experiments. When the knife is dragged through the stratified fluid, vertical motions are suppressed in its turbulent wake and only horizontal motions persist. Effectively, the motion evolves from one which is three dimensional (moving horizontally and vertically), to one that can be thought of as two dimensional (moving strictly horizontally). Although the transition from three dimensional to two dimensional motion is not yet well understood, scaling theory can be applied to “two dimensional turbulence” to predict that large scale, slowly evolving vortices should develop, as observed. Progress is being made in understanding turbulence with the aid of computer models. To this end, scientists are reformulating equations, like that above, into a form that can be calculated numerically. Such methods can only approximate the exact solution because one must ultimately impose a restriction on the smallest sizes of motion that can be resolved by the computer. Computers with greater speed and memory are providing ever more accurate solutions which are only now capable of reproducing some of the observations in laboratory experiments. It is not unreasonable to hope that with improvement in computers and laboratory measurements, further mathematical breakthroughs are just around the corner.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.