4.4 - Viscous Fingering
15. The Hele-Shaw Experiment with Glycerin
16. The Hele-Shaw Experiment with Carrageenan
14. The Fractal Dimensions of Hele-Shaw Patterns
How is pumping oil from the ground related to the development of ulcers? Is the path of a lightning bolt governed by the same laws of physics as the growth of a snowflake? And why can you understand how a lightning rod works by studying random walkers? or by studying the growth of a dust particle?
The experiment described in this section exhibits aspects of the fundamental physics of all the processes described above. This experiment, a study of "viscous fingering,'' was originally performed by H. S. Hele-Shaw, a naval architect, in 1898. The geometry of his apparatus was a little different from the one we use here. Nevertheless, we refer to the apparatus as the Hele-Shaw cell after his original design.
Consider this story. Let's say you have a large number of balloons to inflate. What kinds of balloons are hardest to inflate? Are long balloons harder or easier to inflate than round balloons? And when is any balloon hardest to blow up? Is it harder to blow air into a balloon when it is deflated? Or when it is close to full inflation?
What is the force that resists the air coming into the balloon as it is inflated? After the balloon is inflated, is the air under greater pressure inside the balloon or outside the balloon? If you think it is greater inside, what counter-force keeps the skin of the balloon from rupturing? And how is this problem related to stretching a rubber band?
Surface tension is a very common force which shapes much of nature around us. In effect, it is a force that resists the creation of surface area. Thus, when you fill a glass to its lip, and keep filling it some more, the curved surface (meniscus) that allows you to fill the glass beyond its top is maintained by surface tension. When it suddenly breaks, the flowing liquid has greater surface area. Surface tension keeps a balloon round instead of growing spikes.
As another example, think of a straw with water flowing through it. Does the water flow more quickly near the walls of the straw? Or near the center? Or is the speed the same throughout?
It has been observed that when a fluid flows next to a surface, the speed of the fluid at the surface itself is zero. But at the center of the stream the fluid is moving! How can this be? What happens when you move one surface over another? Is it easy? What will stop a metal block from sliding over a metal surface? If you think it is friction, how do you imagine friction operating at a molecular level? Is heat generated? If so, where does that heat go? Can you apply your analysis to the fluid flowing in a straw, or over the surface of a plane's wing? What is happening at the molecular level that causes adjacent layers of fluid to move at different speeds?
The frictional property of fluids is called viscosity. The greater the viscosity of a fluid, the greater the force necessary to maintain fluid flow through a straw, or over a surface. Suppose two fluids have different viscosities; what differences in the molecules of the two fluids could give rise to the difference in viscosity?
Viscosity is a measure of the resistance of a liquid to flow. At room temperature, honey and molasses do not flow easily: they have high viscosity. Motor oil flows more easily than honey; it has a lower viscosity. Water has a still lower viscosity. Air flows so easily that one might be tempted to say that it has zero viscosity. But it does resist flow a little, as you can prove by trying to breathe through a drinking straw. The viscosity of air is approximately 50 times less than the viscosity of water.
Spread some wax paper or aluminum foil on the table. Pour a small puddle of honey in the middle. Place a plate with a flat bottom on the puddle of honey. Now drag the plate sideways along the surface of the table. It takes a force to keep the dish moving. The viscosity of the honey is related to the force needed to drag the dish over the honey-puddle.
You will want to look up the viscosities of fluids you use in the Hele-Shaw experiments described in the following section. So you need to know about the unit in which viscosity is measured. Viscosity is measured in poise, whose plural is also poise. The viscosity of water is easy to remember; it is one centipoise, that is, one hundredth of a poise. Glycerol has a viscosity 1200 to 1400 times greater than water, or 12 to 14 poise. Air has a viscosity some 50 times smaller than that of water, or approximately 200 micropoise.
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