CBE 30355 - Lecture Notes - Oct. 10, 2024

Announcements

The mid-term exam will be one week from today!

Class notes

• Dimensional Analysis

Goals:

• Use dimensional analysis to determine the number of dimensionless groups involved in a problem.

Reading

• The class notes.
• BS&L, chapter 3

Additional Readings:

In class we talked about the example of estimating the yield of a point explosion through dimensional analysis, solved for early atom bomb tests by G.I. Taylor. A nice discussion of his result is provided here and the complete paper (publication of which caused a bit of a stir, as the yield was still classified at the time) can be found here. It's a good illustration of the power of dimensional analysis to solve complicated problems.

Demonstration:

In class today we looked at the breakup of a drop in a viscous fluid in creeping flow in the absence of surface tension. Theory suggests that while a spherical drop should be neutrally stable at zero Re, any imperfections are swept around to the back and cause it to transition into a toroidal shape. Inertia then causes it to grow in radius, and this larger torus becomes unstable, resulting in two drops forming from the ring. These drops are unstable in turn, leading to a cascade of ever smaller drops. The theory behind this process is described in the paper by Kojima and Acrivos given here.

It actually turns out to be a little tricky to determine the best fluid combinations for a nice demonstration of the effect. When the first demo failed (breakup occurred about the time the drop hit the bottom of the cylinder) I first turned to dimensional analysis to get an idea of the important relationships and then re-analyzed the data of Kojima (and later measurements by Pignatel et al., for clouds of particles) to determine a combination that actually would work. The dimensional analysis is given here and a more detailed analysis, together with some experiments used in optimizing the demonstration, is given here.