I'll admit when I first saw the words "Dimensional Analysis," I felt my skin flush; my heart started beating faster; my mind began racing; and I scanned the exits of the room I was in. Classic fight or flight, with an emphasis on the latter.
But as I read, I realized my first reaction was silly. That said, I couldn't stop thinking that a lot of this was hand-waving. Can we legitimately summarize this discussion (or least summarize the justification) by observing that "in physics, almost everything is continuous" so arguments like this just work?
More precisely, what exactly is "length scale" or "characteristic length" supposed to represent? Is this along the lines of the length of the box everything is contained in, or is this the length of the smallest phenomenon observable/significant? What about in problems with a large container and small phenomena of global significance?
Also, why do we put a bar on the velocity scale U?
Finally, how is Reynolds number in any way well-defined? Can't I just say the scales are approximately this or that and get entirely different values?
Onto the next section, when can we legitimately make the lubrication assumption and get realistic results? I want to say for "slippery" fluids, but what does that even mean?
When we get a time estimate for the length of time needed to remove an adhering object, what assumptions are we making about the way it's pulled off? I feel like this should be clear, but wasn't really for me.
Overall, really cool stuff. I'm amazed that despite the sophistication of the equations, we can get tangible and useful numerical results.
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