 Before we start talking about internal and external flows, we should take a minute to talk about dimensional analysis and one of the things that we can use it to produce, which is non-dimensional numbers. Fluid mechanics is an empirical field. For the most part, we aren't using general governing equations to produce a relationship with the situation in front of us, and certainly not in most practical cases. Most of our equations come from correlations made from experimental data. An experimentalist somewhere performed a set of experiments, drew a correlation from it, and we use it and try to apply it to our situation. As a result, it's important to try to generalize correlations as much as possible. For that, we use dimensional analysis. For example, if you consider the frictional effects on flow through a pipe, we describe the friction as a non-dimensional number, and the flow through the pipe as a non-dimensional number. Describing them this way allows us to make as broad and as general a correlation between how fluids are affected by friction. This is not unlike the way that you describe your performance on exams. You could describe your performance as a number of points achieved, but it's much more useful to describe the number of points achieved divided by how many points were available. The result of that division, a percentage, is a dimensionless quantity. And as a result of it being dimensionless, you can apply generalizations between exams. You could say an 80% regardless of the circumstances is a pretty good performance. Within the field of thermal fluids, we use many non-dimensional variables to try to quantify fluid behavior or heat behavior or thermodynamic behavior. Some of the most useful or most important non-dimensional numbers are given names. Those names are usually the researcher who best demonstrated its effectiveness or created it in the first place. Note on this table the top value Reynolds number is described as important almost always. It is very true for internal and external flow especially. Many of our correlations are going to be drawn from Reynolds number. Generally speaking, dimensional analysis yields two benefits. The first is reducing complexity. We are reducing how many independent variables actually affect a dependent variable. This is particularly important for the people performing the experiments. If your field, your specialization, said that it took 10 experiments to show how one independent variable affected a dependent variable and you were in a situation where you had four independent variables affecting the thing that you were studying and you had to perform 10 experiments for each, that means that you would have 10 different values of x1 to test and for each of those values you would have 10 different values of x2 and for each of those values you would have 10 different values of x3 and for each of those values you would have 10 different values of x4 it would take 10 to the fourth experiments to draw that correlation. And 10 data points to draw a correlation is not very many. If it was 15 or 20, that's 15 or 20 to the fourth power. Now, if you could prove that x1, x2, and x3 were not actually independent of one another, and the thing that actually affected y was the proportion of x1 to x2 times x3, now all of a sudden, you only have two independent variables to worry about. Now it's only 10 squared, or 15 or 20 squared. You are reducing how many experiments you have to perform in order to get a certain confidence in your correlation by two orders of magnitude. The second benefit is scaling laws. We could use dimensional analysis to show how a model would represent or not represent the real thing. Instead of building a whole aircraft, we could build a model to a smaller scale, test that model in a wind tunnel, and draw correlations between the model and the real thing. Better yet, we can remove scale from our analysis entirely. We could try to describe flow flowing through a pipe regardless of the size of the pipe. One of the tools we use in dimensional analysis is named after a researcher named Buckingham. Buckingham, or Buckingham, used a theorem to generate non-dimensional numbers. He abbreviated those non-dimensional numbers with a letter pi, and as a result, that's referred to as Buckingham's pi theorem or the Buckingham pi theorem. The Buckingham pi theorem is an algorithm which generates non-dimensional parameters from a set of independent physical parameters, and it consists of six steps. The first step is to list all the physical variables, then you list all the dimensions of all the variables, then you try to identify how many repeating variables you will need to generate your non-dimensional analysis, then you select which of those variables should be those repeating variables, then you generate pi groups one at a time using each of the non-repeating variables, and then you check that your resulting non-dimensional analysis is actually dimensionless and you write the final form. I think the best way to get a handle of how the Buckingham pi theorem works is to try a couple of examples. Let's do that now.