 Okay, so we're talking about experiments and experimentation and fluid mechanics and we're looking at a technique that makes our experiments more efficient and this technique is referred to, well we'll begin with the theorem, but Buckingham Pi theorem and it's named after Buckingham who developed this in the early 1900s and he came up with Pi parameters and that's why we call it Buckingham Pi. So what we're going to do in this segment is we're going to present the theorem and the methodology behind it and then in the next segment we'll go through the step-by-step procedure and then we'll work some example problems. So what the Buckingham Pi technique does or method is it provides us with a way of finding non-dimensional parameters for a given experiment. We talked about the sphere in the last segment, what Buckingham Pi will enable us to do is determine how the different variables and we talked about force, we talked about velocity, diameter, density and viscosity. How can we combine these together into non-dimensional variables that help us collapse the data and I talked about collapsing data from a Pi parameter one to a Pi parameter two and you might have more than just two Pi parameters but if you could collapse your data into a line or it could be a curve or anything like that, that is good because then that relationship then can characterize many many experiments and as engineers we like having that. So what we're going to do we're going to begin kind of with a generic type of scenario where we have some functional relationship. So imagine we have a scenario where we have a number of parameters and we can have parameter one which we will call our dependent parameter and then the independent parameter is q2 through qn minus one. So let's imagine we have that scenario or we could rewrite that in the following manner so you could write that as another functional relationship yet to be determined. So let's assume that we have a scenario where we have all of these different parameters one through n, one of them is dependent, the others are independent. Looking at the Pi theorem, the Buckingham Pi theorem, so what the Pi theorem says is that our n parameters can be grouped into n minus m independent Pi groups where we could rewrite that function with a new function now. So n was the number of parameters that we had, I haven't defined m yet but I will in a moment, or this could be rewritten with one of our Pi parameters on the left. So we have this relationship that exists now n and m, let's define those. So remember n was characterizing the number of dependent and independent parameters or variables we have and in the example we're looking q1 all the way up to qn and m is the minimum number of dimensions required to characterize those parameters or variables. So by dimensions we mean length, mass, time, temperature if you have non-isothermal scenarios. So we have these functions and the thing that is different from what we're looking at here from this function even. So when we look at this function this had all dimensional variables so that that's, those have dimensions, those have dimensions but when we come to the Pi groupings and so let's look at this one these Pi groupings are all non-dimensional so they all have units of one and and so that's the difference. What we've done is we've collapsed all the variables in our experiment into these Pi groups and so we say that there can be n minus m of these Pi groups so let's take a look at that. Now one thing to note about the Buckingham Pi technique it will not tell us the functional relationship in order to get that we need to conduct experiments. So when we have these non-dimensional Pi groups we will not know the relationship we need to get their relationship from experiments and the other thing is that if you take Pi parameters and combine them together they will not generate a new Pi parameter they need to be independent of one another so let's take a look at that. So if we had Pi groups and we tried to recombine them like this thinking that we're coming up with new Pi groups, Pi 5, Pi 6 were not. Pi 5 and Pi 6 are not independent because they can be formulated by combinations of other non-dimensional Pi groups and consequently we cannot call them new Pi parameters so all Pi groups need to be independent of one another. Okay so what we're talking about here taking a lot of different parameters scaling them down collapsing them into these non-dimensional groups what we will do next is take a look at how to put this all together and we'll look at the technique first and then we'll look at some example problems involving doing Buckingham Pi analysis.