 Okay, we're looking at the boundary layer equations and we're looking at the flat plate boundary layer and what we're going to do in this segment is we're going to look at the solution for the boundary layer equations for a laminar boundary layer. So this solution was provided or first performed by Blanzius. He was a student of Prandtl's and he did this in 1908. So what Blanzius did is he came up with, he transformed the boundary layer equations which are partial differential equations into a series of ordinary differential equations and then he integrated by hand and so things that we could do on the computer today very very quickly took him a great deal of time to do by doing manual hand calculation. But with this he was able to come up with expressions and the main results were the following. He came up with an expression for the boundary layer thickness and we can compare this to the approximation that von Karman obtained using his momentum integral technique which was much simpler in terms of calculation and it was also appropriate for a turbulent flow provided that you have the velocity profile for one a friction coefficient and finally the displacement thickness. So what we can see for all three of these is von Karman's solution really wasn't that far off although it did involve a number of approximations. Another comment that I should make here is when I say that Blanzius was exact it was not a closed form solution and consequently the only way to get the solution was by doing numerical hand integration which is what he did and coming up with a table of values that he could then determine these values that we have here but it was not closed form it was a numerical solution. The final comment for laminar flow is that the drag coefficient on a plate is related to the friction coefficient evaluated at the length of the plate and this is for one side of the plate. So the drag on a plate on one side of the plate is equal to two times the friction coefficient at the end of that plate. So if we have a plate and this here would be x equals l this is x equals zero you'll have a friction coefficient that will vary along the length of the plate but if you evaluate cf at l the drag the drag coefficient on that plate is equal to two times cf at l. So that is one thing that comes out of the analysis for laminar flow on a flat plate and what we'll be doing in the next segment is we'll be taking a look at the parameters that result for a turbulent flow but this was one that could be computed numerically although it was using the boundary layer equations and Blasio did this by hand in 1908.