 In the last lecture what we did is we took a look at the momentum integral technique that was developed by Theodore von Karman and he came up with expressions for both the thickness of the boundary layer as well as the wall shear stress or the friction coefficient derived from that and so what we're going to do now is we're going to look at an example problem enabling or where we go through and we use one of the equations that for the boundary layer thickness. We're going to take a look and see how quickly the boundary layer is growing. So von Karman's equation gave us the boundary layer thickness divided by x that's position from the leading edge of the plate and it was approximate because he assumed a velocity profile. The utility of his technique really was in the fact that it could be applied to either laminar or turbulent. It was not as exact as Blasius's solution which we will look at in this lecture and he obtained this using controlled volume analysis and he for laminar flow assumed a parabolic velocity profile and it was a fairly simple method in terms of the analysis required in order to use it. So what we're going to do is we're going to apply the equation for the boundary layer thickness to an example problem where we are asked to calculate okay so what we've been asked to do is to evaluate the thickness of the boundary layer for two different conditions one for air flowing over a flat plate and the other one for water. We're told that the plate is one meter in length and that the free stream velocity on the outside of the boundary layer is 0.5 meters per second. So what we're going to do we'll use von Karman's thickness and we will calculate the boundary layer thickness. So let's start with air. So what we're going to do in the equation it requires us to evaluate the Reynolds number at the end of the plate and this will also enable us to determine whether or not we're dealing with a laminar or a turbulent flow as a bit of a check. So we get a number 3.3 times 10 to the 4 and this is less than our 5 times 10 to the 5 and therefore we can conclude that it is a laminar boundary layer and consequently we can use the result that we showed on the earlier slide from von Karman's momentum integral analysis. So what we'll do let's write out his equation. We have delta over x is approximately 5.5 divided by Reynolds number evaluated at x equals l to the one half. The x here is going to be the length of the plate. So evaluating that we get 30.1 millimeters or that's the equivalent of about 1.9 inches. So we can see here we have a plate that is one meter in length and the boundary layer is only about 30 millimeters or three centimeters at the end of the plate. So really the boundary layer is not growing that quickly. Let's take a look at what is happening with water. Evaluating the Reynolds number. So we get a number 4.99 times 10 to the 5 which is just barely under our 5 times 10 to the 5 per laminar. We can still see it as laminar flow although we are getting very close to the point where the flow will undergo transition to a turbulent boundary layer. But with this we can now evaluate the boundary layer thickness for water and it is even thinner than it was for air about 0.3 inches or about 8 millimeters. So what this tells us and this is a plate that is one meter long and so it tells us that for this type of flow the boundary layer is growing very very very very very slowly and it is not very thick and so you have to go over a very long distance before you start seeing significant boundary layer thickness. And the implication of this is that there is very slow growth rate of the boundary layer and that knowledge was actually used by Prandtl and Blasius when they went through and derived the boundary layer equations which we will be looking at in this lecture. So let me just make a comment about that. So what this example problem tells us it says even though we're dealing with a scenario where we do have relatively small velocities we're dealing with a one meter long plate the free stream velocity is only 0.5 meters per second so that's fairly small. Even though we have these small velocities and then our plate is one meter in length even though we have this the boundary layer is not really getting thick very quickly and and so that's the boundary layer thickness. So with that what what we do and this is very key to the boundary layer approximation the velocity going normal to the wall is much less than the velocity going in the wall direction. So we can say that u is much greater than v and that is one of the primary assumptions that is used in the boundary layer approximation and formulating the boundary layer equations. And so we will be seeing that in the segments throughout this lecture and we'll come back to that over and over again but the velocity in the stream direction is much greater than anything moving away from the wall where the viscous diffusion is taking place.