 So we're going through the momentum integral analysis that Theodore von Karman performed in 1921 and we've come up with an expression for the shear stress along the wall and we also came up with an expression for the momentum thickness and we saw that they related to the velocity profile on the boundary layer. Now it turns out von Karman had no way of determining exactly what the velocity profile was. However what he did is he would have had access to experimental data and so he had an idea as to what the velocity profile looked like in a laminar boundary layer and he made an approximation and he basically said the velocity profile was a quadratic relationship and then he used some boundary conditions that we'll take a look at in a moment in order to come up with an approximation for the velocity profile in the laminar boundary layer. So this is the velocity profile that von Karman assumed and here we have constants a, b, and c yet to be determined and he also knew that there were certain boundary conditions that the flow would be subject to. First of all he said the velocity at zero I should write that out at y equals zero was equal to due to the no slip it was equal to zero and then the velocity at y equals delta so at the outer edge of the boundary layer was equal to the external flow condition u0. Another thing that he knew now looking at the velocity profile and this he probably would have looked at experimental data but he knew that the velocity gradient or the derivative of the velocity with respect to y at the outer edge of the boundary layer was equal to zero so there is zero slope changes you went away and into the free stream region so he didn't know what that was at the wall because that would be related to the shear stress which was an unknown so looking at the velocity profile we have y this is our wall going in the x direction he had this so once he got to the outer edge of the boundary layer that was y equals delta he knew no slip at the surface here he knew u0 and then out here this is where we have du by dy equals zero so those were the boundary conditions now what we're going to do we're going to apply those we're going to use them to solve for the constants a, b, and c which is what von Karman would have done equip with that we can then go back and look at shear stress and momentum thickness and a boundary layer thickness so to begin with taking the derivative of the velocity profile we get b plus 2 cy and that is equal to zero from that he was able to come up with an expression for b in terms of c we don't know what c is yet but we'll just box that and come back to it in a moment he also looked at what was happening at the wall and so here the velocity was equal to zero along the wall looking at our velocity profile that would be equal to a and so that then told him that a is equal to zero that's one of the constants specified and then at y equals delta this is where the velocity was equal to the free stream velocity now a we can get rid of because we know not zero we can plug in the value of b and from this we can then determine c so that is c determined and knowing c we can go back and solve for b and we get that for b so taking a b and c and plugging them back into the expression that we had earlier for the velocity the one with the quadratic profile and what we end up with is the following so that is the velocity profile that von Karman then worked with and he knew this was probably not exact it wasn't entirely correct but it was a close enough approximation enabling him to replicate a lot of the experimental data that he had probably collected up until that point and then continuing with the analysis so a good coupling between experimental and taking experimental data and interpreting theoretical values and guiding that as you go through and do the theoretical development so armed with this now what we can do is we can go in and we can calculate delta which is the boundary layer thickness as well as the skin friction and that gives us the skin friction coefficient okay so in the next segment what we'll be doing is we'll be taking a look first of all at delta the boundary layer thickness and then a later segment we'll look at the skin friction coefficient so that's where we're going we have a velocity profile we can now continue on with the momentum integral analysis results and determine these values