 In this segment what we're going to do is we're going to take a look at the boundary layer equations. So what we're going to be doing is a little bit of an order of magnitude analysis. And so we take the governing equations and have a Stokes equations and we simplify them using this order of magnitude approach where we look at large terms small terms and we neglect the smaller terms. And so in the analysis what we're going to be doing we're going to make a couple of assumptions. We're going to neglect gravity so the body forces will be neglected and we're going to assume that we're looking at two-dimensional flow and incompressible as well. So what we're going to begin with if you recall from an earlier lecture we were looking at an airfoil and we were saying that you could apply the boundary layer approximation on to any kind of flat surface and if you zoom in on an airfoil on the top of it it almost looks like a flat plate with the exception that there is a pressure gradient external because the pressure is going to change as you go around the airfoil. But what we're going to do we're going to zoom in on this region here and we're going to consider a section normal to the wall so x is going to be in the wall direction and we're going to examine the boundary layer in this region and then outside of that would be the free stream flow but this is going to be the velocity profile and it will be a function of both position in the x direction as well as normal to the wall in the y direction. So we're going to begin with mass conservation so the continuity equation and we're going to write it out in differential form. If you recall when we did the von Karman momentum integral technique we used control volume analysis now what we're going to do we're going to go in and use differential analysis in order to come up with the boundary equations. These equations we'll see were developed and solved earlier before von Karman's momentum integral technique but for the most part it was only able to be applied to laminar flows and and you needed to do very tedious hand calculations von Karman's technique which came later and was a little quicker and could be applied to laminar or turbulent flow as well. Not that this technique cannot apply to turbulent flow it just gets too complex and it can't be solved. Okay so that's mass conservation and then momentum. Okay so we have those as our equations for the 2d incompressible flow and this is where we are beginning and so the goal that we have here so the goal is to solve for u v and the pressure the external pressure subject to the boundary conditions of no slip along the wall so no flow through the wall and no movement to the fluid at the wall and inlet and exit boundary conditions. Now in order to do this properly what we really should be doing is going through and non-dimensionalizing all of the different values within the equation and then doing a order of magnitude analysis that way. I'm going to skip that and do a little bit of a shortcut here but some of the assumptions that we can make in order to simplify the governing equations are the following. The first one is that the velocity normal to the wall is much less than the velocity along the wall. We saw that when we looked at the previous in the previous segment that we saw for the boundary layer thickness and so von Karman's integral analysis kind of tells us that and people like Prandtl would have known this as well by doing experimentation they would have been able to come up with that assumption. Approximation the other one is the derivative in the streamwise direction or along the flow is much different much smaller than what is happening normal to the wall and so that's another approximation and with that you can go through your governing equations and doing a dimensional non-dimensionalizing and order of magnitude analysis and for the y direction momentum equation we can come up with the following and this is actually something that Prandtl came up with in the early 1900s he realized this that the pressure gradient normal to the wall is zero and consequently pressure was a function of distance along the plate only and this is quite significant because what it was saying was that if we have our boundary layer here it is saying that the pressure out here so let's say we have some pressure outer and we move in towards the wall the pressure on the wall is going to be the same and so there is no pressure gradient in the vertical direction and so the pressure of the external field can be imposed and we'll be using that in a moment and that is kind of significant for one of the parts of the analysis of the boundary layer equations so what that means is the following so the implications of that are quite significant and this was the thing if you recall back to the beginning of the previous lecture we had a segment where we talked about long or Prandtl I should say bringing together the theoretical hydrodynamicists and the experimentalists and uniting them because the theoretical hydrodynamicists had mainly studied inviscid flow they did realize that flows were viscous but it was just too difficult to study and so they would study it assuming viscosity not to be significant and inviscid flow and it brought together the experimentalists who were always studying viscous flow because they're studying real-world flows and brought those two worlds together but with this what we can say is potential flow we could do the potential flow calculations on the outside and then bring that pressure into our boundary layer relations and if we recall for potential flow we had that we had the flow being irrotational and irrotational we said that del cross v had to be equal to zero and also incompressible and for that del dot v is equal to zero and from that would result to Laplace equation then you could use a method of superposition but it also enables us to apply the Bernoulli equation everywhere so not just along a streamline but everywhere in the flow and with that we're going to use that for the boundary layer analysis and specifically we're going to use it to address the dp by dx term we already saw that dp by dy was approximately equal to zero there's no pressure gradient normal to the wall but we still need to deal with the dp by dx term and for that we will use Bernoulli's equation and so if we're dealing with an irrotational flow we can say that this is going to apply anywhere in the flow field we're going to neglect the body force and also any kind of change in height so we're going to assume this to be negligible and now what we're going to do we're going to use Bernoulli's to try to evaluate the dp by dx term we're going to play a little bit of a mathematical trick here so we're after dp by dx so we can take the derivative of it but if we look at the second term we can rewrite it in the following manner and if you were to integrate this back you would get the original one half u0 squared and so we then get that expression and we can isolate for dp by dx in this and given that it's not a function of y i can write it as an ordinary differential and that on the right hand side where u0 is the external flow so if we have again drawing a little picture of the airfoil let's say we're examining the boundary layer here u0 would be the external flow outside in the potential flow area and so that gave us an expression for dp by dx which is good and essentially what it tells us is that we can solve the inviscid flow about a body and then bring that pressure into the boundary layer so when pressure is brought in from the external flow a final assumption that is used in coming up with the boundary layer equations is this one the second derivative of the velocity in the streamwise direction with respect to x is less than with respect to y so with these assumptions and approximations and the pressure field brought in we can then arrive finally at the boundary layer equations again this is not really a very rigorous derivation of them there are more rigorous ways of doing it there's kind of a hand waving approach but we get to the same spot in the end anyways so the boundary layer equations for continuity and momentum all we have for momentum is going to be the x direction equation and this is for steady so I'm getting rid of the time rate of change term and then for the pressure gradient term we can plug in what we came up with from the external flow and then for the viscous shear I'm putting it in terms of shear stress and let me expand on that in a moment here so on the shear stress we will have two different options one if we have laminar flow which we can handle and this is what Blasius solved and the other one is if we have turbulent flows remember I said that you could use these equations for turbulent but the problem is is that we got this guy in here now and this was our Reynolds stress term and that related back to the whole turbulence closure problem it's hard to how impossible to come up with what that is without some sort of hand waving argument so anyways we have those two equations really we will look at both of them as we move on here but it's the first one that Blasius was able to come up with a solution for and going back to what we're after we want to solve for the velocity field in an ideal world and v and this is subject to the following boundary conditions so at the wall we know that the velocity we have no slip and no flow no flow through the wall I'll just write it as no slip for now and at the outer edge of the boundary layer that's when we get to the outer stream which would be the external flow and that could change as a function of position and we refer to that as kind of being a patching between the boundary layer equations and the external potential flow solution so those are the boundary layer equations what we are going to do in the next couple of segments we're going to begin by looking at the solution of Blasius came up with in 1908 for the laminar boundary layer and then we'll go and look at the turbulent flat plate boundary layer we won't really be able to come up with a heck of a lot we'll come up with an expression but it will have experimental data embedded within it and that's where that will proceed so that is the boundary layer equation and we'll now move into solving things of engineering interest using the boundary layer equations