 We're now going to take a look at a concept that was developed by Ludwig Prandtl back in 1904 and that is the concept of the boundary layer. Now what Prandtl was able to do is he was able to unite two fairly separate groups that existed in fluid mechanics prior to 1904. And basically what existed was a split between the theoreticians and the experimental fluid mechanists. So the theoreticians sometimes were referred to as being people who were involved with theoretical hydrodynamics. So the people that were involved with theoretical hydrodynamics looked at things such as potential flow, inviscid flow, Euler's equation, very very mathematical. Now they did acknowledge the fact that fluids were viscous but their study was restricted to fluids where the viscosity was assumed to be small and consequently that's why they were looking at things inviscid flow related and it just simplified the math and made it easier for them to study it. And then on the other side were the experimentalists and the experimentalists were fully aware of the fact that the flow was viscous. However what they did is they conducted experiments in order to understand what was happening within the fluid flow. So the people studying hydraulics were typically experimentalists, quite often engineers just coming up with empirical fits that they would fit their experimental data curves to the data and then they could use it in design and for things like that. But these two different groups were really quite separate from one another. They were not combined and it wasn't until Ludwig Prandtl came along with the boundary layer concept and what that enabled was a coupling between the potential flow or the mathematical treatment with the viscous flow and what the experimentalists had been doing. It kind of united the two fields. So what we're going to do we're going to take a brief look at what the boundary layer concept is and then we'll spend a little bit more time studying boundary layer equations. So with this what the boundary layer theory enabled was the acknowledgement of the fact that viscosity was really only important very close to the wall in this layer called the boundary layer and outside of that region for the most part the fluid could be treated as being in viscid and what Prandtl said was that viscosity is important but it's mainly confined to the layer right next to the wall and consequently that was the origin of boundary layer theory and what early boundary layer theory was applied to was the simplest flow that could be imagined and that was that of a flat plate. It was zero pressure gradient so that means that there'd be zero pressure gradient outside. The flow wasn't accelerating nor decelerating due to a pressure gradient and so let me just sketch out what that looked like. So what we have here is a schematic of a typical boundary layer and we begin at the leading edge of the flat plate and the leading edge is right here and that's that location x equals zero and we have the zero slip condition right along the wall and so there is no velocity along the wall but the effect of the wall is felt by the fluid and as you go further and further downstream from the leading edge of the plate viscous diffusion is taking place so the viscosity diffuses out and away from the wall and so this region of velocity deficit or lower than free stream velocity grows and grows and grows and so that's why we say the boundary layer is growing on a flat plate or it grows on an object as as the flow goes over and then what happens is the boundary layer usually begins as laminar unless you trip it you could trip it by putting sandpaper here or a little wire or something like that and you could force a turbulent boundary layer from the beginning that provided that you have the conditions for that but then the boundary layer will transition going through a transition zone and this is actually longer than what I've shown here but then it goes into a turbulent boundary layer where the growth rate would be higher than it is for a laminar boundary layer and you would have a very different velocity characteristic for the turbulent boundary layer versus the laminar boundary layer so that is kind of the concept of the boundary layer and with the analysis that we're going to be doing we're going to be really focusing on the laminar boundary layer part because that's the only place that we can do analysis once you get the turbulent it gets a little bit more complex and not really ideal for this course because this is just introductory now the characteristic dimension or the number that is used here in fluid mechanics to characterize the boundary layer is the Reynolds number and and so for a boundary layer we come up with the new Reynolds number and we put the subscript x because that will denote the spatial dimension but what we have is the density of the fluid multiplied by some characteristic velocity usually the external out stream velocity which I've shown as being u infinity and then our characteristic dimension x so that means that the Reynolds number is a function of position along the plate so the Reynolds number will get larger and larger and larger as we go along the length of the plate from the leading edge and then we divide by the dynamic viscosity of the fluid so that is Reynolds number that we use in a boundary layer so you'll always see re subscript x denoting that and the Reynolds number at which we go through this transition here where we start to go from laminar to turbulent that is referred to as being a critical Reynolds number and critical denotes the fact that you're going through the transition from the laminar flow regime into the turbulent flow regime and that is an approximation in terms of what it would be the rule of thumb is 5 times 10 to the 5 would be what we use and that's for a flat plate so if you recall when we looked at pipe flow we said the critical Reynolds number was around 2100 21 to 2300 is typically what people will use for flat plate flows with zero pressure gradient the critical Reynolds number whereby you go through transition is 5 times 10 to the 5 so what we're going to be doing in the next few segments is taking a look at some of the early theoretical developments that were done in studying the boundary layer and trying to characterize the growth rate as well as the friction that would occur along a body when you have the boundary layer flow going over it and so we'll begin by doing that and then we'll move into more sophisticated analysis using the Navier-Stokes equations although we won't really do that in this course we'll look at a more simplified approach that was developed by Theodore von Karman so that is boundary layer theory on the next segment we move into von Karman's momentum integral method