 Okay, so we're almost there. What we're doing, we're in the process of deriving the Navier-Stokes equations. And what we are going to do is we are going to make the assumption that we're dealing with a Newtonian fluid. And this will lead us to the equation by Navier and by Stokes named after both of them, two individuals. I'm not mistaken, I think one was French and the other was British. Okay, so if you look back to the last segment, we had this equation here, basically taking continuity or sorry linear momentum and applying it to a differential element. We went through, we were fine with the body forces, pressure is okay, right hand side that's our inertia term, but it's the term with the shear stresses in it, which is right in the middle here that we need to deal with. And in order to deal with that, we need to make an assumption about the fluid. And what we're going to do here, we're going to assume we're dealing with a Newtonian fluid. And what we're after is a relationship between velocity and shear. So I'm just going to give you that. And you can see, given that we're using some of these both surfaces, y and the z surface, that means that it is a symmetric stress tensor that we're dealing with. The other thing we're going to do from the linear momentum equation that we had, we had the total derivative or the substantial derivative. So just in terms of shorthand, let's go back where the heck was it. It was right here. With this term here, what I'm going to do is I'm going to condense that down using the total derivative or the material derivative. And so what we're going to do, we're going to say the following. And that's using the material or substantial or total derivative. So with those, we combine them all together. We put in the shear stress and we end up with the following set of three equations. It's a vector equation. So we get that. I'm going to box it. And those are that is the Navier-Stokes equation. This is the equation that governs fluid flow and anything from the blood flowing within your veins to the wind blowing outside to the flow within a pipeline to the flow over an aircraft wing. And the other one, unless you go compressible flow, then you got to modify things a little bit and it gets more complex. But anyways, let's not get too carried away with that. The other one we have here, this is continuity. So how do you solve these? Well, how many unknowns do we have? Let's think about that. Unknowns. What don't we know in this equation? We know the body force. We usually know the density. We should know the viscosity of the fluid we're working with. So when I look at this, the unknowns seem to be the three components of velocity, u, v, w, and pressure. So the unknowns are u, v, w, and pressure at every point in the flow. So there are four unknowns and how many equations do we have? Well, we're not dealing with non-isothermal flow or anything like that. So the energy equation isn't here. But for dealing with isothermal flow where heat transfer is not important and you do not have temperature variation, what we have are the Navier-Stokes equations. So those provide, you get three equations there and we get continuity and continuity provides one equation, therefore four. Now, I don't know about you, but I remember in math they said if you had four equations, four unknowns, you could solve it. It's not so simple here. We're dealing with a partial differential equation. Boundary conditions. We need boundary conditions for these PDEs. What are the boundary conditions that we have? Usually what's going on at the boundaries of our domain that we're looking at. And so we have the fluid at a wall equals the fluid velocity. So you can have no slip and v normal is equal to zero. That would be the no flow boundary condition unless you have suction along the wall. But no slip usually the wall is not moving and consequently the fluid velocity there is zero as well. So it looks pretty clean but really it's not and the reason is because it's a partial differential equation with these constraints. Usually we can only solve this equation for laminar flow and that's in terms of closed form solutions. So laminar flow and even with that there are probably only I don't know what it would be now. It's probably 90 maybe 100 known solutions to the Navier-Stokes equation in terms of analytical relations. Even the flow over a flat plate boundary layer that we'll look at later in the course that was derived by Blasius. He couldn't even solve this closed form. He had to do it numerically back in the day when there was no calculator or computer and so he got his PhD dissertation for that by solving it by hand. But but it's a very very difficult flow to solve and the other thing is that it's non-unique and what I'm referring to is that the flow could be laminar or the flow could be turbulent and both of those are solutions to the equation for the same boundary conditions. Wow that would be slightly different boundary conditions because you'd have a different Reynolds number. But what happens laminar is okay but when you get to turbulent the place where we start running into problems are the inertia terms over on the right hand side here and what happens is you take your mean flow plus a fluctuating value and there are non-linearities on the right hand side and so you get other things that they call apparent stress or Reynolds stress and that leads to all kinds of difficulties in doing analytical modeling. Even for doing computational fluid dynamics modeling it's a challenge. There are turbulence models that exist. They have done direct numerical simulation having very very small grids but that only works for very very basic flows not the things that we would normally look at that would be industrially important. So anyways that's why this is such a rich area of research. People spend their lives studying the Navier-Stokes equations doing computational fluid dynamics and all the other aspects like that but that is the probably the most important set of equations that we will see in the course actually it is the most important but we're not going to spend much more time on them because they're so complex and that would require you going to either graduate school or maybe you'd have it in a tech collective within your undergrad program where you start to delve more deeply into the solutions to these equations. So with that we will conclude the derivation of the Navier-Stokes equations and we will move on to other forms of differential analysis in fluid mechanics.