 We're now going to work an example problem involving the acceleration term that we just looked at from the velocity field. And in this problem what we'll do is compute acceleration in two different ways. One using the Eulerian and then the other using the Lagrangian formulation. So this problem will illustrate a couple of things. So the problem is going to highlight the differences between the Eulerian and Lagrangian description, and it will also show the convective acceleration. So let's begin with the schematic. And essentially what we're looking at is a nozzle in which a fluid is being accelerated as it moves along the length of that nozzle. And so we have the solid walls of the nozzle. Our coordinate system, we will denote x to be going in this direction and y to be in that direction. And the fluid is moving through this nozzle with some velocity v. And we will denote this plane here as x1, which is equal to 0. And the plane at the end of the nozzle as x2 equals to capital L. So what we are given in this problem is we're given the velocity in the nozzle. So we're given the velocity in the nozzle. And we can note a couple of things that this is a steady flow. And the reason we know that is none of the terms are time. So the velocity is not changing as a function of time. And the other thing that we can conclude by looking at this formulation for the velocity is that as the particles move from x1 from this plane, and as they move through to x2, what happens is they speed up or they accelerate. So the flow speeds up or accelerates towards x2. So that is the problem. Now what we want to find in this problem is the following. So we're told to find two things. The first one is the acceleration in the x direction. And the second thing is in a particle formulation. So considering a particle of fluid at x equals 0 and t equals 0, find the following. The first is the position of the particle. And we'll call that x little p as a function of time. So this is the Lagrangian formulation because it's following the particle. The first one is the Eulerian at the ax. And the second thing that they want us to find here is the x component of acceleration of the particle. And that will be denoted by ax. And we'll denote it as being a particle with a little p as a function of time. So what this problem will do is it will illustrate both the Eulerian and the Lagrangian, or the particle-based formulation. So let's begin with the solution. So the solution we will begin with the first part, which was the Eulerian description of the acceleration of the particle. And for that we're going to use our substantial or our material derivative, which we saw in the previous segment. And so if you recall, that was expressed in this format. And we used the large d to denote the substantial or material derivative. And you have a local acceleration term. And then you have a convective acceleration term that we said you could express in terms of the gradient. So I'm just going to expand that out now. So that's what we have. Now examining this, there are a couple of things that we can do right off the bat. First of all, if you look back at the formulation we had for the velocity, there was no time in this. And that means that it was steady, which is what we said here. So going back to our expression, that means that this term goes away. That is a zero. The other thing that we can say, if we look back at our schematic, the velocity is only moving in the x direction. There is no velocity in the y or in the z direction. And consequently what that means is that v is equal to zero and w is equal to zero. And consequently that term and that term disappear. What we're left with for the acceleration then is the following. It's that expression there. And if you recall our expression for velocity, so I will sub that in. And what we then obtain, this would be u. And then we multiply it by du dx. So du dx would be v one over l. And with that we can then write out the acceleration in the x direction. And so this would be the answer to the first part of the problem. And there are a couple of things that we can note. We have ax even though steady flow, fluid convex to region of higher velocity. And this is the Eulerian expression. So if you want to find the acceleration, all you do is you plug in the value of x into that expression. And that would give you the acceleration at x. So that is the acceleration using the Eulerian approach. What we will do in the next segment is we'll take a look at the same example problem. But we will compute the second part of the problem. And the second part asks us to do this, which would be the Lagrangian approach. So that's what we will do in the next segment.