 In the last segment, we took a look at the fluid mechanics associated with free or natural convection. What we're going to do in this segment, we're going to take a look at the governing equations. When we looked at the velocity profile, if you recall, we were drawing a plate, assuming that the plate temperature was hotter than that of the surrounding fluid. We had the condition where T wall is greater than T infinity, and T infinity was the quiescent or ambient fluid outside of the plate. When we did this, what we did is we introduced a coordinate system with x going in this direction and y being normal to the plate. We will continue with that coordinate system as we look at the governing equations. What we're going to do, we're going to begin with the x-momentum equation that has been reduced a little bit, similar to the boundary layer approximations. We're not doing that in this course, but if you look at my fluid mechanics course, you'll find discussion about how to reduce the Navier-Stokes equations into the boundary layer equations. With that, I will write out the x-component of the momentum equation. That is the x-momentum equation. One thing that you may notice immediately, if you compare this to the Navier-Stokes equation, we have a minus g here. The reason for that, the reason why it is negative is because the gravity vector is acting downwards, which with our coordinate system that we've just cast out here, are written there, that would be a negative value, and that's why we have the negative value there. It's negative as x is acting downwards, or as the gravity acts downwards. That is the x-momentum. The other thing that we know, we were showing that the boundary layer starts to grow from the beginning of the plate, and in reality, what's happening is we do have fluid being entrained, and so there would be fluid coming in from below, and that would give us a little bit of a velocity profile at the beginning, but usually what we assume is out beyond the boundary layer, that is when we get into what we call the quiescent region, where quite often we make the approximation that the velocity in that region is zero. With that approximation, what we can do is we can look at the x-momentum equation, and the x-momentum equation then in the quiescent region reduces to the following. This was just minus rho infinity g is equal to the pressure gradient. Now what I'm going to do, I'm going to call the x-momentum equation equation one. I'll call the quiescent region equation equation two, and now what we're going to do, we're going to sub equation two into equation one. With that, the following results, and so we obtain this equation here, and now what we're going to do, we're going to focus in on this term here, how to handle the fact that we have a density difference, and in order to do that, what we're going to do, we're going to introduce the volumetric thermally expansion coefficient, and the volumetric expansion coefficient is defined in the following manner. So beta is related to the change in density with temperature at a constant pressure. And with this, what we can do is we can look at it for a couple of different scenarios, be it a gas or a liquid. And so to begin with, if we're dealing with a gas, if we're dealing with an ideal gas, we can use the ideal gas equation, and with that, we can rewrite beta in the following manner, and then making a substitution for the density. We get one over the temperature in Kelvin is beta for an ideal gas, and if we're dealing with a liquid or a non-ideal gas, in those cases, beta would be from property tables. So that's how we handle beta, which is our volumetric thermal expansion coefficient. Coming back though, what we want to do is we want to be able to find a way to use beta in order to figure out how to handle this term here, where we have the difference in density. And so what we're going to do, we're going to express beta in terms of a finite difference. So that would be beta expressed in terms of a finite difference. Now what I'm going to do, I'm going to solve for rho infinity minus rho, which is what is in our momentum equation. So what we have here is an expression for the difference in density expressed now in terms of the density, the volumetric expansion coefficient, and the difference in temperature. And this is referred to as being the Boozinesq approximation, and it gives us a coupling between the momentum equation and the energy equation. So with that, what I'm going to do is I'm going to rewrite our governing equations. So we have the x momentum equation, we have continuity, and lastly we have the energy equation. So those are the three equations that we have that we need to solve if we want to be able to determine the temperature and the velocity profile in front of or above a plate that has natural convection. So remember y was in that direction, x is in that direction, and this would be a case where the plate wall is greater than t infinity, t infinity being out here. So we have momentum, continuity, and energy, and those would have to be solved. What we're going to do in the next segment, we're not going to solve them, we'll actually rely on a solution that others have come up with, and it turns out that in order to solve this, you get two differential equations that you have to solve numerically, very similar to the Blasius solution actually. If you look at my fluid mechanics course, you'll see a discussion about that process. But what we'll be doing, we're going to be doing some dimensional analysis, we'll non-dimensionalize the equation I should say, and then some non-dimensional numbers will be dropping out of them, and that's what we'll be looking at in the next segment.