 In the last segment, we left off with the governing equations that we can use for looking at a convective free convection on a vertical heated plate. What we're now going to do, we're going to go through a process of non-dimensionalizing the governing equations. And the governing equations that we had, I'll write them out, we had X momentum continuity and energy. So those are the governing equations that we have. Now what we're going to do, we're going to introduce a non-dimensionalizing process. And the purpose of non-dimensionalizing, we kind of often do this in fluid mechanics, but what it enables us to do is determine the order of magnitude of the different terms, which terms are important and which are less important. But in order to do this, what we'll be doing is we will be introducing non-dimensionalized variables and they will be given the star. And then we'll take, for example, the length scale X and we'll divide by some characteristic length scale. We do the same with the Y dimension, we introduce a new variable Y non-dimensionalized by our length scale L. Now for velocity, for free convection or natural convection, we don't really have any specific free stream velocity that we can non-dimensionalize by. So we're just going to do some characteristic velocity U0. And we'll do the same for V star. And then finally, for temperature, we'll do something. We do this in heat transfer quite often. We subtract by the free stream and then T wall minus T infinity. Now sometimes this is T s, so don't get confused. T wall, T s, they're one and the same, the surface or the wall temperature. And in this, what we've done is L is a characteristic length scale and U0 is some arbitrary reference velocity that we have not yet defined. So if we go through the process of non-dimensionalizing, for example, let's take a look at one of the terms that we have here. We have U du by dx. So if I was to non-dimensionalize that, what I would do is I would go through the following process I would take and that would be U star times U0 and then inside of the partial we have a U. So again, we would pull out another U0 and then that would be partial U star. And then in the denominator, that would be partial x star. But we would have a length scale L there. So that would result in the following. It would be U0 squared over L U star partial U star partial x star. So if we do that for all the different terms, we'll result in a new equation. I won't go through all of that. But if we take that final equation that we've non-dimensionalized and we multiply it by the Reynolds numbers squared, a new non-dimensional number will drop out and that number we refer to as being the Grashof number. So with this, what happens is the Grashof number comes out of our x-momentum equation and the Grashof number, we have not seen it yet, but it is expressed in the following manner. So that is the Grashof number where U is equal to our dynamic viscosity divided by the density. So when we look at this, what the Grashof number represents, it is a ratio of buoyancy force to viscous force. And by comparison, when we saw the Reynolds number, recall the Reynolds number was inertia, the inertia force to the viscous force. So the Grashof number is a non-dimensional number that we use quite often for natural or free convection. And it plays a very similar manner to the Reynolds number did for force convection over a flat plate flow. So what we see is that the Grashof number plays a very similar role as the Reynolds number did for force convection over a flat plate in that it is giving us an indication of when we will transition from a laminar free or natural convection flow to a turbulent free or natural convection flow. And consequently, we will have a critical Grashof number. And that is approximately four times ten to the eight. However, anything from ten to the eight to ten to the nine is commonly used. And recall what that means is that if we have our vertical flat plate here, we can have a laminar boundary layer. And then when we hit the transition point, the boundary layer will grow at a higher rate, maybe not that high. But it would go through a transition process. And so here we would have laminar. And then up here would be turbulent. What that means is the convective heat transfer relationships will be different as we go through this transitional point. And X was in that direction. And so this would be the Grashof number evaluated as a function of X at that location. So that is the Grashof number, a non dimensional number, very, very important for free or natural convection. And we'll be using that quite often as we go through the analysis in this chapter.