[MUSIC] We now have at hand the tool of dimensional analysis. And we are going to use it to try and classify the cases of fluid solid interactions. This is the first objective of this course. What we are going to do here is something you would not find in most textbooks because we are looking for general results. We are going to go small steps by small steps together. But once we get to the results, you will see that the view is clear and beautiful. Let us consider, schematically, the fluid here in blue and a solid in red. To make things simpler, we assume that they stand in separate domain of space and that there is no mass transfer between them. As for the case of drag on a sphere, we now need to specify what quantities we want to use to define our problem and what we are looking for. First, the fluid on the left. Let us say that we're looking for the local velocity U in relation to the coordinate of the point we consider X and the time T. This is also going to depend on the viscosity of the fluid, mu, The density of the fluid rho and the gravity, G. Also the result is going to be different if I change the size of the domain. So I say the result depends on the size L. Of course, this velocity is also going to depend on some boundary condition. For instance, an upstream flow velocity that I call U naught. This is the list of quantities I'm considering in a given problem in the fluid. Second, the solid on the right. We might want to know the displacement, csi, at a position X, at a time T. It may depend on E, the stiffness of the solid, (I shall come back to this later) on it's density, rhoS and on gravity G. Again, it will also depend on the size. And there is somewhere the magnitude of this displacement that is set, say cdi naught. As you can see here, I'm quite general in stating what the problem is. Still by stating that this is the list of quantities that I want to relate by my physical law, I'm not that general. For instance, I've excluded the temperature. But we have to choose what is the kind of problems that we want to consider. And this is already quite general. [MUSIC] By the way, what did I mean by the stiffness E and the density rho S for the solid? If this is a continuum that deforms elastically, E is what I call Young's modulus. end rho s , it is the density itself. If the solid is more like a discrete system, what is E and rho s? I can always define a stiffness like quantity and a density like quantity as soon as there is stiffness and mass. For instance, in a mass spring system here we may define a stiffness that would have the same dimensions as a Young's modulus as E equals K over L. We also have an average density based for instance on the total mass divided by the scale of our volumes L cubed. So in a way or another, we shall always be able to define the quantities I've listed. [MUSIC] Let us apply the powerful tool of dimensional analysis to our problem of a fluid and a solid. We are going to start by something very simple. Doing dimensional analysis separately in the fluid and in the solid. Imagine now that what happens in one domain is totally independent of what happens in the other. For instance, in the fluid. This is what you have done in fluid mechanics when you have ignored all possible influence of what happened inside a solid that bounds the fluid. So, we assume that there exist a physical low that relates the fluid velocity with all the other parameters, namely X, T and so on. This means that the flow is not going to depend on the deformation of the solid, because the stiffness E for instance, is not included in there. This is pure fluid mechanics. Let us do the dimensional analysis of this. Here is a law F between the dimensional variables. There are eight. To use pi theorem, I need to build the matrix of dimension exponents. Here it is. X is the coordinate, so it is a length. T is a time. U is a length per time and so on. As soon as you can put some units on these quantities, you can write the dimension exponents. Now, what is a rank of this matrix? We can find three independent vectors. For instance, here. And certainly no more than three because the dimension is three. So the rank is all equal to three. I can conclude that we should be looking for 8 minus 3 equals 5 dimensionless parameters. So, let us write the law we are looking for in the form of one depending on only five dimensionless parameters. What are these dimensionless parameters? We know that we should find five independent ones. I can easily start by defining a dimensional velocity by diving U by U naught. Both are velocities. So the ratio is dimensionless. Second, X divided by L. Third, something I shall explain in a moment. Then, of course, the Reynolds number that combines these four. What else? I haven't used the gravity G so far. So let us use it in a dimensionless number. Here is what is usually called the Froude number combining U naught, G and L. These five members are dimensionless and they are independent. You cannot get one by a combination of the others. Let us go back to the ratio U naught T over L. As all dimensionless quantity, this one can be understood as the ratio of two dimensional quantities, two lengths, two times. I can write this one as T over T fluid where T Fluid equals L over U naught. What is L over U naught? It is just the time taken by a particle of velocity U naught to travel across the distance L. So T fluid is a time scale associated with convection in the fluid. A very important quantity that we shall use later. At this stage, we have just written down the fact that the dimensionless velocity in the fluid is dependent on a dimensionless coordinate, a dimensionless time, the Reynolds number, the Froude number. Let us now do the same for the solid alone. Now, we look for a relation between all quantities on the solid side. F of X,T, csi, E, L, G, tho s chi naught, equals zero. I have singled out the displacement, which is unknown. Let us use again the pi theorem. Here is the matrix of the dimension exponents. We have here, too, 8 quantities, a rank of 3, and so 5 dimensionless parameters to find. What are they? Here is a choice. The dimensionless displacement where I've divided csi by the length L. The dimensionless coordinate or dimensionless time , I will discuss just after, and two other dimensionless parameters. The first one is the ratio between the displacement data csi naught and the length scale of the system. We shall call it the displacement number. When large, the displacements are large with regard to the size. This is what we call usually large displacements. The second one combines gravity, density, length and stiffness and I shall call it the elastogravity number. When it is large, it means that the deformations induced by gravity in the solid are large. For instance, in a jelly cake, the shape is really effected by gravity. Let us go back now to the dimensional time that I introduced. I can write this as T over Tsolid, where Tsolid is L over a velocity C, and this velocity us square root of E over rho s. What is it? It is actually the scale of elastic wave velocities inside the solid. So T solid is the time that an elastic wave takes to go across the solid. [MUSIC] To summarize, before doing some dimensional analysis on the full fluid plus solid problem, we have looked at dimensional analysis for the decoupled problems. By this I meant the fluid alone independent of what happens to the solid, and the solid alone independent of what happens to the fluid. What did I get from my dimensional analysis? I could build the list of dimensionless quantities that can use in either of these two problems. Fluid alone and solid alone. I obtained dimensionless numbers that are the Reynolds number and the Froude number for the fluid, the displacement number and the elastogravity number for the solid. With that, we can do the fluid mechanics and the solid mechanics separately. But this is not what we're looking for. We want to address the case where they are coupled, which means the case where everything may depend on everything. Be patient. We can now go forward and do it. [MUSIC]