[MUSIC] We are now ready to undertake the dimensional analysis of a fully coupled fluid and solid interaction problem. We have done the case of the fluid alone and the case of the solid alone We're going to use exactly the same method but considering the fluid and the solid, simultaneously. We are back to our full list of parameters that define the problem. Let us discuss a bit what these quantities are. Some of them are only defined on the fluid side, or on the solid side. This is, for instance, the case of the viscosity mu in the fluid or the stiffness E in the solid. Others are common to both domains such as the gravity g or the scale of lengths L. What about our variables of interest, those that we want to relate to the parameters? I mean, the velocity U or the displacement csi. One of them is defined in the fluid and the other in the solid. But now, we are going to consider that they are related to all the parameters of the problem without separation. So, I'm looking for a relation between one of the variable of interest, for instance, the fluid velocity and all the other parameters. I write this as a function G of all the quantities equals 0. First of course as before, U depends on the position that I'm considering, x, and the time I'm considering, t. Then it depends on the parameters in the fluid mu, rho, and U naught. And also, on the parameters common to both domains, L and g, but now I assume that the velocity in the fluid may depend on what happens in the solid. And I do this by including the stiffness E and the density rho s, and the displacement data xi 0 in the equation. If you count, you find that we have a set of 11 parameters to consider. The issue now is to apply the pi theorem to get some information on the dimensionless parameters. Let us do this, exactly as before. Here is the matrix of the dimension exponents. You find our 11 quantities. The velocity, the space and time coordinates, the three parameters, the common parameters and the solid parameters. In terms of exponents I know what they are for each of them as I have done the dimensional analysis of the fluid alone and the solid alone before. The only new thing is that they are now considered simultaneously. Again I find the rank of three, so the total number of dimensionless parameters that we expect is P equals N minus R, which is 11 minus 3 so, 8. [MUSIC] What are these dimensionless quantities in such of problem that mixes fluid and solid ? Let us try to give a set of eight independent dimensionless quantities out of the 11 dimensional ones. I'll start with the one I know. U over U naught, x over L, U naught t over L, the Reynolds number and the Froude number. That makes five. Now, I can also use the ones I know from the solid side, combining the three quantities in a solid, and that gives us the displacement number, csi naught over L, and the elastogravity number, G. That makes 5 + 2 equals 7. But from the pi theorem I know I should use eight dimensionless quantities. No less, no more. I miss one. Let us call it A for the moment. What should it be? It cannot be in combination of the other ones because we are looking for a set of independent quantities. It necessarily mixes things from the fluid and the solid side otherwise I would have found it before when doing the uncoupled case. So what is it? What can we imagine as the dimensionless quantity combining fluid and solid dimensional once. The simplest one is the ratio of the two densities. Let us call it the Mass Number, M. This seems a very good choice because it simply tells you that it is different for a solid to interact with air or with water. In the hard-disk drive example, M is the order of 1, air, over 10 to the 4, metal, and so M is of the order of 10 to the minus 4. Conversely, for the dolphin skin, both media have about the same density, and M is the order of 1. Here is another possible choice, the reduced velocity. It is the ratio between our free velocity, U naught, and the velocity of elastic waves in a solid, c. This also seems a good idea because it contains information on the way the two dynamics are related. It would be quite different between two examples I considered before. The inflatable dam and the dolphin's skin. [MUSIC] As possible new dimensionless parameters I have proposed the ratio of two densities, that was the mass number and the ratio of two velocities, that was the reduced velocity. I can also imagine something combining stresses or stiffnes. This here is the Cauchy number. What does it mean? It is the ratio between the fluid loading quantified by the dynamic pressure over a unit square and the stiffness of the solid E. The higher it is, the more the solid is elastically deformed by the flow. That seems quite useful for instance between the hardest drive case or the Tacoma bridge case, the values would certainly be different, because the reading head does not deform much, but the bridge does. So, the new dimensionless number A that I need to use can be any of these three, or others you may find that combine fluid and solid quantities and are independent of the others. These three are actually the most important ones, and are used a lot. Which one should you choose for your problem? Well as I said before, there is no good choice of dimensionless numbers. But there are efficient choices, that would be more helpful in solving a given problem. In this course you will learn how to use all of them in particular cases. [MUSIC] To summarize now, we have shown that doing the dimensional analysis of a coupled fluid and solid problem gives you the dimensionless quantities of each domain plus a new one that combines both domains. But why did we do all this, because we wanted to find a way to classify problems. Remember that in Fluid Mechanics , the Reynold's number allowed to classify the flow patterns between creeping flow at very low Reynolds number, and detached flow at high Reynolds number. So the idea now is simple. Let us use our new dimensionless number, the one specific to spatial interactions, to classify our fluid solid interactions problems. And why classify? Because this will allow building simple models. In practice what we are going to do is to go in different places in the space of dimensionless parameters and derive models that are applicable there. I mean model for all the cases are listed at the beginning of this week and many others. In all the steps towards dimensional analysis I've tried to stay as general as possible. That made things a bit more difficult to understand. I know. But it was necessary to somehow set up the stage of the course. We are now ready to explore the space of our dimensionless numbers and find simple ways to model the various kinds of friction interactions. As you will see, there are a lot of simple models and they are very useful in engineering. Next we are going to see how our dimensional analysis can be combined with what we know of the equations of Fluid and Solid Mechanics. Then, we shall be able to effectively solve problems. [MUSIC]