[MUSIC] So we have defined a condition of small reduced velocity where the problem of the coupled dynamics between the fluid and the solid could be considered as independent of that of the proper dynamics of the fluid. Let us see now how this is going to simplify the equations. First of all, I have to reconsider my choice of dimensionless numbers. I have been using some numbers which contain the velocity U0. This is not a very good idea to keep using them. Because U0, the upstream velocity is just not relevant for the dynamics of our coupled problem as I just said. Where do I see U0 in the numbers ? In the Reynolds number, in the Froude number, and in the Cauchy number. This scale of flow velocity U0, with a speed of viscosity diffusion, with the velocity of gravity waves, and the third one scales the deformation due to the flow. [MUSIC] In place of U0, it would be much wiser to use in the dimensionless numbers or a velocity related to the solid, like c, the velocity of elastic waves. Or equivalently the velocity L over T-solid. Here are our new numbers defined using c in place of U0. They are the Stokes number, the Dynamic Froude number, and the Mass number. They scale now what happens in the solid with the speed of diffusion, the velocity of gravity wave, and finally, just the ratio of masses. You remember that the particular choice of dimensionless parameters is really up to you. As long as you satisfy the result of the Pi theorem. What I have done here is to take the Reynolds, Froude, Cauchy numbers and the reduced velocity, combine them with Ur, which is U0 over c in order to eliminate U0, and obtain a new set. This is much better because now U0 is only in the reduced Ur. Here is the same new set of parameters using L over T-solid in place of c, when I don't have a wave velocity c in the solid. We are now going to use these numbers in our equations. [MUSIC] I recall here the dimensionless equation that I had, where the dimensionless numbers appeared. The fluid, the solid, the interface, and the fluid and solid boundary conditions. The only thing I did so far was to assume that I could set the fluid boundary conditions at U equals 0 because it has a negligible effect on the dynamics of the fluid. But I have to be consistent in terms of the variables that I use in my equations. Here, other dimensionless variables that I used in my dimensionless equations, in the fluid domain, the solid domain and the time, the same for both. There is a problem here. You can see that I'm using U0 to scale the velocity and the pressure. But at the same time, I assume that U0 is just not even relevant for my problem. All I have to do is to make another choice, and replace U0 by a velocity referring to a solid, for instance, again, the elastic wave velocity c. I'll then call these new variables U bar and P bar, and for the sake of clarity, I will name x tilde x bar. What is that going to change in my equations, because there are just changes of fluid variables? Let us look at the fluid equations. In the fluid domain, we have the mass balance and the momentum balance. By substituting U tilde with U bar, and P tilde with P bar, respectively, I get very similar equations, except for the coefficients, which happen to be my new dimensionless numbers, the dynamic Froude and the Stokes number. This is no surprise of course because I have obtained these new dimensionless numbers also by replacing U0 by c. At the interface we can do the same kind of substitution, and similarly we have the Mass and Stokes number that appear. To summarize, here are our new dimensionless equations. Things are simpler now because all the fluid dynamics is governed by the interaction with a solid. [MUSIC] Can we solve these equations in a general case? Not yet. We have to simplify them again by using stronger and stronger assumptions. At this point we have just assumed that the fluid moves very slowly in comparison with the solid. In other terms, the oscillation of this boat when I get on board are not going to depend on the velocity of the river flow. What can we do next to simplify the gain? Well, certainly the case of small motion of a solid, just vibrations, is simpler to solve and very important in practice. Let us go in that direction and see what we can do. [MUSIC]