[MUSIC] Hello, in the first chapter we set up the stage of the course by using dimensional analysis. We know the dimensionless numbers that can be used to model fluid solid interaction problems. We know that dimensionless equations and how our dimensionless numbers appear in these equations. With that, we are now ready to start building models that will be appropriate for various regions of the parameter space. In engineering, we need these models. Where should we start? As I mentioned before, the reduced velocity is a very useful parameter to help in classifying problems because it compares the timescales of the solid dynamics and of the fluid dynamics. Remember, that the reduced velocity reads T Solid over T Fluid. It is also the ratio between the fluid difference velocity U0 and that of elastic waves in the solid, c. So, in the forthcoming chapters we shall explore the effect of this reduced velocity UR. A very important case in practice is when the fluid is apparently not moving. I mean, when its motion seems very slow in comparison with that of the deformation of the solid or very small, we use velocity. For instance, imagine the oscillations of our flexible damn in contact with a very slowly moving water or a small boat that oscillates on a river. Certainly, we can derive model for such cases which are much simpler than the general case. Let us do it. What does small reduced velocity mean? Let us imagine that we follow in time the evolution of a quantity pertaining to the solid. For instance, the displacement somewhere. This quantity evolves with a timescale that I called T Solid. Conversely, the quantity pertaining to the fluid dynamics such as the position of a fluid particle would evolve on a much longer timescale, T Fluid. Intuitively, we may say that they easily meet when these timescales are so apart, such that the dynamics of the solid during T Solid, occurs in the interaction with the still fluid. This is certainly a much simpler framework to model fluid solid interactions. Note, that by stating that the fluid is still, I do not mean that it doesn't move. First, I say that it's own motion that regulated by U0 occurs so slowly that I can neglect it when computing the motion of the solid. But of course, I will have to compute it at some point if needed. Second, and this is the meaning of the small dotted curve here. There is certainly a motion of the fluid that mirror somehow the motion in the solid. How? We shall solve that later. [MUSIC]. All I just said here was essentially an intuition that if the two timescales are so separated, there is going to be a more solvable problem. Let us see how this materializes in the equations. The idea is to take into account the fact that there is a significant difference between the dynamics of the fluid and that of the solid. Let us see that on the boundary conditions on the fluid. You remember, that the fluid domains see two types of boundary conditions. Those at the interface with the solid, and the others, anywhere else. How do these boundary conditions contribute respectively to the dynamics of the fluid? Let us compare the order of magnitude of what they impose on velocity. At the interface, the condition on the velocity is that it is equal to that of the solid U= dxi/dt. Here, schematically, is an evolution of the displacement somewhere at the interface. What is the order of magnitude of this dxi over dt? Well, xi is a displacement in the solid and we have an order of magnitude of this that we called xi0. And xi evolves with the characteristics timescale that we called T Solid. So, the quantity dxi over dt has the order of xi0 over T Solid. And therefore the boundary condition that applies there on the velocity is of the order of xi0 over T Solid [MUSIC] The other boundary conditions also affect the dynamics of the fluid. This is a very general issue, but remember that we have defined a quantity U0 which sets the magnitude of our fluid velocity. For instance, it may be the multitude of the upstream velocity in the river. So, I am going to state that the fluid boundary conditions set a velocity of order U, of course, order of U0. Let us summarize. The dynamics of the fluid is governed on one hand, by the condition of order U0, and on the other hand, by the condition of order xi0 over T Solid. Certainly, if U0 is much smaller than a xi0 over T Solid, the first condition can be set to 0 without changing much the results. This corresponds exactly to what we meant by neglecting the proper fluid condition. The solid moves in an approximately still fluid because the fluid moves so slowly. What is the meaning of this condition, U0 much smaller than xi0 over T Solid? By rearranging it, it also reads U0 T Solid over L much smaller than xi0 over L. And in terms of our dimensionless number this is exactly UR much smaller than D, the displacement number. So to summarize, under this condition we say that to compute the dynamics of the solid in contact with a fluid, we can totally disregard the proper dynamics of the fluid. And we can do this because it is the interaction that governs the dynamics of the fluid. This is exactly what we entreated at the beginning when we imagine that the fluid was apparently still during the motion of the solid. [MUSIC] So, schematically, in the general case we have the fluid dynamics and the solid dynamics. And they are coupled by the kinematic and dynamic conditions at the interface and they evolve simultaneously. But, in our new approximation of small reduced velocity, we have two dynamics, one slow, which is that of the fluid, and one fast, which is that of the coupled still fluid and solid. We have separated the two dynamics by taking advantage of the differences in the timescales. Of course, this is going to be much simpler to solve than the general case. What should we do now? We just need to write down the equations for that simple problem and hopefully, solve them. [MUSIC]