[MUSIC] Well, we have the equations that govern the dynamics of our coupled system. But they relate dimensional quantities. We now want to move to equations relating dimension-less quantities. First of all, we have to define the dimensionless variables we want to use out of the dimensional ones. In the fluid, I have the coordinate x, the velocity u and the pressure p. Let us define the dimensionless coordinate, x tilde, which is the coordinate divided by L. Similarly, I define U tilde equals U over U naught, and for the pressure I can divide by whole unit squared, and I call this the fluid. From now on I shall use the tilde subscript when things are related to fluids as of small wave. On the solid sign now, I can do something similar for the coordinate x the model displacement q and the model force f. And now use the bar subscript For dimensionless quantities in the solid. So let us have X bar the dimensionless coordinate, Q bar the dimensionless model displacement and F bar the dimensionless model force. Note that the model shape Is generally given in a dimensionless form. It is a normalized direction or eigenvector. All these choices are rather arbitrary. But remember that we have some freedom in choosing how we build dimensionless quantities. So far I have not said anything about the dimensions time. Let us do this, in the previous dimensional analysis based only on the parameters. You remember that I could only find the dimension time relevant for the fluid elements and one relevant for the solid elements. They're just referred by the time they were based on. They referred to the convection time, T fluid, or the time of elastic waves to propagate, T solid. In other terms, these two clocks have different references. And the ratio between them is what we call the reduced velocity. At that stage I must choose which one I use in the next steps. Because the time clock needs to be common to both fluid and solids if we want to compare them. Let us choose quite arbitrarily, the reference time of the solid. So our dimensionless time will be called t bar from now on. It is equal to t over T Solid. And here, I have a very simple way to define T Solid based on the model, mass, and stiffness parameters. T solid equals square root of m over k. Later in the course, we shall switch from fluid-based to solid-based time as needed. To summarize, here is our set of dimensionless variables for the fluid and the solid. All we have to do now is to take the equations in the dimension form and substitute these variables to obtain equations between our dimensionless quantities. Simple. Let us do it. [MUSIC] Let us start by the fluid. I had the mass balance and the momentum balance. First, the mass balance. By substituting the variables I just defined, here's what I obtain. The mass balance takes the same form, but now a new tilt. In the momentum balance, the substitution is a bit longer because we have more terms. But this is just a change of valuables. In front of each term, we end up with groups of quantities c over U naught, gL over U naught, U over roe U naught. You already identified these groups of quantities as dimensionless numbers. So, I can write the dimensionless equation where the reduced velocity. The fluid number and the Reynolds number appear. This is no surprise. We have done this before in for instance. When trying to identify the effect of the Reynolds number. What is new here, is that because we use dimensionless time base on the soy dynamics. We have one over ur coefficient in front of the exoneration. For the solid, things are quite simple actually. We just substitute. And we end up with a standard oscillator equation. The frequency of this oscillator is one, because we have chosen, as reference time, the time of the solid, the period. Now, the same operation at interface. We start with a Kinematic condition. We shipped it again to the variables. And we end up with a kinematic condition that contains two-dimensional parameters that we have already defined. The reduce velocity UR, and the displacement number. Finally, the same operation on the dynamic condition. No difficulty. Although, you have to do it carefully. Note that the sum of the whole interface involves also the change of space coordinates. You end up with a form which is similar to a original one. But with coefficients that you can recognize. The quotation number, the Reynolds number, the displacement number. So, these change of variables were a bit long to do, but now we have everything. Here is a set of dimensionless equations, the fluid, the solid. The interface and the other boundary conditions, which we know to be given in model that is here. What is new? Why did we work to have these dimensionless equations? Because now we have equations where the influence of our dimensions numbers is evident. Here are the Reynolds and fluid. Here is the displacement number. Here are our new neighbors, the quotient neighbor and the reduced velocity. Now we expect that for a particular case, when a given demonstrated number is large or small. The placement in terms of the equations will be more or less influencing the result. This is a way to simply find models by neglecting things here and there. This is the next step. So if we go back to the idea of classifying problems and building models. We have a way to do it. First, we have found dimensionless numbers will even fold these kind of problems. Second, we have equations where these numbers appear. We are now ready to explore systematically the effect of dimensionless numbers. While going to first see what is the inference of the reduced velocity UR. This is a key parameter, because it gives the ratio between the two time scales. That of the fluid and that of the solid dynamics. In the next weeks, we shall build models that are efficient for cases where the reduced velocity is small. As for our flexible dam, or conversely, high as for the plane further here. [MUSIC]