[MUSIC] We have defined the dimentionless parameters that need to be used for fluid solid interactions. We have done this without using any of the equations of fluid mechanics or solid mechanics. We just worked on the parameters that would be needed to describe the system. And how to have to some consistency in the description in terms of dimensions. But we actually know much more, we have models for some laws that are satisfied by the variables in each domain. And we know how to write a condition at the interface. So let us do this, and connect it with the dimensional analysis we just made before. What are the equations that we want to use in the fluid, the solid, and at the interface? First, in the fluid. We can write the mass balance, which states that the divergence of the velocity equals zero. We have considered so far that the density is a constant and so the fluid is incompressible. We can write the momentum balance which relates the inertia with the local loads. The first term is inertia, including the convective term. And on the right hand side of the equation, I have the gravity force, the pressure gradient, and the viscous stress. As you know, this set forms the Navier-Stokes equations. For the Solid there are several models that I can use as I mentioned before. The most general one would be Continuum mechanics. It would be very well suited to be combined with the fliud model that I have used which is also continuum mechanics. This can be done but you may not be familiar with continuum mechanics for the solid. So I shall use now on as much as possible a modal approximation, and even in many cases the simplest one, the single mode approximation. What does it mean exactly? [MUSIC] The single mode approximation means that you are going to assume that the displacement in the solid side is a product of a function of time q and a function of space phi. Now q will be called the modal displacement and phi the modal shape. For instance, imagine a mass spring system, in that case the displacement of a solid, here the mass, is the same for every points. And reads q of t times the unit vector of the direction of motion. This is quite simple. But this model also applies to more complex motions such as the bending of a beam here. In that case, the model shape phi is a function of the coordinates. So, this is a really simple framework to describe the kinetics of the motion of a solid. I mean, the way it moves. But we also need equations of motions. Here is the single mode approximation. First of all, we assume that we know the modal shape phi. This may result from experiments or computation. But by stating that the motion follows the shape of phi the only unknown that remains is the modal displacement q of t. I'm going to assume from now on, that the modal displacement in the solid q of t satisfies what is called an oscillator equation m d squared q over dt squared plus kq equals f. Here m is called the modal mass, k the modal stiffness and the f of t is the modal load. What are these? Well, in my mass spring system, the modal mass is the mass. The modal stiffness is the stiffness of the spring and the modal force is the force that applies on it. But this oscillator equation also applies on q in cases such as the vibration of the beam along its first mode here. This is a general result of the modal approach in dynamics and m and k can be defined easily. In that case the modal load is just the sum over the solid of the local loads times the modal shape phi. We say that the forces are projected on the mode. [MUSIC] So to summarize we have the Navier Stokes equations on the fluid side and a modal equation on the solid side. How do we connect them? Well, we have to connect them by the continuity equations at interface. What are they? At the interface between the fluid and the solid, we have two kinds of condition. The first one is called a kinematic condition, because it connects directly the velocities. I will mostly use a condition that the two velocities are equal, meaning that there is no mixing between the two domains. And there is no sliding either. This simply reads that u equals d xi, over d t. Or equivalently here, U equals dq over dt times pi. The second condition states an equilibrium of forces at the interface, we shall call it the dynamic condition. On the fluid-side the acting local force is the result of pressure and viscous forces. On the solid side we only consider the quantity called the modal force. This means that we have to equal to sum of all the fluid forces times the model shape with the modal force f. Here is the modal form of our Dynamic condition. Let us summarize all that we have now. Navier-Stokes equation into fluid, single modal oscillator equation for the solid. Kinematic and dynamic conditions at interface. To have a full picture we should add that there are some boundary conditions that apply on the free domain outside of its interface with the solid. Same thing on the solid side, symbolically because it is included in the normal shapes normally. So, these are the equations that we want to use to model our systems. How can we relate all this to the dimensional analysis we have done before? Well, these equations relate dimensional quantities. In the mass balance, when I write divergence of U equals zero, U is a true velocity with dimensions of lengths per time. So, all I need to do is to write now my equations on dimensionless quantities. Let us do this next. [MUSIC]