[MUSIC] Well. You know that the free surface is going to bring you some stiffness to the fluid dynamics, and therefore there is the possibility of sloshing modes. We obtained the first mode for a rectangular tank and we said that we are going to use it to represent the whole sloshing dynamics of the fluid. That is a bit of an oversimplification, but we want simple models to understand the mechanisms of the interactions. Here is the problem I want to solve. The dynamics of a tank containing sloshing fluid and elastically supported. So this is going to be some combination of the sloshing motion that I've just analyzed and that of a simple mass spring system, the mass of the tank connected to a spring. How can I solve this? My fluid system on the left is defined by capital Q of t, which satisfies MF capital Q double dot plus KF capital Q equals zero. And the solid system on the right is defined by q(t) which satisfied q double dot plus q equals zero. Actually, I know how to solve some coupling. When I did not consider sloshing, I found how a fluid was coupled to a moving solid. That was the issue of added mass, remember. In that particular case, the added mass problem is simple. The fluid just moves with a solid. The added mass is the mass of fluid. So the pressure shape is just minus x, the velocity shape is the pure translation. It is identical to the solid mode shape. And the added mass is just the fluid mass, so here the mass number over two. [MUSIC] First I would like to model the effect of solid motion on sloshing, just like in my glass of wine when I moved it. Let us try building a combined solution using the added mass case and the simple sloshing mode. Here it is, p equals capital Q double dot, capital Phi p, this is the sloshing part, plus Q double dot Phi p, this is the added mass part. This solution satisfies the boundary condition on the sides and at the bottom. Now, on the free surface, I can do exactly as before, writing the free surface condition in a projected form. I write that the free surface quantity times phip summed over the whole free surface equals zero. By inserting the combined form of p into this, I get terms relative to capital Q and terms relative to q. What are they? Those related to capital Q and its time derivatives are exactly those I got before. The sloshing modal mass, MF, and the sloshing modal stiffness, KF. But now, I have a new term that combines phi p and capital Phi p. So to summarize, by using a combined solution based on the pure sloshing solution plus the added mass solution, I obtain the equation governing the dynamics of sloshing as the classical modal oscillator equation, with the modal mass MF and the modal stiffness KF, forced by the acceleration of the solid, q double dot. The coefficient, which I call the solid to fluid coupling mass, can be computed with the pressure shapes of the sloshing and of the added mass solutions. In our problem, I get mSF equals two over pi squared. This is why wine sloshes in my glass, because I accelerate the glass. Now reversely, let us see what is going to be the effect of the motion of the fluid on the dynamics of the solid. This is the issue of the truck overturning. It is the sum on all the interface of the pressure force minus pn It is the sum on all the interface of the pressure force minus pn ������ scalar product with the shape of motion phi. Now, because my pressure here is the sum of two parts, I get two parts in my loading. One is a coefficient times capital Q double dot The other one a coefficient times q double dot. The second one is easy to identify. It is the fluid loading caused by my added mass solution, so it is the added mass effect. The first one gives the effect of the sloshing itself. It is the fluid to solid coupling mass. Let us summarize. The dynamics of this fluid solid system is governed by first, an oscillator equation for the fluid sloshing motion, with an excitation from the motion of the solid. Second, an oscillator equation for the solid motion with an excitation by fluid sloshing. There is also the classical added mass effect, which I have put on the left hand side to focus just on the coupling terms. So what I get is coupled sloshing solid dynamics. Actually, this is very similar to a double mass-spring system, but here, one of the oscillator is solid and the other is fluid. This is a classical case of mode coupling. What does the motion look like? Here we have integrated the two equations in time, and what you see is that the motion of the support affects sloshing. And that sloshing affects the motion of the support [MUSIC]. This is a very important result. We obtained it on a simple geometry, but you can obtain similar results in the general case. The key point is that the force feedback, in relation to the added mass effect, is itself governed by a oscillator equation. So, the fluid here again as an internal time scale, which is that other sloshing mode. This time is scaled by the dynamic Froude number. When we investigate the effect of viscosity, we also add an internal time scale. It was that of viscous diffusion, and it was scaled by the Stokes number. So, oscillation on one side, diffusion on the other. These are the two main mechanisms that may cause the feedback force from the fluid to the solid to be delayed. What happens if the dynamics of the solid need to be approximated by more than one mode? Well, all this is linear mechanics. So, we're going to have coupling matrices in place of coupling coefficients and so on. But the root mechanisms of coupling remain. What happens if the dynamics of the solid needs to be approximated by more than one mode? Again we can extend all this easily and the root mechanisms remain. What you see here corresponds to large amplitude of course. And that's another story. But the underlying mechanics is the same. Let us conclude on all these effects of fluid-solid couplings at low reduced velocities We found that the variety of coupling that exists when the fluid is not moving by itself is already quite large. Added mass, added damping, added stiffness, memory effect, mode coupling. What is nice is that using the dimensionless numbers, you know when you can find them, and what will be the effect on the dynamics of the solid. A very good news also is that to obtain the coefficients in the equations, like the added mass, you only need to solve very simple equations in the fluid domain. Once for all, and not at every instant. And because these equations are so simple, the results exist already in plenty of tables and graphs for many geometries. Again, no need of advanced CFD to get these effects. Here are some of the examples I used when we explore all these effects. As you can easily imagine, reality in all these cases is more complex than the idealized problems we solved together. But the important thing is that you know what is behind all the effects that you may observe. And you know what to expect. Remember that we said that we could use dimensionless analysis to classify problems and eventually build models. This is exactly what we have been doing together. So far, we have explored the domain of very low reduced velocities. This means when the dynamics of the fluid is very slow compared to the dynamics of the solid. But there are plenty of things happening at high reduced velocity when, conversely, the dynamics of the fluid is much faster. For instance, when wings start to vibrate in planes, or when bridges oscillate under wind. This is exactly what you shall explore next: coupling with a fast flow. You will learn how to predict things like wing flutter in planes. [MUSIC]