[MUSIC] Hello, using our dimensionless parameters, particularly the reduced velocity, we have seen that we could build simple models for the interactions between a fluid and a solid. These models were applicable in different ranges of these parameters. In practice, we now have models for interaction with fast flows (very high reduced velocities) . They are efficient to predict coupled model flutter of wings, for instance. We have models for slower flows (intermediate reduce velocities) that are useful for the prediction of galloping of power lines for instance. And we also have models for very slow flows when the fluid velocity is negligible. They are the one used to predict the stability of ships, or the added mass of cables under water. This is nice and useful, but it is a bit uncomfortable to have patches of models. Can we connect them? What happens when a given system sees a progressive increase of the fluid velocity from zero to very large? In other terms how do we jump from one model to another? [MUSIC] We cannot address this in the general case. This is why we have to make models in parts. But there are some simple systems where we can do this because the geometry is simple. We are going to do that on the toy problem, a very classical problem of fluid solid interactions, the garden hose instability. Certainly you have experienced this, a flexible pipe that flutters when you put some flow velocity. Why is it so famous? First, because people soon recognized that it was experimentally rather simple to realize. Second, because a pipe is a simple mechanical system, a kind of beam. And the fluid inside it just follows along the direction of the pipe. So certainly the equations are going to be much simpler. Just to give you a taste of how this problem has fascinated people, here are some pictures taken in the 30s in France by Bourrieres of experiments on the fluid conveying pipe fluttering with air flow or with water flow. And Bourrieres recognized that the shape of flutter was different. And here are experiments of Paidoussis, in the 60s, in England. He then built a full model of the problem. Today, there is a huge body of literature on this phenomenon. For instance, an authoritative book here written by Paidoussis after 50 years of research in this domain. As I've said this problem is interesting to understand the fundamentals and that is why many people worked on it over years. But this research driven by curiosity also found applications in the field of offshore engineering for aspirating pipes, in cooling systems or in dredgings or nanowire dynamics or mining. If you think of it, there are many related systems, sky dancers, thin jets, coiling systems, and so on. Let us go back to the garden hose, or fluid conveying pipe, instability. Here's what happens. Consider a straight pipe with flow inside. Depending on the flow velocity, here's what schematically happens. At zero flow velocity, the perturbation, just pulling and releasing, results in a damped motion of the pipe, more or less damped depending on the pipe material. This is shown here schematically. If the flow velocity is now increased, the same perturbation results in an oscillatory motion of similar form, but more damped. Then, as the flow velocity is further increased, this trend is reversed and eventually, a situation of zero damping arises. Above this critical flow velocity, the pipe starts to oscillate by itself. A true dynamic instability in the sense we defined it before. This is the famous garden hose instability. To summarize, as the velocity is increased in the pipe, we have an increase of damping, then a decrease and eventually the instability. Actually above the critical velocity, the amplitude grows until a regular self sustained motion sets on. This motion, seen here with a fast camera is a bit different from the simple pendulum motion. Further in velocity, other surprising things happened, but this is another story. [MUSIC] How can we relate all these to what we have done before? First, dimensionless parameters, here the mass number is simple. It will be the ratio of fluid mass over solid mass, by unit length of the pipe. Clearly, it will be different for a gas or liquid conveying pipe. Second, the reduced velocity. Quite naturally it may be defined in terms of ratio of times. The time characteristics of the solid dynamics is, for instance, the period of oscillation of the first mode without fluid. T solid equals one over f. The time characterizing the fluid dynamics is for instance, the time a fluid particle takes to go from the entrance to the exit of the pipe. T fluid equals L over U, where U is a mean flow velocity in the pipe section. So UR equals U over fL where f is the frequency of oscillation. In the experiment I was discussing before, increasing the flow velocity means increasing the reduced velocity from zero to critical. Does the behaviour of the pipe resembles what we have seen before? Yes, we have flow induced added damping, and a flow induced dynamic instability. But at this stage, we don't know what precisely the causes are. Moreover, we imagine that as reduced velocity is varied, we somehow switch from one mechanism to another. For instance, we have an increase in damping first, then a decrease. A key aspect of the models we developed so far was the level of approximation of the dynamics of the solid. We used single mode approximations and in some cases, two modes approximations. For instance, for the dynamics of an airfoil, we had pure plunge or pure torsion or simultaneous plunge and torsion. What would be the equivalent for the pipe? A single mode of approximation of the bending of a pipe would be using just the first bending mode like here. Or the second bending mode, or a two modes approximation would combine first mode and second mode, like here. We have at hand all the ingredients to predict how the pipe is going to behave, damped or not and stable or not. Let us now explore the dynamics of these fluid conveying pipes with our models. [MUSIC]