 it's okay yeah so thank you yes indeed I would like to speak about the Faraday instability and mixing zone driven by it and this is a work done in collaboration with Ali Eboadou at Econormal Super de Cachan and I would like to thanks the organizer of this conference and also my colleague from CEA Gérôme and Olivier where a great help about this for this so I think there is a problem with the screen sorry about that but it's not the good yeah there is the bottom missing with this yeah you see so when you put the presentation okay this one is thanks sorry about that okay so let me first introduce you a little bit about the Faraday instability which has been discovered for a long time at the end of 19th century so it consists in applying an acceleration vertical to the interface which is oscillating so here is the expression for the acceleration you have a G-naught which is the mean acceleration and you have the oscillating part which depend on F which is the forcing parameter on omega the forcing frequency so by convention if F equal to 0 you have a stable fluid which means that the acceleration point to the bottom you have a life field on top of EV fluid so it's stable so you don't have instability if F equal to 0 so it can be encountered in many many application it has been studied a lot you can do yourself an experiment by spinning on glass water glass you can you will excite a faraday instability in the water it is well not because it produces a lot of fascinating structure and it has been studied to to look at phase changes in the shape here and I like this picture because it's a way to communicate between alligator so by vibrating their lungs they can produce a faraday waves here at stop of a river so this is a classical problem which is studied the linear analysis use flocaded composition this has been done the first time by Benjamin Russell in 54 and then much later by Kumar and to Kerman in 94 taking into account a viscous effect so you see it's quite recent as a theoretical development for this instability one of the main characteristic is that it is subharmonic which means that the fluctuation generated by the instability oscillate without the frequency of the forcing so this problem the faraday problem has been studied mainly in the context of laminar flows but of course if you easily imagine if you consider miscible fluids if you apply a very strong forcing parameter or also if you have initial very well dissolved initial condition then the standing waves appearing at the surface of the fluid would interact each other non-linearly and create some turbulence so this has been observed for instance in experiment by Westiag et al in GFM 2009 so you see in this nice experiment this is a miscible faraday and you see here the interface and you see the development of turbulence and then you have you can observe the growth of the mixing layer so it has been studied also by the workout but this author look mainly at the onset of the instability and they observe however that as the mixing zone grows then resonance condition triggering the instability are no longer fulfilled and then you have a saturation of the mixing zone so here this is a numerical simulation we perform you have the time evolution of the mixing zone size and here you see some oscillation triggered by the faraday instability if you are lucky we can look at the movie yeah okay so say you we have to wait a little bit but if you pay attention then you see the oscillation due to faraday you have oscillation of the mixing zone it's a subharmonic instability here and then you have the enlargement of the instability and the saturation so the question which is very simple I would like to address in this talk is can we predict here the final size of this mixing zone okay so the first step to do that is well as always to perform dimensional analysis and well we can at first glance say that initial condition and on dissipative phenomenon do not play an important role in deriving the final size of the mixing zone so the final size should depend on the forcing parameter on frequency and also the mean acceleration which is modulated by a which is the atwood number the contrast of density between fluid so we obtain by no dimensional analysis this simple expression which depend on unknown function j as a function of the forcing parameter so the objective the outline of this talk is three parts first I will propose a theoretical framework to study the sensibility then it leads to a system in order to understand how gravity waves interact with the main density profile and we will derive the saturation criterion from this analysis and then we'll verify it against numerical simulation so first the theoretical framework so you have a mixing zone and well we can study it first very simply in the businesque limit you have a question for the concentration of heavy fluids this is for binary mixture and the velocity and the velocity field is incompressible so it is also convenient to introduce a second a second framework by extracting from this equation with Reynolds decomposition the tubulant fluctuation and writing the equation for the tubulant fluctuation and assuming that the mean density profile as an uniform gradient then this is what we call stratified homogenous tubulant which is how tubulant fluctuation oscillate on response to the mean field at the center of a mixing zone so this framework is interesting because you decoupled the mean density profile to tubulant fluctuation and that's why it's easier much easier to do for instance numerical simulation because here you are limited by the size of the domain but not with this setup because the size of the mixing zone can evolve infinitely okay so I will use both I will show you both simulation with mixing zone I call it mz on the stratified homogenous tubulant some will show I will show you that it gives same result at the end so we we use also the rapid acceleration model which has been derived the first for relatively tubulance and it is a very simple model which represents the linear evolution of vertical velocity and the buoyancy coefficient which interact linearly each other and depend which depend also on theta which is the angle between wave vector with the vertical so you recognize here the equation for for a simple stratified stratified solution and what is what is original here is there is the mixing zone size here which interact in this equation so it runs the system nonlinear but in short this is a rapid distortion theory this equation which has been first introduced by Anna Zacchaehunt in 96 and the equation for L which is the equation for an approximation of the mean gradient so we think it's interesting to study this system because it's nonlinear and it will give us some clue how how the mixing zone evolve and how it will saturate at the end so if I work a little bit on this equation I can put it on this form which is a second order differential equation for the buoyancy parameter which which is nonlinear because you have the size of the mixing zone here but also you have an infinite set of this kind of the equation which are a material equation and I will consider this system as a description very rough indeed of how gravity waves interact with the main dead city profile so here is my system which is nonlinear and well the first the first step is to understand how the stability the linear analysis of the system so if we do that then the size of mixing zone does not intervene in this system and I obtain an infinite set of matter equation depending on the angle theta 0p and this is what I've been studied by Benelie and Sumeria in 98 and this is the stability of a stratified flow excited by parametric instability so the stability of this set of equation can be represented on the the material diagram which represents the forcing parameter as a function of the ratio between the Brunweiser frequency and the forcing frequency square so here you have a different tongue in stability tongue corresponding to subharmonic and harmonic instability and in this problem it is interesting to see what will you trigger in this representation for a fixed mixing zone size and because of the sinus theta here it you see that you excite a wall a wall bound of frequency in this system not a single one but a wall bound this is the main difference with a single interface here and well of course you say for instance in this example even if the most amplified frequency are stable here you have a bound here which in sense unstable so you would you would say that this system is unstable so the mixing zone will will grow on a large and this will reduce L and you will go like this so what you would expect from linear analysis is that the final size of the mixing zone correspond to the first transition curve in the material stability diagram okay but this is a linear analysis and well the system is non-linear does it work that's the question so we try to pursue the analysis of the rapid acceleration model by performing a multiple scale analysis and to see if taking into account non-linear effect it indeed saturate to the first the first transition curve so here are my system on what I assume is an expression like this depending on two time okay with a small parameter on here is what is classical for instance in Godrej Malville I assume a weekly non-linear system by supposing that the perturbation is close to the first subharmonic perturbation so using steepest descent method I can obtain a solution for that okay so this is a classical analysis using multiple scale analysis so I have a question taking the correct count the amplitude on to find the equation for the amplitude I look at resonance condition at next order and what I obtain at the end is this relation which correspond indeed at taking into account weekly non-linear effect it indeed saturate to the first transition curve in this method so I will skip all the detail and go directly to the verification of the saturation criterion which is then we I I say that the mixing zone will saturate to the first transition curve so we worked out by doing lot of numerical simulation on here this is an example with stratified homogeneous turbulence and well we perform at least more than 250 numerical simulation in DNS to explore the phase space of initial condition and so this diagram you see the stability tongue the instability tongue here as a small the small symbols correspond to initial condition on the phenol source of the mixing zone correspond to the big symbols and you see that mainly you saturate to the theoretical prediction which is the first transition curve of the mixing zone ok so it works but I will I would like to show you a movie of the evolution of this instability on which explain the main mechanism this one so if you pay attention you see first the structure during the further instability are elongated along the vertical direction and then you have a transition because then the structure bent and you look in the direction and then it's recovered the main direction vertical direction at the end so what you have in this example is a transgender transition harmonic subharmonic because in in the material diagram you have two tongues harmonic and subharmonic which can be triggered at the same time depending on the size initial size of mixing zone so at the beginning in this simulation here you have both harmonic and subharmonic instability and then at the end you have only the subharmonic instability so also pay attention to the size of the structure with bent because the mixing zone evolve and then the main the most amplified angles are not correspond does not correspond to the vertical at the beginning but to a bent solution ok I think I have to stop here ok I have one minute so I say at first that it does not depend on initial condition this is mainly true but here we perform a different experiment to take into account the effect of initial fruit number which is the effect of if I put at initial condition a stronger strong turbulence but so you see that it's mainly true that it does not depend on initial condition but if initially the turbulence is strong then the saturation is beyond the line here the transition curve because at first I put in turbulence as a stronger a strong impact and then it will go beyond beyond the theoretical prediction escape to the conclusion so we perform weekly non-linear analysis describing interaction between gravity weights on the mean density profile we derived the saturation criterion for mixing zone driven by faraday instability and we verified the prediction of saturation over 300 mixing zone simulation and we we found a nice harmonics of harmonic transition to which is which is coherent with the theory derived in this in this problem thank you for your attention yeah it's a j yeah I call it at first for the faraday instability is for interfaces then they call it also faraday instability with miscible fleets so you have a mixing zone and you cannot you cannot identify the very neat waves in this problem so after you have a turbulence mixing zone on your study it's characteristic and it's what is proposed here yeah okay