 Hello. Hello everybody. Thank you for coming. I'm Francisco Cobos and I work with Gustavo Gochuk, who is my PSD advisor. We are from the University of Castilla, Mancha, Spain and we study hydrodynamic flows behind ripple shocks and rarefaxion waves. Here we present some linear results of the tiny evolution of the two fluid rich major mesc of life flows and we compare them with experiments and simulations. Well, first of all we with some physics valid either for shock-reflective case or rarefaxion-reflective case. The typical picture is that rich major mesc of instability develops when a planar shock waves collides with a corrugated contact surface, separating two different fluids. Its fluid is characterized by its density and its equation of state. We assume ideal gases so we give the ratio of a specific heat gamma a and gamma b. The strength of the compression is determined by the incident shock mass number. So, when the incident shock has completely disappeared, a transmitted and reflected fronts are formed. A shock is always transmitted and a shock or a rarefaxion wave is reflected back in the first fluid. It will depend on the four zero-order pre-shock parameters that are both gamas, the initial density ratio across the interface and the incident shock mass number. We can go continuously for a situation where a shock is reflected to a situation where a rarefaxion is reflected, only changing the initial density ratio for given values of gamma and ni. So, for some specific value of r0 equal to r0 dt, this one, we have that no front is reflected back. This situation is total transmission. If the initial density ratio is greater than this value r0 dt, a shock is always reflected. On the contrary, if r0 is lower than this value, a rarefaxion fan is reflected. This value of r0 is equal to one or equivalent at good number equal to zero when gamas are equal. But if gamas are different, it can be greater or lower than unity. So, since the contact surface is corrugated, the transmitted and the reflected fronts are also corrugated. The shock ripple, the shock corrugation induced by the conservation of the tangential momentum and initial velocity shear deposited at the interface, at time equal to zero plus. Beside, the shock corrugation induced or generate perturbation in the pressure, in the density, in the velocity, in the entropy and in the vorticity. So, for these two effects, the initial circulation and the perturbation, the contact surface ripple start to grow in time. So, we have an instability. We have two kinds of perturbation. Evanescent zone wave and vorticity and entropy. Perturbation which are frozen to the fluid element for considering inviscid flow. So, in our calculation we take some assumption. The first one is that the contact surface ripple is much smaller than the wavelength. This makes that the perturbation values are much smaller than the background quantities. So, this is linear theory. We also assume ideas gal, but we can use another equation of states as well. As we say vorticity and heat conduction are neglected, the upstream flows are homogeneous and we consider a single mode ripple. So, mathematically we need to solve the linear wave equation in the space between the interface and the wave fronts in both fluids. If we make this variable chain, the solution can be written as a series of vessel functions. Beside, it is convenient to work with this auxiliary function H, which sees a combination of the partial derivatives of the pressure perturbation. In order to solve this problem analytically, we take the Laplace transfer in the variable R. Then, we need boundary condition. For a shock reflected case, we have that the boundary condition at the shock are the linear ranking Eugoniot equation, which relates the function H and P at the shock surface. Of course, we need at the interface the continuity of pressure and normal velocity and the initial condition is that the pressure perturbation at T equals 0 plus R0. So, if we match both fluids across the interface, we arrive to a capital system of functional equation for the Laplace transfer of the pressure perturbation. We arrive at that system. This is for shock reflected case. When a refaction is reflected, we need to change the boundary condition at the refaction tail. So, this functional equation system will be not the same, but is functional too. So, if we iterate it, we can obtain successive approximation, like P0, P1, P2, etc. We extend to the shock solution by increasing the iteration order. So, we can choose the accuracy of the solution only selecting the number of iterations we do. One time we have the PS, we take the inverse Laplace transfer to have information of the shock pressure perturbation as a function of space and time. And how we'll see later knowing them, the pressure perturbation at the shock, we can describe all the physical quantities downstream. So, the shock correlation not only gives pressure perturbation, but also velocity and vorticity perturbation. The linearized wave of equation can combine into wave equation for each velocity component, where the non-mogenius term is essentially the back vorticity and its derivative. The vorticity is generated by shocks, by the conservation of the tangential momentum. So, as long as we assume inviscid fluids, the vorticity at some point is proportional to the shock pressure perturbation at the instant of time the shock arrives at that point. So, if we know the pressure perturbation through the functional equation system, and hence the vorticity, we can describe the velocity fields. Now we move to the syntoptic regime when t quantine tend to infinity. So, when the shocks separate from the contact surface, the ripple decreases as than the pressure field. So, when the shocks are far away, they regain planar shape and no more pressure perturbation exist in the compressed fluid. However, a velocity rotational field remains in the fluid. So, for time tend to infinity, the velocity components satisfy this ordinary differential equation in which the inhomogeneity is the back vorticity and its derivative again. To solve this system, we have boundary condition. The first one is that the normal vorticity must be continuing at the interface. And the second one is a relationship between normal and tangential velocity at the interface. So, this last condition is all we need for any rich major mass cost life flow in order to have bounded the perturbation at infinity. So, the different of the velocity are function are equal to this function we call f. This f is an wasted average of the vorticity generated inside the fluid. So, we plot here the typical behavior of the quantity f as a function of the shock strength. We see that for weak shock, it scales at the shock strength to the third power. So, for very weak shock usually when incident mass is less than 1.5. We can neglect this quantity and hinge the vorticity. In that case, the three velocities at the interface have the same value. And the velocity profile will be well-approximate by exponential as in the impassive mode. Now, we show some result for the shock reflected case. So, the first result is the syntotic velocity spatial profile which are the solution of that equation. Here on top we plot the velocity component, the normal and the tangential for the particular choice of the pre-shock parameters. And we see that in the reflected fluid and exponential description is enough because vorticity is... vorticity is not important here. But on the contrary in the transmitted side we see that the exponential is valid only very near the contact surface because the velocity field is soon dominated by the oscillation of the vorticity. Also we realize that the normal and the tangential velocity of the transmitted side are quite different. So, in fact here below we plot the extreme light of the velocity. We clearly see that there are alternating vortices in both fluids. We see that inside the transmitted fluid there are many vortices in comparison to the reflected. It's because the vorticity, the length of the vortices are strongly dependent of the shock compression level. If we increase the shock intensity these vortices become smaller and more intense. So the vortice length has arranged between infinity for which shock limit to a minimum less than the wavelength for strong shocks. In additionally we have calculated the weak shock limit of the length of the third vortex and need a scale of inverse power of the square root of the shock intensity multiplied by a factor which is different in each fluid. So, we need to know the normal asymptotic velocity at the interface also known as growth rate. This is the sum of two terms. The first term is that it will be considered an impulsive term because it only takes into account the initial velocity shear deposited by the shocks at t equal to zero plus. And neglect it after compressibility effects. What's more in this work, in this work they realize that this term is exactly the result of a true gravity impulsive model. And the second term takes into account the vorticity quantified by the quantities f a and f b we showed before. These quantities are proportional to the vorticity spread and it depends on the whole compression history of the problem. So, this term is a good approximation for weak shock and this term needs to be added when the compression becomes important. In order to study when we need to consider this term we plot the relative error if we only consider the first term instead of the complete formula and see that for weak shock or low compressibility situation is a good approximation because the error is less than 20%. But when compression becomes important this error increases a lot. The same deduction we can arrive if we plot the ratio between the velocities, the velocity at the interface. For weak shock the normal tangencia velocity at very similar but when we increase the shock strength the ratio becomes greater than unity. We propose an easy to use formula derived from the first iteration of the functional equation that does a very good job also in high compression situation. Also we extract a analytical formula in every physical limit for example in weak shock we can expand the growth rate in terms in a series of mi-1 to get different approximation. We realize that this first term is exactly the risk measuring passive formula. But this expansion has a very small convergence radius less than 1.5. This makes that not very useful, this kind of weak shock limit. In the strong shock we see that the growth rate scales as constant plus and it scales as the inverse power of the square of the shock intensity. Once we have the velocity field we can describe the ripple of the contact surface. In linear theory we can distinguish two periods. The first period is a transient compressive state where oscillations are noticed induced by the acoustic sound wave. And a second linear state where this formula is valid where the asymptotic ordinate is different from the initial box shock amplitude where the evolution start plus the growth rate multiplied by T. So both parameter this amplitude this asymptotic amplitude at the growth rate we can calculate analytically. This asymptotic ordinate is the shaded area here. The difference between the set history of the normal velocity and the difference between this asymptotic value. So we can analyze this asymptotic well I have not I sorry. We show some comparison with experiment. This is the impact model this is our set linear solution the blue line and the asymptotic linear. And we clearly see this the period of growth the initial transient the linear situation at the non-linear state. We compare with more experiment. Well sorry only one highlight in the in the rarefaction reflected case the main difference is that the rarefaction will not generate vorticity. But we need to have to take into account the vorticity in the transmitted fluid. So we have the same result and for the shock reflected case. And we also shown this comparison with experiment. We also see the different state the different period of growth. Well the initial transient the linear saturation and then the non-linear growth. So well I have no time sorry about well it's a little crazy. That's all if we have interest there are more limits in these references and more detail can be found in it. Well thank you very much yeah yeah we can use other question of state yeah. Yeah we don't try it but the question of state will change the curvature of the unit curvature. So well yes the result were different yeah yeah.