 z vsi in poslednjih vseh. Pravih je to je z vseh, ki ga vzelo z vseh. Tudi, ne bo izlečiti zelo. Tudi, da se pobila. Tudi, da se pobila. To je pa nisi. Tudi, da se pobila. To je nekako vzelo. To je vzelo. Tudi je vzelo. Tudi, da se pobila. Na vseh, kako se pobila, da se predstavimo, nekako vzelo. 15-20 minut. Hydrodynamic instability is controlled variety of processes and fluids, plasmas and materials, including formation of hotspots and inertial confinement, fusion, compression of applause in Z pinches, core collapse, supernova, propagation of nuclear flame front and thermonuclear flames, supersonic flows, premixed combustion, as well as detonation to deflagration transition. Material dynamics is often characterized by strong acceleration, sharply and changing, sharp changes of scalar and vector flow fields, and by relatively small influence of dissipation and diffusion. This leads to formation of interfaces or fronts that separate flow heterogeneities or faces at macroscopic continuum scales. In my personal view, the formation of a phase and the transport of a multi-phase flows is maybe one of the most interesting and productive areas of research for the future. To capture the transport of mass, momentum and energy, a boundary value problem should be solved. And in this work, while still working in a far field approximation, and while looking at a field separating incompressible ideal fluids of different densities, we identified two quantitatively different behaviors of the front, stable and unstable, depending upon whether the energy flux produced by the perturbed front is large or small compared to the flux of kinetic energy across the planar front. The solution for the landauder instability is consistent with one of these cases. In principle, the front evolution can be studied in a continuous approximation, and this is what is commonly done in a diffusive approximation. In this case, initial value problem is being solved. Solution is relatively easy to find. However, to describe the dynamics reliably detailed information on the structure of the front is required, and this was the discussion presented today in the lecture of Professor Golov. This information is often a challenge to obtain and quantify in experiments and simulations. Therefore, the models necessarily use adjustable parameters, and the use of these adjustable parameters may influence predictive capability of the model and bound the parameter regime. On the other hand, the front evolution can be studied in a sharp boundary approximation. In this case, a boundary value problem is being solved. Solution is usually hard to find. Flow diagnostics, but the advantage of this approach is that the flow diagnostics is conducted away from the interface or from the front. Adjustable parameters are not employed. The model obeys the conservation laws, has high predictive capability and can be applied in a broader parameter regime when compared to diffusing models. We consider the evolution of a front that has a mass flow across the interface, and for us, the front is a boundary separating two fluid phases. So, we are not talking about a flame, even though this type of dynamics is usually studied in combustion, actually, consider it in the dynamics of the front and in premixed flame. There was a lot of work done on premixed combustion and especially on the stabilities and instabilities, because the first works that have been appeared, maybe dated by 1944, by Landau who considered the flame front as a hydrodynamic discontinuity and found that the front is unconditionally unstable, and the growth rate of this Landau instability is proportional to the velocity of the heavy fluid and inversely proportional to the wavelength. This result contradicted some experiments, because the stable front exists and absorbs in the laboratory. Several approaches were applied to resolve the paradox, including the seminal works of Markstein, Sivashinsky, Williams, Klawine, Peterson, many others. And indeed, there were extensive studies and seminal works, which are listed here, and it's only probably a very small list of the papers that we would like to refer. And, in fact, a number of stabilization mechanisms in Landau, the reinstability have been proposed, and one of them has been discussed today in Professor of Gold lecture. However, as a rule, the idea is such that the complex transports of the temperature and mass at the interface, as well as various diffusive mechanisms, may potentially stabilize the front, and if the front is thick, then it is stable. So, in our work we, again, consider the two incompressible ideals, so in a sense it's a model mathematical problem, but we justify it with the use of a far-field approximation. We are not staying in the vicinity of the interface. We don't look at it like that, we look at it like that. So, in this case, the conservation of mass momentum and energy governs the dynamics. It's a, in principle, three-dimensional motion with an unsteady end. There is a density, velocity, pressure, and energy fields that describe the dynamics energy. It involves internal energy plus V squared, where W is a specific enthalpy. Here we introduce a local scalar function that describes the interface, presuming that the first derivative of this function exists, and that the light fluid is located in one region, and the heavy fluid is located in the other region. In the bulk, the flow fields have this form. There is no, there is a mass flow across the interface, therefore the standard transkin-gubanian conditions should balance fluxes of mass to components of momentum and energy across the interface. So, the front is presumed to be planar and steady. In the moving frame of reference there are the boundary conditions at the outside boundaries of the domain, and the initial conditions include perturbation of the flow fields at the interface. As I mentioned, the fluids ideal and compressible have constant internal energies. We, of course, may linearize these dynamics, consider small perturbations around the uniform flow fields, and reduce our unkin-gubanian conditions to zero-order terms, which are presented here, as well as to the first-order terms, which include conditions for the velocities in the bulk, as well as the conditions at the interface, which are given here. Then the structure of the solution is actually based on the observation and was proposed by Landau, such that the velocity of the heavy fluid is potential, whereas velocity of the light fluid has a potential in the vertical component, and there is an interface perturbation. So, we essentially have four independent variables. So, potential of the heavy fluid, potential of the light fluid, as well as the interface perturbation. And there are four different equation, conservation of mass, normal and tangential components of momentum, and energy. So, we may introduce the wave vector, substitute all these expressions into our governing equations, count what would be the typical characteristic wavelengths of the vortex field, calculate what would be the pressure perturbations, and then all this is reduced to a certain linear four-by-four matrix, where r, where actually the r is represented, is built from the functions containing the potentials of the heavy light fluid interface perturbation and vertical component of the heavy field. Definitely it's a four-by-four matrix. We can identify its eigenvectors, eigenvalues. So, even though this is kind of straightforward approach, Landau used somewhat different approach. He considered the boundary conditions that balance mass to components of momentum at the interface, and he didn't consider a question for energy transfer. How are four variables still required for equations? And referring to the constant of the flame velocity, Landau employed additional condition at the front, which was called the continuity of the normal component of perturbed velocity, or this. So, even though it's written in a very simple mathematical format, really long as a long physical description. As soon as we employ this condition, it would immediately lead to continuity of perturbed pressure at the front, and it would identify the boundary conditions in the following way. So, as a result, if we would consider that r large is a ratio between the fluids, we are actually from the system of our governing equations, which is involved this and this equation. We may derive the characteristic eigenvalues, one of which corresponds to the Landau re-instability, which is here. There are a few interesting things here, in the previous talk, in the talk on Wednesday, that in fact, despite we have four variables, and four by four matrix, there are actually only two conditions, which are classically considered. In fact, actually there is a sort again value, which is not presented here for simplicity, because it has zero perturbation fields. However, still the number of eigenvalues appears to be smaller than the dimension of the system. And in fact, Landau system is indeed degenerate, because one of the conditions specifically this condition is a kinematic condition, rather than dynamic condition, on the side of physics and on the side of mathematics, it actually, this degeneracy can be seen from the form of the matrix. On the other hand, if we look at the Landau re-instability and actually look at the structure of the flow field, we will see that actually this solution has three important properties. First, it's only coupled potential and vertical components of the dynamics interface perturbation. That characteristic land scale of this vortex actually this vertical field is very large and in fact, actually it approaches infinity when density ratio approaches one. It also approaches infinity when density ratio approaches infinity. And it has a typical maximum size of the land scale at this specifically value of the density ratio. However, a remarkable property is that that therefore ideal incompressible fluid with the Landau solution is incompatible with the condition for energy balance across the front. And this can be found by means of simple calculations because if we substitute these equations into the equation for energy balance and by performing some calculations we will realize very quickly that the reason indeed our energy equation is reduced to the requirement of the zero perturbed enthalpy jump which is here, and given that our internal energy does not change it essentially would mean that our p divided by rho should have a zero jump at the interface, but because p is constant and rho is not constant, this condition actually appears to be incompatible with the momentum balance. Thus, Landau solution requires energy imbalance across the front and probably for the flame this is okay because flames are indeed known to produce the energy. And in fact, if we will be trying to see what kind of scaling can be applied to quantify this energy imbalance we see that there are two possible mechanisms because one of them is a kind of traditional scaling which is presumed that our enthalpy perturbations scale is actually the zero values of the enthalpy and in this case in fact, accounting for that our delta omega large would be proportional to delta t essentially would be proportional to delta in the density ratio it would be might be something in the order of r minus 1 on the other hand, we might apply the scaling which is associated with the pressure and in this case our jumping in enthalpy might be scaled like this, so it might be scaled with the kinetic energy. Dimensional parameters can be introduced and can differentiate between these cases. To see importance of the energy balance for the front stability, we may further analyze the conserved system which balances all transport including mass to components of momentum and energy and it's indeed can be found that is stable because there are the modes that include oscillatory modes that has these typical eigenvectors as well as there are the modes which describe the steady solution and while both for the system and for these solutions there is an unstable mode it has zero perturbation fields so in this case we have a completely different vertical dynamics and I will not be just balancing it, I just would like to point attention that in this case this vertical component is effectively decoupled from the interface perturbation. Now, if you would like to model essential effect of energy imbalance produced by the perturbed front within the frame of an ideal incompressible fluids we just need to introduce additional source term which is producing energy which is done with this term and then we essentially derive our equation governing equation for the eigenvalues which can be considered in two different limiting cases so in fact actually what is this it's a term which is coming from the balance dynamics what is this, it's a term which is coming from landauder instability so when f is small this term is essentially done and we have a balanced dynamics so this is actually oscillatory stable oscillations when f is large become very large approaching infinity then this term should be very small leading to generation of landauder instability and this type of questions can be essentially found and analyzed in certain limiting cases including large density, contrast, large energy perturbation, small density contrast, small density perturbation, etc. What is more important is that what I would like to point your attention to the effect of gravity on the balance dynamics rather than balance dynamics in this case we may actually turn on the gravity field repeat the same exercise analyze the stability again because it is just a linear problem is the same structure of the flow fields and what we actually will derive that the balance dynamics which is a potential dynamics it has a very interesting flow structure because first of all it has a zero component for the vertical field both here and here on the other hand it has coupled components of the potential motion to the light and heavy flow to the interface perturbation but what is interesting it has a completely different growth rate, completely different dispersion relations that is described in the classical book Landau and Liffschitz and in fact what we can see in this situation that I mean for this solution which is omega 3 marked here some vertical dynamics can be implemented with a special choice of the dynamic of the initial conditions however just for standard for general initial conditions that we may observe here there is the gravity induced dynamics which are solution omega 1 and omega 2 here are essentially potential dynamics and this potential dynamics becomes unstable only when the gravity field increases certain threshold which depends on the density ratio in this particular form in other words the larger the density ratio the bigger gravity is required for us to destabilize in a sense actually this type of approach might be applied further to model ablative Rayleigh Taylor instability I think probably the gravity induced dynamics can be first of all it's interesting that despite we consider the interface that has a mass flow across the interface and we would not probably expect it to be similar to classical Rayleigh Taylor instability but in fact it appears to be quite similar to classical Rayleigh Taylor instability in that sense that the flow is potential in both cases so we don't have any bulk varticity compared to Landau de Reinstability on the other hand it's quite qualitatively quantitatively different from the classical Landau de Reinstability at least in the values of the growth rate which is given here and the fluids with highly contrasting densities are mega as this particular form and typical gravity that has a threshold value is given by this expression the fluids with similar densities we have a somewhat different expression which also can be calculated so we see that when we balance in the dynamics of the interface that has a mass flow across it and it's in gravity field then this interfacial dynamics is unstable when the gravity value is large enough or for a fixed value of g when the perturbation wavelength is large enough and in this sense actually this offers us great opportunities for stabilization of the front on the other hand if we will still keep on going with an imbalanced dynamics such as Landau de Reinstability in a gravity field we may apply similar mechanisms as it was discussed before and consider again limiting cases when our energy imbalance is very small which is described by the letter f and in this case we have our limiting expressions that they have been just been discussing or consider another case when the energy perturbations produced by the perturbed front is very large and in this case we are transferring to the Landau de Reinstability and this is exactly that expression that is given in volume 6 of Landau Leveschitz course it is interesting though that this imbalanced solution which is classical Landau de Reinstability in a gravity field it has a somewhat different physics properties because first of all it is imbalanced for any gravity value it does not have any stabilization mechanism it does not have a cutoff or a threshold value of gravity in contrast to what the balance solution has and while the imbalanced solution in gravity field looks like Rayleigh Taylor instability it has a completely different flow field specifically for this solution the potential and vertical components of the motion are strongly coupled and there is a large scale vortex in the bulk of the light fluid behind the front we definitely may consider various limiting cases as I would do it here and I probably will proceed to the discussion and I would mention that we consider essentially a linear problem of the effect of the energy flux as well as the mass flux on the stability of an interface with and without gravity field and we identified some interesting properties that have been not discussed before and again in a far field approximation we found two different behavior of a hydrodynamic discontinuity which is stable and unstable we found that the flow stability depends on the energy flux produced by the perturbed flow and Landau de Reinstability is consistent with one of these cases