 All right. My name is Dan Ilyan. I will be talking today about the stability and structure of fields in a flow with a hydrodynamic discontinuity. The motivation for this project is that interfacial mixing and transport, they control a broad range of processes in fluids, plasmas, and materials. For example, they control supernova infusion, planetary convection, reactive flows, formation of phases in supercritical fluids, also material transformations, and material transport across the interface is often characterized by sharp and rapid changes of the flow fields and relatively small effects of dissipation and diffusion. All this may lead to a formation of discontinuities that separate the phases of the fluid at macroscopic scales. In this project, we looked at the far field approximation. We considered the far field approximation of the evolution of a hydrodynamic discontinuity that separates incompressible ideal fluids of different densities and has a mass flow across it. We solved the boundary value problem. We find the fundamental solutions of the linearized dynamics and we directly link the interface stability to structure of the flow fields. We identify a degenerate and singular character of the classic solution by Lundau and show that eliminating this degeneracy leads to a neutrally stable solution whose vertical field can seed the instability. In addition, we find that the interface stability is linked to the magnitude of energy fluctuations at the interface. Specifically, the interface is unstable if the energy fluctuations are large compared to the kinetic energy flux and it is stable if the energy fluctuations flux at the interface is small. And Lundau solution is consistent with the case where the flux is large. So the schematics of the flow look like this. We have light fluid on top with density rho L and velocity VL and a heavy fluid on the bottom with corresponding density and velocity. Then in the middle, we have a perturbation of characteristic length lambda and we consider the perturbation velocity, the velocities at the interface to be a sum of the bulk velocity and the perturbation velocities. So the definitions are fairly standard. The density is rho. The velocity is V. The pressure is P. The specific internal energy is given by E and its physics definition. In addition, we define energy and specific enthalpy and we introduce a continuously differentiable local function theta that is less than 0 in the light fluid and greater than 0 in the heavy fluid. This allows us to define unit normal and tangential vectors to the interface as shown there and to describe the mass flux across the interface. Now, the fluids we're dealing with are incompressible and ideal so the specific internal energy does not change. So it's derivative so what time and space are also 0. The flow that we are considering is two-dimensional as shown on this picture. It's the flow is along z and the perturbation is along x and it's periodic with wavelength lambda. The governing equations are the conservation of mass momentum and energy. Thanks to the function theta, we can describe the flow field in the entire domain by using the heavy side function as a sum of the flow, the fields in the heavy and in the light fluid. The governing equations are the conservation of mass momentum and energy and the boundary conditions for those equations at the interface are the balance of flux of mass, tangential and normal components of momentum and energy at the interface, also known as the Rankine-Huginio boundary conditions. The outside boundary conditions as shown on the slide with the schematic are just the bulk velocities in the heavy and light fluid. The characteristic length scale is given by 1 over k where k is the wave vector of the perturbation and the time scale is also given on the screen. So in the leading order, the flow is uniform. We can define the function theta to be just minus z and we will get these boundary conditions. Now if we look at a small perturbation, we will add a perturbation term to the velocity, the pressure, the enthalpy, and the flux. And these perturbations terms, of course, will be small. We can then define our actual perturbation as part of theta and then we obtain the following boundary conditions. So the structure of the solution that we're looking for as per experimental observations as per Lindau's theoretical framework is such that there is a potential velocity field in the heavy fluid and there is a potential end vertical velocity field in the light fluid. The fluid potentials have this form. They are oscillating along x as expected because of the perturbation and they are decaying exponentially along z. The interface perturbation also has an exponential form. It oscillates along x and the pressure perturbations can be derived as follows. Of course, the vertical field does not contribute to the pressure perturbations and the k tilde gives us a characteristic length scale for the vortex. It's useful to define dimensionless values, such as the dimensionless characteristic frequency and the density ratio. Our solution has the form given on the screen. It's a vector. And as shown on the previous slide, all of these values have an exponential term with time. So when we plug them into the boundary conditions, we are able to obtain this form. And because of how the time comes into play in this solution, we are able to simplify this expression down to a matrix because we're differentiating the with respect to time here. In a non-degenerate case, the matrix p has an inverse. And this will come into play in just a few slides. And so then we can express the pr dot equals sr as that. And so we then can find the fundamental solutions by finding the eigenvalues of that system. The final solution will be a linear combination of the fundamental solutions. And their general form is given here. So now, going to the actual system, if we balance the fluxes of mass, tangential, normal, components of momentum, and energy across the interface, we will find these boundary conditions. The matrix that I described on the previous slide has this form. And its eigenvalues are given here. We have two eigenvalues that correspond to a stable solution. One eigenvalue and the third eigenvalue corresponds to an unstable solution. The fourth eigenvalue is formally stable. So now, we will look at the velocity fields of this solution. In this case, we're looking at the first solution, which is stable. There is a phase difference between the velocity fields of the light and heavy fluids. You see the velocity fields are slightly out of phase with each other. And the pressure fields are in anti-phase. The region of high pressure in the light fluid corresponds to a region of low pressure in the heavy fluid and vice versa. If we look at the complex conjugate of this solution, we see a very similar picture. It is the velocity fields are slightly out of phase. The pressure fields are in anti-phase. And now, the third solution, which is omega equals, it's formally unstable, but it is not a trivial solution because it has a non-zero eigenvector. And the vertical and potential components of the light fluid are non-zero as well. It just so happens that together, they cancel each other out. And this leads to zero perturbation fields in both the light and heavy fluid. The stable solution described here has a vertical component that increases as we go away from the interface. And in order to satisfy the boundary conditions, we must set the constant of integration for the solution to zero. Now, going to the classic Landau system, instead of balancing the energy flux across the interface, the fourth boundary condition is that of the continuity of the normal component of perturbed velocity. This is equivalent to a zero perturbed mass flux across the interface. The matrix is given here. And this matrix only has three eigenvalues. This corresponds to the Landau-Durier instability described in this paper. And this solution is stable. And this solution is formally unstable. But as we will see, it is zero, just like in the case with the conservative system. So about the three eigenvalues, the system has four independent degrees of freedom, and it has four equations. However, there are only three fundamental solutions. So the case must be the generate. And indeed, if we look separately at the two matrices, S and P, where L is given by S minus omega P, we see that the matrix P does not have an inverse, because it has a row of its fourth row is all zeros. This is due to the boundary condition to the fourth boundary condition that says that the normal component of perturbed velocity across the interface is continuous. This same condition implies that the perturbed mass flux at the interface is constant and zero. In order to lift this degeneracy, we may modify this condition to a more general form and just set it equal to a constant. This will correspond to the partial derivative of mass flux with time to be equal to zero at the interface. And so, we call this the dynamic Lundau system, because the boundary condition is dynamic. So we have a time derivative in the fourth condition here now. And now the matrix is given by this. And as you can see, we have omegas on the fourth row, which implies that the matrix P does not have all zeros. Which implies that the matrix is invertible and the solution will not be degenerate. And indeed, it has four eigenvalues. The first three are the same as before. This corresponds to the unstable solution. It corresponds to the Lundau derivative instability. This is a stable solution, as we'll see in two slides. It will have to be equal to zero because of the same vortex at infinity. And this solution is formally unstable, but it has a zero field. And the last solution, it's closely related to the first solution in how the velocity fields are structured. So let's look at that a bit more closely. The large scale for the first solution, there is a large scale of vortex in the bulk of the fluid. It may be a bit difficult to see here, but there is a larger picture in a few slides when I will talk about the fourth solution. But still, one can see that there is a vortex here, as opposed to no large scale of vortex in the bulk in the heavy fluid. There is only potential motion in the heavy fluid. The pressure fields are in phase at the interface, unlike for the conservative system. So the region of high pressure corresponds to the region of high pressure in the heavy fluid corresponds to the light fluid as well. The second solution, which is formally stable, it has a vertical field that increases away from the interface. And in order to satisfy the boundary conditions, the integration constant for this solution must be set to zero. The third solution, which is identical to the third solution in the conservative system, also gives a zero for the same reason. Its eigenvector is not zero. But the omega equals r, the vertical and potential fields cancel each other out in the light fluid. And now the fourth solution, it is neutrally stable. And it has a large scale of vortex. Now if we look at both the first solution and the fourth solution, the first is on the left, the fourth is on the right, we see that while both of them have a large scale of vortex, the first solution has a finite length vortex. And the neutrally stable solution has an infinite vortex. A side of that difference, these solutions are fairly qualitatively similar. They both possess a vertical field that is in phase in the heavy and light fluids at the interface. They have a thin layer between the bulk and the interface motion, and the motion in the heavy fluid is potential. So the scale of the vortex in the solution that is the Landau-Derrier instability, that vortex has a, the scale of that vortex is related to the energy. It's related to the density ratio of the two fluids. But as I mentioned, in the fourth, in the neutrally stable solution, that vortex is infinite. This is interesting because the vortex in the infinite vortex may serve as a seed for the Landau-Derrier instability. Now I will briefly talk about the role of energy fluctuations in the conservative system. So recently it was found that the classic Landau solution is not compatible with energy conservation at the interface. And so the question we asked is what is the effect of energy fluctuations on interface stability? So the interface, at the interface fluids undergo a phase transition. They may transform into one another. And this process may be accompanied by energy fluctuations. So if we introduce an additional energy source at the interface, we can model this effect. So these boundary conditions are identical to the ones for the conservative system, except for this term here, which models the energy flux at the perturbed interface. The matrix is given here. The eigenvalues can be found from this equation and they are fairly complex. So I will not show them here. However, I will show this graph which demonstrates the behavior of the behavior of each of the four eigenvalues as the value of fluctuations goes from 100 to 100. And also I will remind that this chart is obtained for a low value of fluctuations, but it is identical to the one we obtained in the conservative solution. And the one on the right is obtained for a large value of fluctuations, but it is identical to the graph of the eigenvalue graph for the Lendau solution. So to summarize, we have considered in a far-field approximation the evolution of the hydrodynamic discontinuity. We've solved the boundary value problem and we've provided fundamental solutions for the linearized dynamics and showed a link between interface stability and the structure of flow fields. We identified the degenerate and singular character of the classic Lendau solution and showed that if we eliminate this degeneracy, we can obtain a neutrally stable solution whose infinite vortex may seed the Lendau durian stability. And we showed a link between the flux of energy fluctuations at the interface and interface stability. Thank you.