 My name is Abeer Bax and PhD student at the University of Science and Technology. My advisor is Rabi Sametani and we work in fluid and physical simulation laboratory in KAUS. So my talk will be presenting the linear stability analysis of magneto hydrodynamic Rikman-Myshkov instability in cylindrical geometry. The outline are given by introduction, linear simulation of compressible MHT RMI, then we will present the linear analysis of incompressible MHT, then we will just give the main point of the numerical inverse Laplace transform why we are using this inverse method, then we will show the results. So as it will known, the Rikman-Myshkov instability occurs when a shockwave impossibly accelerates operative interface between two different fluids, which have many important subjects such as understanding the light curve and supernova explosions, mixing in scramjet engine combustors and the initial confinement fusions. So the RMI promotes the mixing between these capsule material and fuel and limit the possibility of achieving the energy which is produced in the ICL. So the previous work in numerical simulation showed that in planar geometry the presence of the magnetic field will inhibit the instability and the suppression of RMI caused by alven fronts or MHT shocks that carry the vorticity away from the interface. In cylindrical geometry there is an extensive study in the, sorry, confusing, okay, an extensive study in non-linear RMI, cylindrical and spherical geometry for various initial seed of magnetic field configuration also, they also showed that the suppression of RMI and RT instabilities will occurs in the presence of the magnetic field. This movie shows the MHT shocks that splits across the interface and they move the instability, remove the vorticity from the interface. For the incompressible model there is a study was done by Wheatley et al. in 2014 and 2005 will be presented later. So this showed the study of linear incompressible model and incompressible simulations of MHT RMI in the presence of transverse magnetic field showed that the amplitude of the interface will oscillate in time causing the suppression of the instability. In curved geometry for hydrodynamic case incompressible theory and compressible simulations also for RMI show that the growth rate of the interface exists because of the initial impulsive deposition of the vorticity and due to the radial motion. So our objective is to show that in cylindrical geometry we will investigate the suppression of these instabilities in the, due to linear simulation for compressible MHT and we will propose incompressible model to compare the numerical linear results. So the physical setup for the compressible MHT we have a cylindrical geometry with as a muscle on axial perturbation on the density interface which is initialized at the distance R naught and we have a magnetic field initially exists either in normal or as a multidirection with the, with the string beta and the shock wave of Mach number m interacts with the, with the contact discontinuity. So we will present here the governing equations of compressible MHT in conservative form where W, F, G and H are the compressible W is the solution vector F, G and H are the fluxes containing the mass momentum magnetic field and the energy and S is the source term arise in the cylindrical coordinates. We close the system by writing the equation of the state and we have the solenoidal, solenoidal constraint on the magnetic field. So in this system has some numerical errors close to the origin because of the, because of the magnetic field depends on the radial, radial on the space. Have some dependence on R. So we rewrite again the magnetic field as a sum of B star plus B1 and we substitute this in radial momentum equation to cancel these, the terms that proportionate to 1 over RQ which cause the numerical errors as the interface go close to the, to the origin. So we rewrite again the system regarding to W tilde. Linearizing our system by writing the solution as a sum of the base state time divinting base state plus the perturbed state and we substitute in the previous equations. We split the, we split them in the two, two, two equations, base state equations and perturbed state equation which are hyperbolic PDEs. And these matrices are the singular Jacobian matrices of the fluxes. Then we are focusing in our study on the growth rate of history of the perturbed interface which is normalized by using the asymptotic growth rate. This asymptotic growth rate was proposed by Lamborghini in the cylindrical geometry. We are showing the previous result we, this, this work was already published for in the, in the normal, in the, in the presence of normal magnetic field. The first one showing the purely azimuthal perturbation. The second one is in the case of purely axial perturbation for large wave numbers and different values of beta. The first one showing the hydrodynamic case where the RMI show the similar results as the planar case. It's oscillate around one, around unity then because of the converging geometry we have the Rayleigh-Teylor phase which control the instability. Similar in the second case then by applying, applying the magnetic field we can see the separation of these instabilities. So the var, the var, we are showing here the varoclinic vorticity, the vorticity that exists on that interface. For the hydrodynamic case we can see the vortex sheet exactly on the position of the contact discontinuity. While in the presence of the magnetic field we have two vortex sheets split away from the contact discontinuity. This, we can, physically we, we say that two alven fronts remove these vorticity and transport them away from the interface which cause the suppression of the instabilities. The second case was presented for azimuthal magnetic field. The same thing we studied hydrodynamic and the presence of the magnetic field for different at-width number, the ratio of the interface, the densities of the interface. So for the positive at-width number and negative at-width number. The first one shows the same results. Here I should mention that there is a phase change in the Rayleigh-Tiler instability while for the second case there is a phase change in the RMI. Again, applying the magnetic field will suppress the instabilities. Here we can see that the amplitude of the growth rate decreases with, with decreasing the, the strength of the magnetic, the strength of the magnetic field. As it's go up, so the oscillation will occurs very quickly and the, the frequency of the, the, these oscillations also will increase. Here we are showing the two, two, the vorticity 2D plot at the interface which is corresponding the arrows position on the growth rate plot here. So on the, on either side of the, on either side of the interface here is the position of the interface. We can see the vorticity are transported away by, by waves traveling parallel and antiparallel to the, to the interface. Okay. So they, they should alternate in signs because of the, because of the constructive and destructive, sorry, interference of the vorticity. So they will internail, they will change in signs. Sorry, it's not working this time. Okay. So, so this changing on the, in the, in the signs will make the growth rate of the interface will oscillate in time. Let's see. Next I will explain what we have done in the linear analysis of incompressible MHD. The previous work was presented in planar geometry by weekly at all where they proposed an incompressible model to examine the impulsively accelerated interface. They have a normal magnetic field. This interface was, has initial impulse which is, which moves in constant velocity. They linearized the initially the, the incompressible equations and solve them using Laplace transform. And they obtain a singular oddies which was easy to solve analytically. The result that they said that the initial growth rate of the interface is unaffected in the presence of the magnetic field. Then two alvein fronts carry the vorticity and propagate away from the interface causing the separation of the instability. So, we started to, we are proposing the similar method by applying that in cylindrical geometry. So, we have incompressible model of two conducting fluids separated by impulsively accelerated interface. As we, as I mentioned before, the interface will move with a constant velocity in our model. We assume here the magnetic field exists in either normal direction or azimuthal direction. We linearize the incompressible ideal MHD equations about the base flow. Then we will use the Laplace transform to reduce these equations into one oddie, singular oddie. Then we, we are trying to solve this oddie numerically and apply the numerical inverse Laplace transform while we are using the numerical algorithm here because our oddie is very complicated than the planar geometry. So, this is our oddie in the presence of normal magnetic field. It's a fourth order oddie containing the exact solution of Cartesian geometry which was presented by Wheatley including our correction in cylindrical geometry. So, we solved this equation numerically. The boundary conditions states that as the perturbations, as the distance from the interface goes to the infinity, there are no incoming or outgoing incoming perturbations, sorry, incoming waves from that side. At the interface, the velocity must be continuous and the magnetic field also must be continuous. So, we have these continuity equation across the interface. The second case when we have azimuthal magnetic field, we have a second order oddie. Again, we have the first part represent the planar case. Then we have the correction of the cylindrical case. Solving this equation, we will use the central difference approximation for spatial discretization. We are using LU decomposition. After solving the system, we are going to apply the numerical inverse Laplace using Gaver-Stefes method which approximates the perturbed velocity by sequence of functions given by this formula. The main thing I should mention that the integration of the promised contour, the result of that integration will be in the real part of the complex plane. We are not dealing with the singularities of that contour using the numerical inverse Laplace transform. So, the verification of the results in the planar case we started to solve the original equation oddie and these initial conditions represent the case of Mach number equal 1.2. So, the results in our simulations are given in the left tab plus where the original plus represents the analytical solution that presented in their work, in Wheatley's work. So, as you can see, it's almost in the same position. This represents the velocity on the interface. This is the initial impulse and after some time, we can see that velocity is split by two elven fronts and they are moving by local elven speeds. Another thing that we cannot capture the exact peak of these velocities because of the dispersion errors in the numerical. So, our solution for our oddie is in cylindrical geometry. This is our initial condition. This is also the results of the velocity at the interface. As you can see, the initial velocity is unaffected by the magnetic field and we have two elven fronts moving in each fluid, in each side of the interface. The most important thing that we have some difference, the difference between these results and the previous result. In the planar case, at the interface, the velocity goes to zero and they stay at zero. While in our result, the velocity here goes to values less than zero, which correspond to the oscillation around zero. We compare the results in compressible model and our linear simulations. We can see that there is a lot of agreement in the early time of the growth rate. The second case is presented for the azimuthal magnetic field. As you can see, also, the incompressible model matches the growth rate in the early time as the simulations, as the American simulations of compressible MHD. To conclude our work, we show that in linear simulation of compressible MHD, the presence of azimuthal or normal magnetic field shows the separation of RMI and RTI. The solution of the proposed incompressible model also is obtained and inverse Laplace transform is used. The results show a good agreement with the linear simulations results. Thank you for your attention. Thank you.