Lie group analysis and evolution of weak waves for certain hyperbolic system of partial differential equations

In the present thesis, we study the applications of Lie group theory to system of quasilinear hyperbolic partial differential equations (PDEs), which are governed by many physical phenomena and having various important physical significance in the real life. Our primary objective in this thesis is to identify the symmetries of system of PDEs in order to obtain certain classes of group invariant solutions. The investigations carried out in this thesis are confined to the applications of Lie group method to the system of quasilinear hyperbolic PDEs arising in magnetogasdynamics, two phase flows and other scientific fields. We organize the whole thesis into 7 chapters, described as follows. First chapter is introductory and deals with a short background history of Lie group of transformations and symmetries along with some of their important features which are of great importance in the work of proceeding chapters and the motivation behind our interest. In the second chapter, we obtain exact solutions to the quasilinear system of PDEs, describing the one dimensional unsteady simple flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. Lie group of point transformations are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities. The next chapter deals with system of PDEs, governing the one dimensional unsteady flow of inviscid and perfectly conducting compressible fluid in the presence of magnetic field. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system whose simple solutions provide nontrivial solutions of the original system. Using this exact solution, we discuss the evolutionary behavior of weak discontinuity

Meshless hydrodynamic simulations of young supernova remnants

The majority of massive stars end their lives by ejecting their outer envelopes in a corecollapse supernova explosion. The collision of their ejecta with the surrounding circumstellar medium results in the formation of supernova remnants that have been detected at all wavelengths, from radio to gamma-rays. For several dozen supernova remnants, very-long-baseline radio interferometers have spatially resolved the interaction region and directly measured the expansion rates of the shocked gas; many show evidence of the interaction of supernova ejecta with the dense slow winds characteristic of the red supergiant progenitors. Understanding the dynamics and morphology of the interaction region, particularly in young supernova remnants leads to estimates of the total mass of the circumstellar medium, as well as its density distribution around the star given the value of the wind velocity. Here we studied the interaction of the supernova ejecta with different circumstellar environments to investigate the hydrodynamic evolution of young supernova remnants in the SedovTaylor phase. We used the massively parallel, multi-physics magneto-hydrodynamics (MHD) and gravity code, GIZMO, for our simulations. We chose GIZMO for its flexibility in allowing the user to choose different methods to solve the fluid equations, i.e., new Lagrangian Godunovtype schemes, e.g., Meshless Finite Volume (MFV) and Meshless Finite Mass (MFM), as well as various flavors of smoothed particle hydrodynamics (SPH), or Eulerian fixed-grid schemes. Since the majority of previous studies used the latter, we focused on an extensive comparison of all the meshless methods in solving the Sedov-Taylor blastwave test, a problem for which there is an exact solution. For our given compute resources, we found the parameters (e.g., smoothing length, number of neighbours, artificial viscosity, and particle resolution) for each meshless method that gave the best agreement with the exact solution. We then carried out 2D and 3D simulations of the hydrodynamic interaction of the supernova ejecta with varying density profiles assumed for the circumstellar medium, namely: a 1/r 2 density profile, for a typical, spherically symmetric red supergiant stellar wind, and an axisymmetric torus profile, inspired by the observation of a dense, dusty torus of the circumstellar material around the red supergiant, WOH G64 (Ohnaka et al., 2008). Radially assembled Hierarchical Equal Area isoLatitude Pixelization (HEALPix) shells were used to set-up the initial density and velocity profiles for the ejecta, which is marked by a flat inner core and a steeply declining outer edge. The Weighted Voronoi Tessellation code was used to produce the 1/r 2 and axisymmetric torus density distributions. We showed that the growth of Richtmyer-Meshkov instabilities in the 2D and 3D 1/r 2 profiles are visible as early as 20 yrs into the evolution of the remnant and become increasingly unstable up to 100 yr. While 2D simulations of 1/r 2 profiles show the presence of the Richtmyer-Meshkov instabilities in the hot shell of a contact discontinuity, in 3D we see large bubbles and filamentary structure of the instabilities. Our results for the numerical approaches to simulating the systems for the 1/r 2 density cases were broadly consistent with previous studies in the literature where stationary grids were used. Two scenarios with different torus-cavity density contrasts were considered in which we found that the instability rolls along the half-opening angle takes ∼ 40 yr to develop in the axisymmetric torus with smooth density drop, whereas the axisymmetric torus with steep density drop does not develop any instability rolls up to the end of the simulation. We concluded with a discussion of the implications of our models for the morphology of supernova remnants and their expected levels of multi-wavelength emission

Theoretical and computational studies of extreme-size dust in plasmas

The effects of spherical particles (“dust grains”) in plasmas are investigated. The importance of dust grains in plasmas is highlighted, especially with regards to fusion energy production in magnetically confined plasmas. The investigation focuses on dust grains that are at the extremes of scale compared to the Debye length. Large dust grains, i.e. dust grains much larger than the Debye length, are investigated by the use of a simple fluid model, which is similar to compressible gas dynamics. Professor John Allen was the first to draw attention to the similarity of the model to compressible fluid dynamics in his 2007 paper [Allen, 2007]. The equations derived, which resemble those of compressible fluid dynamics are solved numerically with the help of a code written specifically for this purpose. The results are similar to PIC code results, with some differences in the shape of the downstream disturbance; more specifically, the downstream disturbance generated by our code is more elliptical than conical, and similar to the disturbance caused by a sphere in neutral fluids at moderate Reynolds numbers. This is to be contrasted with the results in the literature which are conical in shape, especially for low values of tau (Ti / Te). This may be an indication that the difference in shape is due to the ion pressure or the electron inertia, both of which we are neglecting in our assumptions. Small dust grains are investigated using a kinetic model. The model is a continuation and evolution of the model used by Filippov [Filippov et al, 2007], to include plasma flow. The equations of the model are solved analytically and the results reveal the presence of upstream structures, even in the case of supersonic flow, a result not commented on before in the relevant literature. The work also reviews relevant analytic theories, such as ABR and OML. ABR is extended by the author to include finding the geometrical width of the sheath. This extension, if confirmed, could be used for predicting the position of the sheath edge in relation to the dust grain. In addition, the work on deriving the Bohm criterion for a spherical dust grain is investigated, using a similar approach to the one taken in the literature for a planar wall. The result indicates that there is no such limitation in the spherical case.Open Acces

Group-kinetic theory and modeling of atmospheric turbulence

A group kinetic method is developed for analyzing eddy transport properties and relaxation to equilibrium. The purpose is to derive the spectral structure of turbulence in incompressible and compressible media. Of particular interest are: direct and inverse cascade, boundary layer turbulence, Rossby wave turbulence, two phase turbulence; compressible turbulence, and soliton turbulence. Soliton turbulence can be found in large scale turbulence, turbulence connected with surface gravity waves and nonlinear propagation of acoustical and optical waves. By letting the pressure gradient represent the elementary interaction among fluid elements and by raising the Navier-Stokes equation to higher dimensionality, the master equation was obtained for the description of the microdynamical state of turbulence

Interaction of particles with complex electrostatic structures and 3D clusters

Particles of micrometer size externally introduced in plasmas usually find their positions of levitation in the plasma sheath, where the gravity force is compensated by the strong electric field. Here due to electrostatic interaction they form different structures, which are interesting objects for the investigation of strongly coupled systems and critical phenomena. Because of the low damping (e.g. in comparison to colloidal suspension) it is possible to measure the dynamics up to the relevant highest frequency (e.g. Einstein frequency) at the most elementary level of single particle motion. The task of this work was to analyze the three dimensional structure, dynamical processes and the limit of the cooperative behavior in small plasma crystals. In addition to the study of the systems formed, the immersed particles themselves may be used for diagnostics of the plasma environment: estimation of parameters or monitoring of the processes inside plasma. The laboratory experiments are performed in two radio-frequency (RF) plasma reactors with parallel plate electrodes, where the lower electrode is a so-called "adaptive electrode". This electrode is segmented into 57 small "pixels" independently driven in DC (direct current) and/or RF voltage. When RF voltage is applied to one of these pixels, a bright localized glow, "secondary plasma ball", appears above. Three dimensional dust crystals with less than 100 particles are formed inside this "plasma ball" - the ideal conditions for the investigation of the transition from cluster systems to collective systems. The investigation of the particle interactions in crystals is performed with an optical diagnostic, which allows determination of all three particle coordinates simultaneously with time resolution of 0.04 sec. The experimental results are: 1. The binary interaction among particles in addition to the repelling Coulomb force exhibits also an attractive part, which is experimentally determined for the first time. 2. Analysis of the dynamical evolution shows the tendency of the systems to approach the state with minimum energy by rearranging particles inside. 3. The measured 63 particles' crystal vibrations are in close agreement with vibrations of a drop with surface tension. This indicates that even a 63 particle crystal already exhibits properties normally associated with the cooperative regime. The possibility to use levitated particles as a new powerful diagnostic of the plasma sheath region is proposed. The existence of different equilibrium positions of microparticles suspended in an Oxygen discharge provides evidence of a structured electronegative plasma sheath, a feature so far only mathematically and numerically investigated

Hadron models and related New Energy issues

The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of lowenergy reaction (coldfusion) phenomena. The authors explored some new approaches to describe novel phenomena in particle physics. M Pitkanen introduces his nuclear string hypothesis derived from his Topological Geometrodynamics theory, while E. Goldfain discusses a number of nonlinear dynamics methods, including bifurcation, pattern formation (complex GinzburgLandau equation) to describe elementary particle masses. Fu Yuhua discusses a plausible method for prediction of phenomena related to New Energy development. F. Smarandache discusses his unmatter hypothesis, and A. Yefremov et al. discuss Yang-Mills field from Quaternion Space Geometry. Diego Rapoport discusses theoretical link between Torsion fields and Hadronic Mechanic. A.H. Phillips discusses semiconductor nanodevices, while V. and A. Boju discuss Digital Discrete and Combinatorial methods and their likely implications in New Energy research. Pavel Pintr et al. describe planetary orbit distance from modified Schrödinger equation, and M. Pereira discusses his new Hypergeometrical description of Standard Model of elementary particles. The present volume will be suitable for researchers interested in New Energy issues, in particular their link with alternative hadron models and interpretation