Similarity solutions for cylindrical shock wave in rotating ideal gas with or without magnetic field using Lie group the
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Similarity solutions for cylindrical shock wave in rotating ideal gas with or without magnetic field using Lie group theoretic method G. Natha
, Sumeeta Singhb
Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh 211004, India Received: 13 March 2020 / Accepted: 16 November 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The propagation of a cylindrical shock wave in rotating ideal gas under adiabatic flow condition is investigated using Lie group of transformation method. Both the cases of shock without magnetic field and under the influence of axial magnetic field are considered separately. The density and azimuthal fluid velocity in the case of shock without magnetic field and density, azimuthal fluid velocity and axial magnetic field in the case of shock under the influence of magnetic field are assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the local Lie group of transformations bring about two different cases of solutions, i.e. with a power law and exponential law shock paths. Exact solutions are obtained in the case of power law shock path for both the cases of cylindrical shock with and without magnetic field. It is not possible to obtain the exact solution in the case of exponential law shock path. In this case, the numerical solutions can be obtained by using the respective boundary conditions. Distribution of gasdynamical and magnetogasdynamical flow quantities are illustrated through figures.
1 Introduction The occurrence of shock waves in the interstellar medium is conventional due to diversity of supersonic motions and energetic events, such as bipolar outflow from young proto-stellar objects, supernova explosions, powerful mass losses by massive stars in a late stage of their evolution (stellar winds), cloud-cloud collisions, phenomena in the central part of star burst galaxies, etc (see [1]). Investigating the cylindrical shock wave under the influence of magnetic field has various applications (see [2]). Taylor [3,4] and Sedov [5] have obtained the numerical solutions for self-similar problems associated with shock waves independently. The formulation of self-similar problems and the examples describing the adiabatic motion of non-rotating gas models of stars were considered by Sedov [5], Zel’dovich and Raizer [6], Lee and Chen [7] and Summers [8]. Rotation of the planets or stars significantly affects the process taking place in their outer layers. Therefore, question connected with the explosions in rotating gas atmospheres are of definite
a e-mails: [email protected]; [email protected] (corresponding author) b e-mail: [email protected]
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astrophysical interest. Shock waves often arise in nature because of the balance between wave-breaking nonlinear and wave-damping dissipative forces (see [6]
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