Lie symmetry analysis and optimal system for shock wave in a self-gravitating rotating ideal gas under the effect of magnetic field and monochromatic radiation. (English) Zbl 1539.85003
Summary: The Lie group invariance method is used to study a cylindrical shock wave in a self-gravitating, rotating perfect gas in the presence of monochromatic radiation and an azimuthal or axial magnetic field. The density of the ambient medium is taken as variable according as the law of shock path. For the system of equations of motion, the one-dimensional optimal system of subalgebra is determined by using Lie group analysis. We have utilized optimal classes of infinitesimal generators to acquire the flow variable transformation and the similarity variable, which are important prerequisites for obtaining the system of ordinary differential equations from the system of partial differential equations. In detail, we have numerically solved and discussed the results in two cases: with power and exponential laws shock path. The effects of variation of the rotational parameter, gravitational parameter, Alfvén Mach number, adiabatic exponent, and dimensionless parameter that characterize the interaction between incident radiation flux and gas are studied in depth. A comparative study is done between power law and exponential law in respect of the strength of shock wave and the flow variables distribution in the flow-field region behind the shock front. The shock is stronger with an axial magnetic field in a power law case; whereas the shock is stronger with an azimuthal magnetic field in an exponential law case. The shock strength is observed to decline when the adiabatic index of the gas or the Alfvén Mach number increases. The shock decays with the rotational parameter in case of exponential law, but its strength is enhanced in case of power law. Also, the rotational parameter and gravitational parameter have an exact opposite impact on the strength of shock in power law and exponential law cases.
MSC:
| 85A25 | Radiative transfer in astronomy and astrophysics |
| 76M60 | Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics |
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 76U05 | General theory of rotating fluids |
References:
| [1] | Sedov, L., Similarity and Dimensional Methods in Mechanics (Academic Press, New York, 1959). · Zbl 0121.18504 |
| [2] | Carrus, P. A., Fox, P. A., Haas, F. and Kopal, Z., The propagation of shock waves in a stellar model with continuous density distribution, Astrophys. J.113 (1951) 496. |
| [3] | Sakurai, A., Propagation of spherical shock waves in stars, J. Fluid Mech.1 (1956) 436-453. · Zbl 0073.46005 |
| [4] | Ojha, S. and Nath, O., Self-similar flow behind a spherical shock with varying strength in an inhomogeneous self-gravitating medium, Astrophys. Space Sci.129 (1987) 11-17. |
| [5] | Purohit, S. C., Self-similar homothermal flow of self-gravitating gas behind shock wave, J. Phys. Soc. Japan36 (1974) 288-292. |
| [6] | Nath, G., Approximate analytical solution for the propagation of shock waves in self-gravitating perfect gas via power series method: Isothermal flow, J. Astrophys. Astron.41 (2020) 1-19. |
| [7] | Larson, R. B., Numerical calculations of the dynamics of a collapsing proto-star, Mon. Not. Roy. Astron. Soc.145 (1969) 271-295. |
| [8] | Penston, M., Dynamics of self-gravitating gaseous spheres — III: Analytical results in the free-fall of isothermal cases, Mon. Not. Roy. Astron. Soc.144 (1969) 425-448. |
| [9] | Shu, F. H., Self-similar collapse of isothermal spheres and star formation, Astrophys. J.214 (1977) 488-497. |
| [10] | Nath, G., Cylindrical shock wave propagation in a self-gravitating rotational axisymmetric perfect gas under the influence of azimuthal or axial magnetic field and monochromatic radiation with variable density, Pramana95 (2021) 1-16. |
| [11] | Nath, G. and Devi, A., Cylindrical shock wave in a self-gravitating perfect gas with azimuthal magnetic field via Lie group invariance method, Int. J. Geom. Methods Mod. Phys.17 (2020) 2050148. · Zbl 07813580 |
| [12] | Nath, G. and Singh, S., Similarity solutions using Lie group theoretic method for cylindrical shock wave in self-gravitating perfect gas with axial magnetic field: Isothermal flow, Eur. Phys. J. Plus135 (2020) 1-15. |
| [13] | Nath, G., Vishwakarma, J. P., Srivastava, V. K. and Sinha, A. K., Propagation of magnetogasdynamic shock waves in a self-gravitating gas with exponentially varying density, J. Theor. Appl. Phys.7 (2013) 1-8. |
| [14] | Vishwakarma, J. P. and Singh, A. K., A self-similar flow behind a shock wave in a gravitating or non-gravitating gas with heat conduction and radiation heat-flux, J. Astrophys. Astron.30 (2009) 53-69. |
| [15] | Nath, G., Pathak, R. and Dutta, M., Similarity solutions for unsteady flow behind an exponential shock in a self-gravitating non-ideal gas with azimuthal magnetic field, Acta Astronaut.142 (2018) 152-161. |
| [16] | Hartmann, L., Accretion Processes in Star Formation, Vol. 32 (Cambridge University Press, 2000). |
| [17] | Balick, B. and Frank, A., Shapes and shaping of planetary nebulae, Annu. Rev. Astron. Astrophys.40 (2002) 439-486. |
| [18] | Lerche, I., Mathematical theory of one-dimensional isothermal blast waves in a magnetic field, Aust. J. Phys.32 (1979) 491-502. |
| [19] | Lerche, I., Mathematical theory of cylindrical isothermal blast waves in a magnetic field, Aust. J. Phys.34 (1981) 279-302. |
| [20] | Shang, J., Recent research in magneto-aerodynamics, Prog. Aerosp. Sci.37 (2001) 1-20. |
| [21] | Christer, A. and Helliwell, J., Cylindrical shock and detonation waves in magnetogasdynamics, J. Fluid Mech.39 (1969) 705-725. · Zbl 0183.55803 |
| [22] | Greenspan, H. P., Similarity solution for a cylindrical shock-magnetic field interaction, Phys. Fluids5 (1962) 255-259. · Zbl 0111.39601 |
| [23] | Pai, S.-I., Magnetogasdynamics and Plasma Dynamics (Springer Science & Business Media, 2012). |
| [24] | Greifinger, C. and Cole, J. D., Similarity solution for cylindrical magnetohydrodynamic blast waves, Phys. Fluids5 (1962) 1597-1607. · Zbl 0117.21203 |
| [25] | Chaturani, P., Strong cylindrical shocks in a rotating gas, Appl. Sci. Res.23 (1971) 197-211. · Zbl 0222.76068 |
| [26] | Vishwakarma, J. P., Maurya, A. K. and Singh, K., Self-similar adiabatic flow headed by a magnetogasdynamic cylindrical shock wave in a rotating non-ideal gas, Geophys. Astrophys. Fluid Dyn.101 (2007) 155-168. · Zbl 1505.85005 |
| [27] | Hishida, M., Fujiwara, T. and Wolanski, P., Fundamentals of rotating detonations, Shock Waves19 (2009) 1-10. · Zbl 1255.76146 |
| [28] | Nath, G., Magnetogasdynamic shock wave generated by a moving piston in a rotational axisymmetric isothermal flow of perfect gas with variable density, Adv. Space Res.47 (2011) 1463-1471. |
| [29] | Khudyakov, V. M., The self similar problem of the motion of gas under the action of monochromatic radiation, Sov. Phys. Dokl.28 (1983) 853-855, trans. American Institute of Physics. · Zbl 0557.76078 |
| [30] | Marshak, R., Effect of radiation on shock wave behavior, Phys. Fluids1 (1958) 24-29. · Zbl 0081.41601 |
| [31] | Sachs, R., Some properties of very intense shock waves, Phys. Rev.69 (1946) 514. |
| [32] | Thomas, L., The radiation field in a fluid in motion, Q. J. Math.1 (1930) 239-251. · JFM 56.1361.02 |
| [33] | Elliott, L., Similarity methods in radiation hydrodynamics, Proc. R. Soc. Lond. A, Math. Phys. Sci.258 (1960) 287-301. |
| [34] | Taylor, G., The formation of a blast wave by a very intense explosion. I. Theoretical discussion, Proc. R. Soc. Lond. A, Math. Phys. Sci.201 (1950) 159-174. · Zbl 0036.26404 |
| [35] | Bhatnagar, M. and Kushwaha, R., Propagation of intense shock waves in stellar envelopes, Ann. Astrophys.24 (1961) 211. · Zbl 0097.23702 |
| [36] | Deb Ray, G. and Bhowmick, J., Similarity solutions for explosions in radiating stars, Indian J. Pure Appl. Math.7 (1976) 96-103. · Zbl 0374.76054 |
| [37] | Rao, M. R. and Ramana, B., Unsteady flow of a gas behind an exponential shock, J. Math. Phys. Sci.10 (1976) 465-476. · Zbl 0364.76052 |
| [38] | Nath, O., Propagation of cylindrical shock waves in a rotating atmosphere under the action of monochromatic radiation, Il Nuovo Cimento D20 (1998) 1845-1852. |
| [39] | Nath, O. and Takhar, H., Propagation of cylindrical shock waves under the action of monochromatic radiation, Astrophys. Space Sci.166 (1990) 35-39. |
| [40] | Zedan, H. A., Applications of the group of equations of the one-dimensional motion of a gas under the influence of monochromatic radiation, Appl. Math. Comput.132 (2002) 63-71. · Zbl 1024.76042 |
| [41] | Vishwakarma, J. P. and Pandey, V. K., Self-similar flow under the action of monochromatic radiation behind a cylindrical MHD shock in a non-ideal gas, Appl. Math. Comput.2 (2012) 28-33. |
| [42] | Bluman, G. W. and Cole, J. D., Similarity Methods for Differential Equations (Springer Science-Verlag, New York, 1974). · Zbl 0292.35001 |
| [43] | Olver, P. J., Application of Lie Group to Differential Equations (Springer Science, New York, 1986). · Zbl 0588.22001 |
| [44] | Bluman, G. W. and Kumei, S., Symmetries and Differential Equations (Springer, New York, 1989). · Zbl 0698.35001 |
| [45] | Ovsiannikkov, L. V., Group Analysis of Differential Equation (Academic Press, New York, 1974). |
| [46] | Hydon, P. E., Symmetry Methods for Differential Equations: A Beginner’s Guide (Cambridge University Press, London, 2000). · Zbl 0951.34001 |
| [47] | Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations, Vol. 197 (Wiley, Chichester, 1999). · Zbl 1047.34001 |
| [48] | Logan, J. D. and Parez, J. D. J., Similarity solution for reactive shock hydrodynamics, SIAM J. Appl. Math.39 (1980) 512-527. · Zbl 0493.76060 |
| [49] | Nath, G. and Singh, S., Similarity solutions for magnetogasdynamic cylindrical shock wave in rotating ideal gas using Lie group theoretic method: Isothermal flow, Int. J. Geom. Methods Mod. Phys.17 (2020) 205-123. · Zbl 07811500 |
| [50] | Ovsiannikov, L. V., Group Analysis of Differential Equations (Academic Press, 2014). |
| [51] | Ibragimov, N. H., Torrisi, M. and Valenti, A., Preliminary group classification of equations \(v_{t t}=f(x, v_x) v_{x x}+g(x, v_x)\), J. Math. Phys.32 (1991) 2988-2995. · Zbl 0737.35099 |
| [52] | Johnpillai, A. and Khalique, C., Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients, Commun. Nonlinear Sci. Numer. Simul.16 (2011) 1207-1215. · Zbl 1221.35338 |
| [53] | Nath, G. and Devi, A., Exact and numerical solution using Lie group analysis for the cylindrical shock waves in a self-gravitating ideal gas with axial magnetic field, Int. J. Appl. Math. Comput. Sci.7 (2021) 1-20. · Zbl 1490.76129 |
| [54] | Levin, V. A. and Skopina, G. A., Detonation wave propagation in rotational gas flows, J. Appl. Mech. Tech. Phys.45 (2004) 457-460. · Zbl 1049.76037 |
| [55] | Wambura, K. O., Oduor, M. and Aminer, T., Lie symmetry analysis and the optimal systems of nonlinear fourth order evolution equation, Int. J. Appl. Sci. Eng.5 (2019) 1-6. |
| [56] | Sahoo, S. M., Raja Sekhar, T. and Raja Sekhar, G., Optimal classification, exact solutions, and wave interactions of Euler system with large friction, Math. Methods Appl. Sci.43 (2020) 5744-5757. · Zbl 1454.35269 |
| [57] | Rosenau, P. and Frankenthal, S., Equatorial propagation of axisymmetric magnetohydrodynamic shocks, Phys. Fluids19 (1976) 1889-1899. · Zbl 0353.76075 |
| [58] | Nath, G. and Vishwakarma, J. P., Similarity solution for the flow behind a shock wave in a non-ideal gas with heat conduction and radiation heat-flux in magnetogasdynamics, Commun. Nonlinear Sci. Numer. Simul.19 (2014) 1347-1365. · Zbl 1457.35042 |
| [59] | Nath, G. and Sahu, P., Similarity solution for the flow behind a cylindrical shock wave in a rotational axisymmetric gas with magnetic field and monochromatic radiation, Ain Shams Eng. J.9 (2018) 1151-1159. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
