A Study of One-dimensional Weak Shock Propagation Under the Action of Axial and Azimuthal Magnetic Field: An Analytical Approach

Authors

  • A. Husain Department of Mathematics, School of Engineering, University of Petroleum and Energy Studies, India https://orcid.org/0000-0003-3016-2898
  • S. A. Haider Department of Mathematics, Shia P. G. College, India
  • V. K. Singh Department of Applied Sciences, Institute of Engineering and Technology, India
Volume: 12 | Issue: 6 | Pages: 9503-9509 | December 2022 | https://doi.org/10.48084/etasr.5277

Abstract

The present paper presents an analytical study of the one-dimensional weak shock wave problem in a perfect gas under the action of a generalized magnetic field subjected to weak shock jump conditions (R-H conditions). The magnetic field is considered axial and azimuthal in cylindrically symmetric configuration. By considering a straightforward analytical approach, an explicit solution exhibiting time-space dependency for gas-dynamical flow parameters and total energy (generated during the propagation of the weak shock from the center of the explosion) has been obtained under the significant influence of generalized magnetic fields (axial and azimuthal) and the results are analyzed graphically. From the outcome, it is worth noticing that for an increasing value of Mach number under the generalized magnetic field, the decay process of physical parameters (density, pressure, and magnetic pressure) is a bit slower, whereas the velocity profile and total energy increase rapidly with respect to time. Moreover, for increasing values of Shock-Cowling number the total energy grows rapidly with respect to time.

Keywords:

weak shock waves, analytical solution, Rankine- Hugoniot conditions, magnetogasdynamics

Downloads

Download data is not yet available.

References

B. M. Johnson, "Closed-form shock solutions," Journal of Fluid Mechanics, vol. 745, Apr. 2014, Art. no. R1. DOI: https://doi.org/10.1017/jfm.2014.107

T. Gegechkori, G. Mamniashvili, A. Peikrishvili, V. Peikrishvili, and B. Godibadze, "Using Fast Hot Shock Wave Consolidation Technology to Produce Superconducting MgB2," Engineering, Technology & Applied Science Research, vol. 8, no. 1, pp. 2374–2378, Feb. 2018. DOI: https://doi.org/10.48084/etasr.1690

W. Zhang, L. Zou, X. Zheng, and B. Wang, "Numerical study on the interaction of a weak shock wave with an elliptic gas cylinder," Shock Waves, vol. 29, no. 2, pp. 273–284, Feb. 2019. DOI: https://doi.org/10.1007/s00193-018-0828-y

A. Ferrari, "Analytical solutions for one-dimensional diabatic flows with wall friction," Journal of Fluid Mechanics, vol. 918, Jul. 2021, Art. no. A32. DOI: https://doi.org/10.1017/jfm.2021.278

L. Saidi, S. Mekroussi, S. Kherris, D. Zebbar, and B. Mébarki, "A Numerical Investigation of the Free Flow in a Square Porous Cavity with Non-Uniform Heating on the Lower Wall," Engineering, Technology & Applied Science Research, vol. 12, no. 1, pp. 7982–7987, Feb. 2022. DOI: https://doi.org/10.48084/etasr.4604

A. G. da Silva Jr, J. A. Martins, and E. C. Romao, "Numerical Simulation of a One-Dimentional Non-Linear Wave Equation," Engineering, Technology & Applied Science Research, vol. 12, no. 3, pp. 8574–8577, Jun. 2022. DOI: https://doi.org/10.48084/etasr.4920

C. Greifinger and J. D. Cole, "Similarity Solution for Cylindrical Magnetohydrodynamic Blast Waves," The Physics of Fluids, vol. 5, no. 12, pp. 1597–1607, Dec. 1962. DOI: https://doi.org/10.1063/1.1706571

N. Geffen, "Magnetogasdynamic Flows with Shock Waves," The Physics of Fluids, vol. 6, no. 4, pp. 566–571, Apr. 1963. DOI: https://doi.org/10.1063/1.1706774

C. K. Chu, "Dynamics of Ionizing Shock Waves: Shocks in Transverse Magnetic Fields," The Physics of Fluids, vol. 7, no. 8, pp. 1349–1357, Aug. 1964. DOI: https://doi.org/10.1063/1.1711380

T. S. Lee and T. Chen, "Hydromagnetic interplanetary shock waves," Planetary and Space Science, vol. 16, no. 12, pp. 1483–1502, Dec. 1968. DOI: https://doi.org/10.1016/0032-0633(68)90061-5

A. H. Christer and J. B. Helliwell, "Cylindrical shock and detonation waves in magnetogasdynamics," Journal of Fluid Mechanics, vol. 39, no. 4, pp. 705–725, Dec. 1969. DOI: https://doi.org/10.1017/S0022112069002424

G. D. Ray, "Similarity solutions for cylindrical blast waves in magnetogasdynamics," The Physics of Fluids, vol. 16, no. 4, pp. 559–560, Apr. 1973. DOI: https://doi.org/10.1063/1.1694381

L. A. Bertram, "Magnetogasdynamic shock polar: exact solution in aligned fields," Journal of Plasma Physics, vol. 9, no. 3, pp. 325–347, Jun. 1973. DOI: https://doi.org/10.1017/S0022377800007534

D. Summers, "An idealised model of a magnetohydrodynamic spherical blast wave applied to a flare produced shock in the solar wind," Astronomy and Astrophysics, vol. 45, pp. 151–158, 1975.

I. Lerche, "Mathematical Theory of Cylindrical Isothermal Blast Waves in a Magnetic Field," Australian Journal of Physics, vol. 34, no. 3, pp. 279–302, 1981. DOI: https://doi.org/10.1071/PH810279

V. D. Sharma, L. P. Singh, and R. Ram, "The progressive wave approach analyzing the decay of a sawtooth profile in magnetogasdynamics," The Physics of Fluids, vol. 30, no. 5, pp. 1572–1574, May 1987. DOI: https://doi.org/10.1063/1.866222

B. Vrsnak and S. Lulic, "Formation Of Coronal Mhd Shock Waves – I. The Basic Mechanism," Solar Physics, vol. 196, no. 1, pp. 157–180, Sep. 2000. DOI: https://doi.org/10.1023/A:1005236804727

J. S. Shang, "Recent research in magneto-aerodynamics," Progress in Aerospace Sciences, vol. 37, no. 1, pp. 1–20, Jan. 2001. DOI: https://doi.org/10.1016/S0376-0421(00)00015-4

M. Pandey, R. Radha, and V. D. Sharma, "Symmetry analysis and exact solutions of magnetogasdynamic equations," Quarterly Journal of Mechanics and Applied Mathematics, vol. 61, no. 3, pp. 291–310, Aug. 2008. DOI: https://doi.org/10.1093/qjmam/hbn011

S. Murata, "New exact solution of the blast wave problem in gas dynamics," Chaos, Solitons & Fractals, vol. 28, no. 2, pp. 327–330, Apr. 2006. DOI: https://doi.org/10.1016/j.chaos.2005.05.052

L. P. Singh, A. Husain, and M. Singh, "An Analytical Study of Strong Non Planer Shock Waves in Magnetogasdynamics," Advances in Theoretical and Applied Mechanics, vol. 3, no. 6, pp. 291–297, 2010

D. I. Pullin, W. Mostert, V. Wheatley, and R. Samtaney, "Converging cylindrical shocks in ideal magnetohydrodynamics," Physics of Fluids, vol. 26, no. 9, Sep. 2014, Art. no. 097103. DOI: https://doi.org/10.1063/1.4894743

L. P. Singh, D. B. Singh, and S. D. Ram, "Growth and decay of weak shock waves in magnetogasdynamics," Shock Waves, vol. 26, no. 6, pp. 709–716, Nov. 2016. DOI: https://doi.org/10.1007/s00193-015-0607-y

M.. J. Siddiqui, R. Arora, and A. Kumar, "Shock waves propagation under the influence of magnetic field," Chaos, Solitons & Fractals, vol. 97, pp. 66–74, Apr. 2017. DOI: https://doi.org/10.1016/j.chaos.2016.12.020

G. Nath and S. Singh, "Approximate analytical solution for shock wave in rotational axisymmetric perfect gas with azimuthal magnetic field: Isothermal flow," Journal of Astrophysics and Astronomy, vol. 40, no. 6, Dec. 2019, Art. no. 50. DOI: https://doi.org/10.1007/s12036-019-9616-z

P. Gupta, R. K. Chaturvedi, and L. P. Singh, "The propagation of weak shock waves in non-ideal gas flow with radiation," The European Physical Journal Plus, vol. 135, no. 1, Jan. 2020, Art. no. 17. DOI: https://doi.org/10.1140/epjp/s13360-019-00041-y

M. Devi, R. Arora, and D. Singh, "Blast waves propagation in magnetogasdynamics: power series method," Zeitschrift für Naturforschung A, vol. 75, no. 12, pp. 1039–1050, Dec. 2020. DOI: https://doi.org/10.1515/zna-2020-0202

A. Sakurai, "On the Propagation and Structure of the Blast Wave, I," Journal of the Physical Society of Japan, vol. 8, no. 5, pp. 662–669, 1953. DOI: https://doi.org/10.1143/JPSJ.8.662

A. Husain, S. A. Haider, and V. K. Singh, "Efficient exact solution of blast waves in magneto-gas-dynamic flow at stellar surfaces," Advances and Applications in Mathematical Sciences, vol. 20, no. 8, pp. 1599–1608, Jun. 2021.

P. Gupta and L. P. Singh, "On the evolution of magnetic shock wave in the mixture of gas and small solid dust particles," Chinese Journal of Physics, vol. 77, pp. 1912–1926, Jun. 2022. DOI: https://doi.org/10.1016/j.cjph.2021.12.027

P. Gupta, L. P. Singh, and R. Singh, "Riemann problem for non-ideal polytropic magnetogasdynamic flow," International Journal of Non-Linear Mechanics, vol. 112, pp. 6–12, Jun. 2019. DOI: https://doi.org/10.1016/j.ijnonlinmec.2019.02.012

P. Gupta, R. K. Chaturvedi, and L. P. Singh, "Solution of Riemann problem of conservation laws in van der Waals gas," Waves in Random and Complex Media, pp. 1–19, Jan. 2022. DOI: https://doi.org/10.1080/17455030.2021.2017068

S.-I. Pai, Magnetogasdynamics and Plasma Dynamics. Vienna, Austria: Springer, 1962. DOI: https://doi.org/10.1007/978-3-7091-8083-9

G. B. Whitham, Linear and Nonlinear Waves. New York, NY, USA: Wiley, 1974.

V. P. Korobeinikov, Problems in the Theory of Point Explosion in Gases. Providence, RI, USA: American Mathematical Society, 1976..

L. P. Singh, S. D. Ram, and D. B. Singh, "Exact solution of planar and nonplanar weak shock wave problem in gasdynamics," Chaos, Solitons & Fractals, vol. 44, no. 11, pp. 964–967, Nov. 2011. DOI: https://doi.org/10.1016/j.chaos.2011.07.012

J. P. Chaudhary and L. P. Singh, "Exact Solution of the Weak Shock Wave in Non-ideal Gas," International Journal of Applied and Computational Mathematics, vol. 4, no. 6, Oct. 2018, Art. no. 136. DOI: https://doi.org/10.1007/s40819-018-0570-2

Downloads

How to Cite

[1]
Husain, A., Haider, S.A. and Singh, V.K. 2022. A Study of One-dimensional Weak Shock Propagation Under the Action of Axial and Azimuthal Magnetic Field: An Analytical Approach. Engineering, Technology & Applied Science Research. 12, 6 (Dec. 2022), 9503–9509. DOI:https://doi.org/10.48084/etasr.5277.

Metrics

Abstract Views: 654
PDF Downloads: 654

Metrics Information