Propagation of shock wave in a rotational axisymmetric ideal gas with density varying exponentially and azimuthal magnetic field: isothermal flow

Abstract

In the present paper, the non-similarity solution for unsteady isothermal flow behind the cylindrical shock wave in a rotational axisymmetric perfect gas in the presence of azimuthal magnetic field is investigated. The ambient medium is assumed to have axial, azimuthal and radial components of fluid velocity. Solutions are obtained for MHD shock in a rotating medium with the vorticity vector and its components in one-dimensional flow case. The numerical solutions are obtained using Mathematica software and Runge–Kutta method of the fourth order. The Alfven Mach number, time and adiabatic exponent effects are worked out in detail. It is obtained that in the presence of magnetic field at the piston (inner expanding surface), the pressure and density vanish and hence a vacuum is formed at the line of symmetry, which is an excellent conformity with conditions to produce the shock wave in laboratory. Also, without magnetic field, the shock strength increases with an increase in time, whereas time has reverse affects on the shock strength in the presence of magnetic field. Our solutions are valid for arbitrary values of time. A comparison is also made between the behavior of non-rotating and rotating medium solutions in the presence or absence of magnetic field.

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Abbreviations

\(A\) :

Angular velocity

\(a_{\text{isoth}}\) :

Isothermal sound speed

\(B\) :

Dimensional constant

\(b\) :

Alfven speed

\(C\) :

Dimensional constant

\(c\) :

Speed of sound in gas

\(D\) :

Function of a variable \(\xi\)

\(\overrightarrow {E}\) :

Electric field vector

\(E_{\text{T}}\) :

Total energy of disturbance

\(F\) :

Radiation heat flux

\(H\) :

Function of a variable \(\xi\)

\(\overrightarrow {H}\) :

Magnetic field vector

\(h\) :

Azimuthal magnetic field

\(h^{\prime }\) :

Reduced azimuthal magnetic field

\(i\) :

Dimensional constant

\(J\) :

Abbreviation

\(L\) :

Abbreviation

\(\bar{L}\) :

Characteristic length

\(l\) :

Dimensional constant

\(l_{\text{r}}\) :

Non-dimensional radial vorticity

\(l_{\theta }\) :

Non-dimensional azimuthal vorticity

\(l_{z}\) :

Non-dimensional axial vorticity

\(M_{\text{A}}^{ - 2}\) :

Alfven Mach number

\(P\) :

Function of a variable \(\xi\)

\(p\) :

Fluid pressure

\(p^{\prime }\) :

Reduced fluid pressure

\(R_{\text{e}}\) :

Reynolds number of the flow

\(R_{\text{m}}\) :

Magnetic Reynolds number

\(r\) :

Independent space coordinate

\(r_{\text{s}}\) :

Shock radius

\(r^{\prime }\) :

Reduced distance

\(\overline{r}\) :

Inner boundary of the disturbance

\(T\) :

Temperature of the gas

\(t\) :

Independent time coordinate

\(t_{0}\) :

Period of almost instant explosion

\(U\) :

Function of a variable \(\xi\)

\(\bar{U}\) :

Characteristic velocity

\(U_{\text{m}}\) :

Internal energy per unit mass

\(u\) :

Radial component of fluid velocity

\(u^{\prime }\) :

Reduced radial fluid velocity

\(\overrightarrow {V}\) :

Fluid velocity

\(v\) :

Azimuthal component of fluid velocity

\(v^{\prime }\) :

Reduced azimuthal fluid velocity

\(W\) :

Function of a variable \(\xi\)

\(W_{\text{s}}\) :

Shock velocity

\(w\) :

Axial component of fluid velocity

\(w^{\prime }\) :

Reduced axial fluid velocity

\(\rho\) :

Fluid density

\(\rho_{0}\) :

Dimensional constant

\(\rho^{\prime }\) :

Reduced fluid density

\(\alpha\) :

Dimensional constant

\(\beta\) :

Density ratio across the shock front

\(\Gamma\) :

Gas constant

\(\gamma\) :

Ratio of specific heats

\(\lambda\) :

Constant

\(\sigma\) :

Electrical conductivity of the medium

\(\mu\) :

Magnetic permeability

\(\nu\) :

Kinematic coefficient of viscosity

\(\phi\) :

Function of a variable \(\xi\)

\(\xi_{0}\) :

Constant

\(\eta_{\text{m}}\) :

Magnetic diffusivity

\(\overrightarrow {\varsigma }\) :

Vorticity vector

\(\varsigma_{\text{r}}\) :

Radial component of vorticity vector

\(\varsigma_{\theta }\) :

Azimuthal component of vorticity vector

\(\varsigma_{z}\) :

Axial component of vorticity vector

\(\varOmega\) :

Constant

\((r,\,\,\theta ,\,z)\) :

Cylindrical coordinates

\(1\,\,\) :

Immediately ahead of the shock front

\(2\,\) :

Immediately behind the shock front

\(T\) :

Process of constant temperature

References

  1. [1]

    L I Sedov Similarity and Dimensional Methods in Mechanics (Mascow: Mir Publishers) (1982)

  2. [2]

    G Nath Adv. Space Res. 47 1463 (2011)

  3. [3]

    M H Rogers Astrophys. J. 125 478 (1957)

  4. [4]

    P Rosenau and S Frankenthal Phys. Fluids 19 1889 (1976)

    ADS  Article  Google Scholar 

  5. [5]

    O Nath, S N Ojha and H S Takhar Astrophys. Space Sci. 95 99 (1983)

    Article  Google Scholar 

  6. [6]

    J P Vishwakarma and A K Yadav Eur. Phys. J. B. 34 247 (2003)

    ADS  Article  Google Scholar 

  7. [7]

    D D Laumbach and R F Probstein J. Fluid Mech. 35 53 (1968)

    ADS  Article  Google Scholar 

  8. [8]

    W D Hayes J. Fluid Mech. 32 305 (1968)

  9. [9]

    B G Verma and J P Vishwakarma Nuovo Cimento. 32 267 (1976)

    Article  Google Scholar 

  10. [10]

    G Deb Ray Bull. Cal. Math. Soc. 66 27 (1974)

  11. [11]

    J P Vishwakarma Eur. Phys. J. B. 16 369 (2000)

  12. [12]

    G Nath Res. Astron. Astrophys. 10 445 (2010)

  13. [13]

    G Nath J. Theor. Appl. Phys. 8 831 (2014). https://doi.org/10.1007/s40094-014-0131-y

  14. [14]

    G Nath, J P Vishwakarma, V K Srivastava and A K Sinha J. Theor. Appl. Phys. 7 15 (2013) https://doi.org/10.1186/2251-7235-7-15

    ADS  Article  Google Scholar 

  15. [15]

    G Nath and J P Vishwakarma Acta Astronatica 123 200 (2016)

    ADS  Article  Google Scholar 

  16. [16]

    G Nath Acta Astronatica 162 447 (2019)

  17. [17]

    T S Lee and T Chen Planet Space Sci.16 1483 (1968)

    ADS  Article  Google Scholar 

  18. [18]

    Ya B Zel’dovich and Yu P Raizer Physics of Shock Waves and High Temperature Hydrodynamic Phenomena Vol. II (New York: Academic Press) (1967)

  19. [19]

    D Summers Astron. Astrophys. 45 151 (1975)

  20. [20]

    P Chaturani Appl. Sci. Res. 23 197 (1970)

  21. [21]

    J P Vishwakarma and G Nath Phys. Scr. 81 045401(9 pp) (2010)

  22. [22]

    J P Vishwakarma and G Nath Commun. Nonlinear Sci. Numer. Simul. 17 154 (2012)

    ADS  Article  Google Scholar 

  23. [23]

    G Nath Ain Shams Eng. J. 3 393 (2012)

  24. [24]

    G Nath Shock Waves 24 415 (2014)

  25. [25]

    G Nath Meccanica 50 1701 (2015)

  26. [26]

    G Nath Indian J. Phys. 90 1055 (2016)

  27. [27]

    G Nath Acta Astronatica 148 355 (2018)

  28. [28]

    J S Shang Prog. Aerosp. Sci. 21 1 (2001)

  29. [29]

    R M Lock and A J Mestel J. Fluid Mech. 74 531(2008)

    Google Scholar 

  30. [30]

    S W H Cowley, J A Davies, A Grocott, H Khan, M Lester, K A McWilliams, S E Milan, G Provan, P E Sandholt, J A Wild and T K Yeoman, Philos. Trans. R. Soc. Lond. A 361 113 (2003)

    ADS  Article  Google Scholar 

  31. [31]

    G C Valley Ap. J. 168 251(1971)

  32. [32]

    Y Q Lou and R Rosner Ap. J. 309 874 (1986)

    ADS  Article  Google Scholar 

  33. [33]

    L. Hartmann Accretion Processes in Star Formation (Cambridge: Cambridge University Press) (1998)

  34. [34]

    B Balick and A Frank Annu. Rev. Astron. Astrophys. 40 439 (2002)

    ADS  Article  Google Scholar 

  35. [35]

    B P Rybakin, V B Betelin, V R Dushin, E V Mikhalchenko, S G Moiseenko, L I Stamov and V V Tyurenkova Acta Astronaut. 119 131 (2016)

  36. [36]

    B P Rybakin and V Goryachev Lobachevskii J. Math. 39 562 (2018)

    MathSciNet  Article  Google Scholar 

  37. [37]

    D I Pullin, W Mostert, V Wheatley and R Samtaney Phys. Fluids 26 097103 (2014)

    ADS  Article  Google Scholar 

  38. [38]

    W Mostert, D I Pullin, R Samtaney and V Wheatley J. Fluid Mech. 793 414 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  39. [39]

    W Mostert, D I Pullin, R Samtaney and V Wheatley J. Fluid Mech. 811 R2-1 (2017)

    Article  Google Scholar 

  40. [40]

    D D Laumbach and R F Probstein Phys. Fluid 13 1178 (1970)

    ADS  Article  Google Scholar 

  41. [41]

    P L Sachdev and S Ashraf J. Appl. Math. Phys. (ZAMP) 22 1095 (1971)

    Article  Google Scholar 

  42. [42]

    V P Korobeinikov Problems in the theory of point explosion in gases. In: Proceedings of the Steklov Institute of Mathematics, Vol. 119. American Mathematical Society, Providence (1976)

  43. [43]

    T A Zhuravskaya and V A Levin J. Appl. Math. Mech. 60 745 (1996)

    MathSciNet  Article  Google Scholar 

  44. [44]

    D D Laumbach and R F Probstein J. Fluid Mech. 40 833 (1970)

    ADS  Article  Google Scholar 

  45. [45]

    Yu P Raizer Zh. Prikl. Mekh. Tekh. Fiz. 4 49 (1964)

  46. [46]

    R Grover and J W Hardy Astrophys. J. 143 48 (1966)

    ADS  Article  Google Scholar 

  47. [47]

    J P Vishwakarma and G Nath Meccanica 42 331 (2007)

    MathSciNet  Article  Google Scholar 

  48. [48]

    I Lerch Aust. J. Phys. 32 491(1979)

  49. [49]

    I Lerch Aust. J. Phys. 34 279(1981)

  50. [50]

    V A Levin and G A Skopina J. Appl. Mech. Tech. Phys. 45 457 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  51. [51]

    G B Whitham J. Fluid Mech. 4 337 (1958)

  52. [52]

    G Nath and J P Vishwakarma Commun. Nonlinear Sci. Numer. Simul. 19 1347(2014)

  53. [53]

    G Nath and J P Vishwakarma Acta Astronatica 128 377 (2016)

    ADS  Article  Google Scholar 

  54. [54]

    G Nath Indian J. Phys. (2019). https://doi.org/10.1007/s12648-019-01511-w

  55. [55]

    MC Kelley The Earth’s Ionosphere Plasma Physics and Electrodynamics, Vol. 96, IInd Edition (Academic Press) (2009)

  56. [56]

    M A Liberman and A L Velikovich Physics of Shock Waves in Gases and Plasma (Berlin: Springer) (1986)

    Google Scholar 

  57. [57]

    S I Pai Magnetogasdynamic and Plasma Dynamics (Wien: Springer-Verlag) (1986)

  58. [58]

    R A Freeman and J D Craggs J. Phys. D Appl. Phys. 2 42 (1969)

    Article  Google Scholar 

  59. [59]

    M N Director and E K Dabora Acta Astronautica 4 391(1977)

    ADS  Article  Google Scholar 

  60. [60]

    P Rosenau Phys. Fluids 20 1097 (1977)

  61. [61]

    G J Hutchens J. Appl. Phys. 77 2912 (1995)

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Nath, G. Propagation of shock wave in a rotational axisymmetric ideal gas with density varying exponentially and azimuthal magnetic field: isothermal flow. Indian J Phys 95, 163–175 (2021). https://doi.org/10.1007/s12648-020-01684-9

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Keywords

  • Shock wave
  • Rotating medium
  • Non-similarity solution
  • Magnetogasdynamics and electro-fluid mechanics

PACS Nos.

  • 47.40.–X
  • 47.32.Ef
  • 47.65.–d
  • 52.30.Cv
  • 52.65.Kj