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

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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

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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 (2020). 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