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Table 1 Variation of the density ratio \(\beta \left( =\dfrac{\rho _{a}}{\rho _{n}}\right)\) across the shock front and the position of the piston surface \(\xi _{p}\) for different values of \(M_{A}^{-2}\), \(\gamma\) and \(\dfrac{\sigma }{i}\)

From: Flow behind an exponential shock wave in a rotational axisymmetric perfect gas with magnetic field and variable density

\(M_{A}^{-2}\) \(\gamma\) \(\beta\) Position of the piston surface \(\xi _{p}\)
Isothermal flow Adiabatic flow
\(\dfrac{\sigma }{i} = 1\) \(\dfrac{\sigma }{i} = 1.5\) \(\dfrac{\sigma }{i} = 1\) \(\dfrac{\sigma }{i} = 1.5\)
0 \(\dfrac{4}{3}\) 0.142857 0.899886 0.834074 0.956562 0.951338
\(\dfrac{5}{3}\) 0.25000 0.822819 0.690086 0.917366 0.908795
0.01 \(\dfrac{4}{3}\) 0.165804 0.892594 0.835098 0.934578 0.925728
\(\dfrac{5}{3}\) 0.261039 0.823930 0.705373 0.904250 0.892942
0.1 \(\dfrac{4}{3}\) 0.296396 0.840903 0.800793 0.856906 0.836709
\(\dfrac{5}{3}\) 0.34838 0.808796 0.744513 0.842153 0.820695