Shock wave structure in non-ideal dilute gases under variable Prandtl number

Abstract

This paper investigates the structure of normal shock waves for a planar steady flow of non-ideal dilute gases under variable viscosity and thermal conductivity using the Navier–Stokes–Fourier approach to the continuum model. The gas is assumed to follow the simplified van der Waals equation of state along with the power-law temperature-dependent coefficients of shear viscosity, bulk viscosity, and thermal conductivity. A closed system of nonlinear differential equations having a variable Prandtl number ( $$\Pr $$ ) is formulated. Exact analytical solutions of the shock wave structure in non-ideal gases are derived for $$\Pr \rightarrow \infty $$ and $$\Pr \rightarrow 0$$ limits, and the corresponding profiles for velocity and temperature are obtained. For $$\Pr \rightarrow 0$$ , an isothermal shock is encountered for high Mach numbers. It appears sooner in non-ideal gases. The solution profiles for $$\Pr =2/3$$ are obtained numerically and compared with the corresponding profiles for $$\Pr \rightarrow 0$$ , 3/4, and $$\infty $$ under the same initial conditions. Qualitative agreement is obtained with the theoretical and experimental results for the shock wave structure. The inverse shock thickness is computed for different values of $$\Pr $$ , and it is found that the inverse shock thickness increases with an increase in the Prandtl number. The bulk viscosity, the non-idealness parameter, the specific heat ratio, the power-law index, and the pre-shock Mach number have a significant effect on the shock wave structure.