Shock Waves https://doi.org/10.1007/s00193-020-00972-x ORIGINAL ARTICLE Shock wave structure in non-ideal dilute gases under variable Prandtl number D. Khapra 1 · A. Patel 1 Received: 27 April 2020 / Revised: 4 September 2020 / Accepted: 13 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 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 →∞ and Pr → 0 limits, and the corresponding profiles for velocity and temperature are obtained. For Pr → 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 → 0, 3/4, and ∞ 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. Keywords Shock wave · Navier–Stokes–Fourier model · Non-ideal gas · Prandtl number 1 Introduction A shock wave is a transition zone across which the flow variables such as velocity, density, pressure, temperature, and entropy of the medium undergo a rapid change when a pressure front moves at supersonic speed. When there are no dissipative effects in the medium, such as viscosity and heat conduction, this transition layer is very thin and is con- sidered as a discontinuity surface. But in the presence of viscosity and heat conduction, the thickness of the transi- tion layer or shock wave is considerable and that requires the mathematicians to investigate the internal structure of shock waves. The internal structure of shock waves depends on the inherent dynamics between momentum diffusion and thermal diffusion. The ratio of momentum diffusion to ther- mal diffusion during a thermo-physical process is defined Communicated by D. Zeitoun. B A. Patel arvindpatelmath09@gmail.com 1 Department of Mathematics, University of Delhi, Delhi, India as Prandtl number (Pr) [1]. It is a dimensionless number depending only upon the characteristic of the fluid and was introduced by Ludwig Prandtl around 1910. For air, at room temperature, Pr = 0.71 and most common gases have similar values. Small values of the Prandtl number (Pr ≪ 1) mean that thermal diffusivity dominates, whereas large values (Pr ≫ 1) imply that momentum diffusivity dominates the behavior. Thus, Pr ≈ 0.02 for mercury means that in mer- cury heat conduction is more pronounced than viscous effects while oils generally have a large Prandtl number (≈ 10,000), making convection the predominant means of transferring energy as compared to pure conduction. Thus, taking Pr as a variable, the shock wave structure for all fluids can be deter- mined for a given equation of state. Owing to the wide application of shock wave structure in many physical problems such as molecular dissociation, vibrational excitation, and high-temperature gas dynamics, there has been a lot of experimental as well as theoretical work done by mathematicians to study the internal structure of shock waves. Riemann first introduced the fundamen- tal concept of shock waves in the mid-nineteenth century. Becker [2] provided an exact analytical solution for the pla- 123