Shock wave structure in non-ideal dilute gases under variable Prandtl number
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ORIGINAL ARTICLE
Shock wave structure in non-ideal dilute gases under variable Prandtl number D. Khapra1
· A. Patel1
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 considered as a discontinuity surface. But in the presence of viscosity and heat conduction, the thickness of the transition 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 thermal diffusion during a thermo-physical process is defined Communicated by D. Zeitoun.
B 1
A. Patel [email protected] 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 ther
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