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Regularized extended-hydrodynamic equations for a rarefied granular gas and the plane shock waves

M. H. Lakshminarayana Reddy and Meheboob Alam*

  • Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India

  • *meheboob@jncasr.ac.in
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Phys. Rev. Fluids 5, 044302 – Published 6 April, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.044302

Abstract

The regularized versions of extended-hydrodynamic equations for a dilute granular gas, in terms of 10-, 13-, and 14-moments, are derived from the inelastic Boltzmann equation. The regularization is achieved by adding higher-order gradient terms that are obtained following a Chapman-Enskog-like gradient-expansion [H. Struchtrup, Stable transport equations for rarefied gases at high orders in the Knudsen number, Phys. Fluids, 16, 3921 (2004)]. For both granular and molecular gases, the resulting moment equations are found to be free from the well-known finite Mach-number singularity (that occurs in the Riemann problem of planar shock waves) since the regularized gradient terms yield parabolic equations in contrast to the hyperbolic nature of original moment equations. In order to clarify the advantage of these regularized equations, the 10-moment model for the plane shock-wave problem is solved numerically for both molecular and granular gases; the calculated hydrodynamic profiles compare favorably with previous simulation results for molecular gases. For a granular gas, both regularized and nonregularized equations predict asymmetric density and temperature profiles, with the maxima of both density and temperature occurring within the shock layer, and the hydrodynamic fields are found to be smooth for the regularized equations for all Mach numbers studied. It is demonstrated that, unlike in the case of molecular gases, a “second” regularization of the regularized equations must be carried out in order to arrest the unbounded growth of density within the shock layer in a granular gas.

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Phys. Rev. Fluids 5, 044302– Published 6 April, 2020
  • Received 29 November 2019
  • Accepted 5 February 2020
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