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Higher-order moment theories for dilute granular gases of smooth hard spheres

Published online by Cambridge University Press:  12 December 2017

Vinay Kumar Gupta Open the ORCID record for Vinay Kumar Gupta [Opens in a new window] ,
Priyanka Shukla Open the ORCID record for Priyanka Shukla [Opens in a new window]  and
Manuel Torrilhon
Vinay Kumar Gupta*
Affiliation:
SRM Research Institute and Department of Mathematics, SRM Institute of Science and Technology, Chennai 603203, India
Priyanka Shukla*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
Manuel Torrilhon
Affiliation:
Center for Computational Engineering Science, Department of Mathematics, RWTH Aachen University, Schinkelstr. 2, D-52062 Aachen, Germany
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

Grad’s method of moments is employed to develop higher-order Grad moment equations – up to the first 26 moments – for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff’s law while the other higher-order moments decay on a faster time scale. The nonlinear terms of the fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have a negligible effect on Haff’s law. Furthermore, an even larger Grad moment system, which includes the fully contracted sixth moment, is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of the scalar sixth-order moment into the Grad moment system has a negligible effect on Haff’s law. The constitutive relations for the stress and heat flux (i.e. the Navier–Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via the Chapman–Enskog expansion and via computer simulations. The linear stability of the homogeneous cooling state is analysed through the Grad 26-moment system and various subsystems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution, while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber, in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment theory is in excellent agreement with that obtained through existing theories.

JFM classification

Granular media: Granular media Rarefied Gas Flow: Kinetic theory Rarefied Gas Flow: Rarefied Gas Flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 451 - 501
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Copyright
© 2017 Cambridge University Press 

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