Evaluation of Grad's Second Problem Using Different Higher Order Continuum Theories

Abstract

In our earlier work (Jadhav, and Agrawal, 2020, “Grad's second problem and its solution within the framework of Burnett hydrodynamics,” ASME J. Heat Transfer, 142(10), p. 102105), we proposed Grad's second problem (examination of steady-state solution for a gas at rest upon application of a one-dimensional heat flux) as a potential benchmark problem for testing the accuracy of different higher order continuum theories and solved the problem within the framework of Burnett hydrodynamics. In this work, we solve this problem within the moment framework and also examine two variants, Bhatnagar–Gross–Krook (BGK)–Burnett and regularized 13 moment equations, for this problem. It is observed that only the conventional form of Burnett equations which are derived retaining the full nonlinear collision integral are able to capture nonuniform pressure profile observed in case of hard-sphere molecules. On the other hand, BGK–Burnett equations derived using BGK-kinetic model predict uniform pressure profile in both the cases. It seems that the variants based on BGK-kinetic model do not distinguish between hard-sphere and Maxwell molecules at least for the problem considered. With respect to moment equations, Grad 13 and regularized 13 moment equations predict consistent results for Maxwell molecules. However, for hard-sphere molecules, since the exact closed form of moment equations is not known, it is difficult to comment upon the results of moment equations for hard-sphere molecules. The present results for this relatively simple problem provide valuable insights into the nature of the equations and important remarks are made in this context.