Study of the Onsager's-reciprocity-principle-consistent approach used in the derivation of higher-order transport equations

Failure of the Navier-Stokes equations leads to utilizing the Chapman-Enskog and Grad moment method to propose higher-order transport equations to extend the applicability of continuum formulations into the continuum-transition regime. However, these equations based on only mathematical principles failed to cover the whole transition regime of flow due to the non-consideration of the non-equilibrium thermodynamics in their derivation process. That is why the derivation of higher-order transport equations is still an essential topic in the research field. In the present work, we propose to incorporate the Onsager's reciprocity principle, a cornerstone of linear irreversible thermodynamics, into our derivation process. We focus on the derivation procedure of higher-order transport starting from the Boltzmann equation of the Kinetic theory. Two different relaxation times have been used to ensure the correct value of  Prandtl number. For their closure, the derivation procedure involves the evaluation of unknown higher-order tensors present in the evolution equations for stress tensor and heat flux vector and constitutive expressions for stress and heat flux vector for the Grad-like and Burnett-like equations, respectively. Using this approach, we have proposed new higher-order transport equations known as OBurnet and O13  equations. These O13 equations are shown to be unconditionally stable for any wavelength and frequency and consistent with Onsager's symmetry principle and H-theorem. The OBurnett are reported to be stable for any disturbances and produce smooth shock structures, showing the existence of heteroclinic trajectory and positive entropy generation inside the shock at all Mach numbers. Therefore,   transport equations obtained using this approach are expected to cover a larger envelope of Knudsen number. This work will present these exciting equations and the results obtained by solving them for various benchmark problems.