Analytical analysis of the isothermal pressure-driven plane Poiseuille flow using the derived third-order super-OBurnett equations

In the past, higher-order transport equations that are a superset of the Navier-Stokes equations have been derived using the Chapman-Enskog and moment methods. The resulting Burnett and Grad equations were found to be plagued by several limitations, such as linear instability, inability to capture the Knudsen layer, and being limited to a small range of Knudsen number. The recently proposed Onsager's-principle-consistent approach, which is an alternative approach to the earlier two approaches, is significant in this context. Hence, it has been employed in the present work to derive the generalized additive-invariance-compliant and second-order accurate representation of the extended single-particle distribution function. This density function is further used to evaluate the constitutive relationships for the stress tensor and heat flux vector. Consequently, using these relationships, the closure of the super-OBurnett equations has been obtained, which provides third-order accuracy in terms of Knudsen number. The stability analysis conducted on the proposed equations reveals unconditional stability for two-dimensional flows across all wave numbers. Moreover, using this set of equations, we derive closed-form solutions for the pressure and velocity fields for the pressure-driven plane Poiseuille flow problem in a two-dimensional isothermal gaseous flow within a long microchannel. To validate our derived analytical solution, we compare the solution with direct simulation Monte Carlo (DSMC) results and observe a close agreement even in the transition flow regime. This suggests that the derived equations offer improved descriptions of flow physics by capturing the Knudsen layer, particularly for large Knudsen number. Moreover, this work holds significance as it successfully tackles benchmark problems that were previously challenging due to the complexity of higher-order transport equations.