The equivalence between BE-LSE and NS-LSEs under continuum assumption

Abstract

The linear stability theory, which is based on the Navier–Stokes equations, has been extensively studied in the field of hydrodynamic stability. However, due to the continuum assumption, there exist various limitations when attempting to solve problems that involve rarefaction effects. In contrast, based on kinetic theory, the Boltzmann equation is well-suited for analyzing linear stability throughout the entire regime, ranging from continuous to rarefied flows. In this paper, a linear stability equation derived from the Boltzmann–Bhatnagar–Gross–Krook (Boltzmann-BGK) equation is introduced, and the relationship between small perturbations of the velocity distribution function and macroscopic physical variables is established. Under the continuum assumption, the linear stability equation based on the Boltzmann equation (BE-LSE) and the linear stability equations based on the Navier–Stokes equations (NS-LSEs) have the same numerical solutions, which indicates that the microscopic BE-LSE can recover to macroscopic NS-LSEs theoretically. Nevertheless, there is still a lack of mathematical proof for this theoretical outcome. To address this issue, the research on the equivalence of BE-LSE and NS-LSEs under continuum assumption is carried out in this paper, which can establish the theoretical relationship between BE-LSE and NS-LSEs. These efforts lay a solid theoretical foundation for stability research based on the BE-LSE.