Theoretical description of the gaseous Knudsen layer in Couette flow based on the second-order constitutive and slip-jump models

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The Knudsen layer, found in the region of gas flow very close (in order of a few mean free paths) to the solid surfaces, plays a critical role in accurately modeling rarefied and micro-scale gases. In various previous investigations, abnormal behaviors at high Knudsen numbers such as nonlinear velocity profile, velocity gradient singularity, and pronounced thermal effect are identified to exist in the Knudsen layer. However, some behaviors, in particular, the velocity gradient singularity near the surface and higher temperature, remain elusive in the continuum framework. In this study, based on the second-order macroscopic constitutive equation recently derived from the kinetic Boltzmann equation via the balanced closure and cumulant expansion [R. S. Myong, “On the high Mach number shock structure singularity caused by overreach of Maxwellian molecules,” Phys. Fluids 26(5), 056102 (2014)], the macroscopic second-order constitutive and slip-jump models that are able to explain qualitatively all the known non-classical and non-isothermal behaviors are proposed. As a result, new analytical solutions to the Knudsen layer in Couette flow, in conjunction with the algebraic nonlinearly coupled second-order constitutive and Maxwell velocity slip and Smoluchowski temperature jump models, are derived. It was shown that the velocity gradient singularity in the Knudsen layer can be explained within the continuum framework, when the nonlinearity of the constitutive model is morphed into the determination of the velocity slip in the nonlinear slip and jump model. Also, the smaller velocity slip and shear stress are shown to be caused by the shear-thinning property of the second-order constitutive model, that is, vanishing effective viscosity at high Knudsen number.