Knudsen layer

Abstract: Derivation of the lattice Boltzmann method from the continuous kinetic theory [X. He and L. S. Luo, Phys. Rev. E 55, R6333 (1997); X. Shan and X. He, Phys. Rev. Lett. 80, 65 (1998)] is extended in order to obtain boundary conditions for the method. For the model of a diffusively reflecting moving solid wall, the boundary condition for the discrete set of velocities is derived, and the error of the discretization is estimated. Numerical results are presented which demonstrate convergence to the hydrodynamic limit. In particular, the Knudsen layer in the Kramers' problem is reproduced correctly for small Knudsen numbers.

Abstract: Accurate modeling of gas microflow is crucial for the microfluidic devices in MEMS. Gas microflows through these devices are often in the slip and transition flow regimes, characterized by the Knudsen number of the order of 10−2~100. An increasing number of researchers now dedicate great attention to the developments in the modeling of non-equilibrium boundary conditions in the gas microflows, concentrating on the slip model. In this review, we present various slip models obtained from different theoretical, computational and experimental studies for gas microflows. Correct descriptions of the Knudsen layer effect are of critical importance in modeling and designing of gas microflow systems and in predicting their performances. Theoretical descriptions of the gas-surface interaction and gas-surface molecular interaction models are introduced to describe the boundary conditions. Various methods and techniques for determination of the slip coefficients are reviewed. The review presents the considerable success in the implementation of various slip boundary conditions to extend the Navier–Stokes (N–S) equations into the slip and transition flow regimes. Comparisons of different values and formulations of the first- and second-order slip coefficients and models reveal the discrepancies arising from different definitions in the first-order slip coefficient and various approaches to determine the second-order slip coefficient. In addition, no consensus has been reached on the correct and generalized form of higher-order slip expression. The influences of specific effects, such as effective mean free path of the gas molecules and viscosity, surface roughness, gas composition and tangential momentum accommodation coefficient, on the hybrid slip models for gas microflows are analyzed and discussed. It shows that although the various hybrid slip models are proposed from different viewpoints, they can contribute to N–S equations for capturing the high Knudsen number effects in the slip and transition flow regimes. Future studies are also discussed for improving the understanding of gas microflows and enabling us to exactly predict and actively control gas slip.

Abstract: A three-dimensional laser-keyhole welding model is developed, featuring the self-consistent evolution of the liquid/vapor (L/V) interface together with full simulation of fluid flow and heat transfer. Important interfacial phenomena, such as free surface evolution, evaporation, kinetic Knudsen layer, homogeneous boiling, and multiple reflections, are considered and applied to the model. The level set approach is adopted to incorporate the L/V interface boundary conditions in the Navier-Stokes equation and energy equation. Both thermocapillary force and recoil pressure, which are the major driving forces for the melt flow, are incorporated in the formulation. For melting and solidification processes at the solid/liquid (S/L) interface, the mixture continuum model has been employed. The article consists of two parts. This article (Part I) presents the model formulation and discusses the effects of evaporation, free surface evolution, and multiple reflections on a steady molten pool to demonstrate the relevance of these interfacial phenomena. The results of the full keyhole simulation and the experimental verification will be provided in the companion article (Part II).