An extension to the Navier-Stokes-Fourier equations by considering molecular collisions with boundaries

In this paper we propose a model for micro gas flows consisting of the Navier-Stokes-Fourier equations (NSF) extended by a description of molecular collisions with solid boundaries and discontinuous velocity slip and temperature jump boundary conditions. By considering the molecular collisions with the solid boundaries in gas flows we capture some of the near wall effects that the conventional NSF with linear stress/strain-rate and heat-flux/ temperature-gradient relationships seem to be unable to describe. The model that we propose incorporates the molecular collisions with solid boundaries as an extension to the conventional definition of the average travelling distance of molecules before experiencing intermolecular collisions (the mean free path). By considering both of these types of collisions we obtain an effective mean free path expression, which varies with distance to surfaces. The effective mean free path is proposed to be used to obtain new definitions of effective viscosity and effective thermal conductivity, which will extend the applicability of NSF equations to higher Knudsen numbers. We show results of simple flow cases that are solved using this extended NSF model and discuss limitations to the model due to various assumptions. We also mention interesting ideas for further development of the model based on a more detailed gas description

Conditions aux limites dans un gaz raréfié: loi de réflexion à la paroi, saut de température, vitesse de glissement, couche de Knudsen Boundary conditions in rarefied gas flows: scattering kernal, temperature jump, slip velocity, Knudsen layer problem

This thesis deals with the problem of gas/wall interaction and boundary conditions in rarefied gas flows. Recent developments in microsystems and atmospheric re-entry flight let appear new flow fields where boundary conditions are very important. These boundary conditions should be basically derived from gas kinetic theory. During this thesis, we developed a model of kinetic boundary conditions for unstructured and structured molecules gas flows in the gas surface interaction topic. The proposed kinetic boundary conditions were based on some mathematical integral formulations of the problem, supported by phenomenological descriptions. Then, the kinetic boundary conditions were used to describe hydrodynamic boundary conditions through the problem of temperature jump and slip velocity at the solid body. The Knudsen layer (which is a thin layer close to the wall) is also briefly described. Finally, the proposed kinetic boundary conditions are used in drag coefficient calculations, for higher altitude hypersonic flows in the free molecular regime, and in some particular flow predictions. Comparisons are made with other models and experiments

A Navier-Stokes model incorporating the effects of near-wall molecular collisions with applications to micro gas flows

We propose a model for describing surface effects on micro gas flows. This model consists of the Navier-Stokes equations (NS) with discontinuous velocity slip boundary conditions and a description of a geometry-dependent and effective viscosity due to special consideration of the molecular collisions with solid boundaries. By extending NS with an effective viscosity we obtain a non-linear stress/strain-rate relationship which captures some of the near-wall effects that the conventional NS are unable to describe. We show results of NS extended by using our effective viscosity applied with Maxwell's boundary condition as well as a second order boundary condition achieved by partly incorporating higher order methods, the Maxwell-Burnett boundary condition proposed by Lockerby et al. (2004). With this proposed model the simple isothermal planar channel case of 2D Poiseuille flow is solved. The results of our proposed model are compared with the conventional NS using similar boundary conditions, the BGK-method and experiments. On the one hand it is seen that our extended NS model yields results that are asymptotic to the results of conventional NS for large flow scales. On the other hand, when comparing results on the micro scale, we see that our extended NS model yields results that are closer to the results of the BGK-method and the experiments than the conventional NS. Our extended NS-model shows signs of capturing the physics of the flow to a certain rarefaction degree where it does not predict the mass flow minimum shown by the BGK-method and the experiments