The concept of mass-density in classical thermodynamics and the Boltzmann kinetic equation for dilute gases

In this paper we discuss the mass-density of gas media as represented in kinetic theory. It is argued that conventional representations of this variable in gas kinetic theory contradict a macroscopic field variable and thermodynamic property in classical thermodynamics. We show that in cases where mass-density variations exist throughout the medium, introducing the mass-density as a macroscopic field variable leads to a restructuring of the diffusive/convective fluxes and implies some modifications to the hydrodynamic equations describing gas flows and heat transfer. As an illustration, we consider the prediction of mass-density profiles in a simple heat conduction problem between parallel plates maintained at different temperatures

Wall temperature jump in polyatomic gas flows

This article deals with the calculations of the temperature jump at the wall for gas flows in the slip regime. The analytical calculations are based on kinetic boundary conditions developed especially for polyatomic molecules. When compared to an expression previously obtained for unstructured molecules, the polyatomic molecule temperature jump reveals supplementary terms of bulk viscosity type due to the internal mode excitation. These terms may be important in high speed flows or in gas flows displaying significant relative density variation at the wall

Temperature jump and slip velocity calculations from an anisotropic scattering kernel

This article deals with the problem of temperature jump and slip velocity at the wall in gas/surface interaction. A consistent modelling of an impermeable surface involving an anisotropic scattering kernel developed in previous works is used to establish boundary conditions in unstructured molecule gas flows. Thus a temperature jump relation is derived in which the gas viscous effects at the wall and the mean velocity gradients appear. Likewise, a slip velocity relation is obtained in which both the slip coeffcient and the thermal creep coeffcient depend on the wall-to-gas temperature ratio. Moreover, both the temperature jump and the slip velocity relations involve not only one accommodation coeffcient as in usual expressions, but also the gas/surface information through the various (notably normal and tangential) accommodation coeffcients of the momentum components

Transition regime analytical solution to gas mass flow rate in a rectangular micro channel

We present an analytical model predicting the experimentally observed gas mass flow rate in rectangular microchannels over slip and transition regimes without the use of any fitting parameter. Previously, Sone reported a class of pure continuum regime flows that requires terms of Burnett order in constitutive equations of shear stress to be predicted appropriately. The corrective terms to the conventional Navier-Stokes equation were named the ghost effect. We demonstrate in this paper similarity between Sone ghost effect model and newly so-called ‘volume diffusion hydrodynamic model’. A generic analytical solution to gas mass flow rate in a rectangular micro channel is then obtained. It is shown that the volume diffusion hydrodynamics allows to accurately predict the gas mass flow rate up to Knudsen number of 5. This can be achieved without necessitating the use of any adjustable parameters in boundary conditions or parametric scaling laws for constitutive relations. The present model predicts the non-linear variation of pressure profile along the axial direction and also captures the change in curvature with increase in rarefaction

Analysis of the thermomechanical inconsistency of some extended hydrodynamic models at high Knudsen number

There are some hydrodynamic equations that, while their parent kinetic equation satisfies fundamental mechanical properties, appear themselves to violate mechanical or thermodynamic properties. This article aims to shed some light on the source of this problem. Starting with diffusive volume hydrodynamic models, the microscopic temporal and spatial scales are first separated at the kinetic level from the macroscopic scales at the hydrodynamic level. Then we consider Klimontovich’s spatial stochastic version of the Boltzmann kinetic equation, and show that, for small local Knudsen numbers, the stochastic term vanishes and the kinetic equation becomes the Boltzmann equation. The collision integral dominates in the small local Knudsen number regime, which is associated with the exact traditional continuum limit. We find a sub-domain of the continuum range which the conventional Knudsen number classification does not account for appropriately. In this sub-domain, it is possible to obtain a fully mechanically-consistent volume (or mass) diffusion model that satisfies the second law of thermodynamics on the grounds of extended non-local-equilibrium thermodynamics

Scattering kernel for polyatomic molecules

A polyatomic scattering kernel phenomenologically presented in a previous article is derived from an integral operator formulation. The five parameters involved in the scattering kernel expression are shown to be equal to the accommodation coefficients of various fluxes at the wall, namely: the fluxes of the three components of the momentum and the fluxes of the rotational and vibrational energies of molecules. Under its present form the model is especially convenient for the diatomic molecules