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 micro channels over slip and transition regimes without the use of any fitting parameter. Previously, Sone [1] 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

Molecular dynamics studies of anomalous transport in rarefied gas flows

We investigate the thermodynamically non-equilibrium gas dynamics by measuring molecular free path distribution functions, inter-molecular collision rates and wall dependent mean free path (MFP) profiles using the molecular dynamics (MD) method. The simulations cover a wide range of fluid densities for single-wall case, parallel walls cases and a cube with all periodic walls. The simulations are validated by deducing the theoretical unconfined MFP values at standardpressure and temperature conditions. The free path MD measurements of individual molecules convey that conventional exponential distribution function is not valid under rarefied conditions and molecules follow L´evy type flights, irrespective of the presence of a wall. MFP profile measurements for confined planar surfaces in the transition flow regime show sharp gradients close to the wall, while theoretical models predict shallower gradients. As gas transport properties can be related to the MFP through kinetic theory, our MD data may help to modify the constitutive relationships, which may then be fed into the Navier-Stokes equations for better effective modeling of micro gas flows in the transition flow regime

Behaviour of microscale gas flows based on a power-law free path distribution function

We investigate a power-law form for the probability distribution function of free paths of dilute gas molecules in a confined region. A geometry-dependent effective molecular mean free path (MFP) model is then derived for a planar wall confinement, by taking into account the boundary limiting effects on the molecular paths. The power-law based effective MFP is validated against molecular dynamics simulation data and compared with exponential effective MFP models. The Navier-Stokes constitutive relations are then modified according to the kinetic theory of gases i.e. transport properties can be described in terms of the free paths which the molecules describe between collisions. Results for isothermal pressure-driven Poiseuille gas flows in micro-channels are reported, and we compare results with conventional hydrodynamic models, solutions of the Boltzmann equation and experimental data

A Note on the Dimensional Reduction of Axisymmetric Spacetimes

We investigate the dimensional reduction of 3+1 vacuum axisymmetric Einstein's equations to 2+1 dimensional Einstein-wave map system and observe that the resulting system is 1) not asymptotically flat, 2) its geometric-mass diverges and 3) the energy of wave map also diverges. Subsequently, we discuss the consequences of these issues