Variational approach to gas flows in microchannels on the basis of the Boltzmann equation for hard-sphere molecules

Abstract

This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.The objective of the present paper is to provide an analytic expression for the first- and second-order velocity slip coefficients. Therefore, gas flow rates in microchannels have been rigorously evaluated in the near-continuum limit by means of a variational technique which applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator. The diffuse-specular reflection condition of Maxwell’s type has been considered in order to take into account the influence of the accommodation coefficient on the slip parameters. The polynomial form of Knudsen number obtained for the Poiseuille mass flow rate and the values of the second order velocity slip coefficients found on the basis of our variational solution of the linearized Boltzmann equation for hardsphere molecules are analyzed in the frame of potential applications of classical continuum numerical tools (as lattice Boltzmann methods) in simulations of microscale flows