RESEARCH PAPER I. A. Graur J. G. Me´olans D. E. Zeitoun Analytical and numerical description for isothermal gas flows in microchannels Received: 24 December 2004 / Accepted: 6 July 2005 / Published online: 14 October 2005 Ó Springer-Verlag 2005 Abstract Analytical solutions for the pressure and the velocity profiles in a microchannel are derived from the quasi gasdynamic equations (QGD). An expansion method according to a small geometric parameter e is undertaken to obtain the isothermal flow parameters. The deduced expression of the mass flow rate is similar to the analytical expression obtained from the Navier- Stokes equations with a second order slip boundary condition and gives results in agreement with the mea- surements. The analytical expression of the pressure predicts accurately the measured pressure distribution. The effects of the rarefaction and of the compressibility on pressure distributions are discussed. The numerical calculations based on the full system of the QGD equations were carried out for different sizes of the microchannels and for different gases. The numerical results confirm the validity of the analytical approach. 1 Introduction A systematic research effort in micro mechanics devices started in the late 1980’s. Microducts, micronozzles, mi- cropumps, microturbines and microvalves are the exam- ples of small devices involving liquid and gas flows. The characteristic length scale of these devices is less than 1 mm but more than 1 l. That means, that even at the atmospheric conditions, the ratio of the mean-free path to the characteristic dimension can not be neglected and in the flow dynamic associated with microelectromechanical systems (MEMS) the rarefied gas phenomena become apparent. If a significant pressure drop appears in the channel, the compressibility effects will be also exhibited. In the last 15 years, various calculation methods were developed and applied to microfluidic gas flows, gener- ally considered as isothermal. Kinetic discrete methods solving the Boltzmann equation [1], DSMC and IP methods [2], and lattice Boltzmann methods [3] allowed to investigate the isothermal flows in a large Knudsen number spectrum. But whatever the efficiency of these methods, the need for qualitative knowledge about the significant parameters governing the rarefied microflows appears dominant today as also the problem of choosing pertinent boundary conditions on the solid surface. The analytical approaches are generally more suited to clarify such questions. Therefore this article is mainly devoted to explore and extend the validity domain of the continuum equations and to derive analytical methods from it. The analytical investigation on the microchannel flows has been opened by the work of Prud’homme et al. [4] who have presented a one-dimensional solution of the Navier-Stokes (NS) equations based on the perturbation analysis. The rarefaction effects were not taken into account but the significance of the compressibility on the pressure distribution was shown. The work of van den Berg et al. [5] was undertaken in circular capillaries to analytically derive the nonlinearity of the pressure dis- tribution from the hydrodynamic conservation equa- tions. In addition, the negligible value of the radial velocity component was pointed out. Harley et al. [6] presented experimental, analytical and numerical meth- ods to investigate microflows with a trapezoidal cross- section, based on the two-dimensional NS equations and on the isothermal flow assumption. The authors found that the pressure may be assumed to be uniform in the conduit cross-sections perpendicular to the flow direc- tion and again that the transverse velocity can by ne- glected. In their study, the influence of the slip-velocity boundary conditions was taken into account. A more detailed investigation of microflows was performed by Arkilic et al. [7]. These authors used a perturbation analysis of the full compressible two dimensional NS equations in Cartesian coordinates involving a first I. A. Graur (&) J. G. Me´olans D. E. Zeitoun Universite de Provence-Ecole Polytechnique Universitaire de Marseille, UMR CNRS 6595, 5 rue Enrico Fermi, 13453 Marseille, France E-mail: irina.graour@polytech.univ-mrs.fr Microfluid Nanofluid (2006) 2: 64–77 DOI 10.1007/s10404-005-0055-6