Microbearing gas flow modeling by fractional derivative for entire Knudsen number range

Abstract

Rarefied gas flow is encountered in many technical applications as well as in scientific inquiry. It may appear in low-pressure or vacuum environmental conditions and, on the other hand at standard atmospheric conditions. First category include gas flow in the devices used in hypersonic space vehicles and in several types of vacuum instruments, while the second category relates to the gas flow in micro/nano-electro-mechanical-systems (MEMS, NEMS) with characteristic dimension of the order of and . In these systems the ratio between the mean free path of the molecules and the characteristic length, which is defined as the Knudsen number (Kn), is not negligible and continuum approach breaks down. As a consequence gas slips along the wall and classical no-slip boundary conditions are no more valid. In the range , known as the slip flow regime, gas flow still obeys continuum i.e. Navier-Stokes equations, but now with slip boundary conditions at the walls. In the range (transitional flow regime) more complex Burnett equations have to be applied. The accuracy of the Burnett equations is of the order and they are solved under the boundary conditions of the same second order accuracy. Besides, the individual particle-based direct simulation Monte Carlo (DSMC) approach might be employed. Finally, for the gas flow is considered as a free molecular flow amenable to the methods of kinetic theory of gases. In this paper rarefied compressible two-dimensional gas flows in microbearings that are often part of MEMS and NEMS are treated. Instead of different approaches for slip velocity for the three rarefied gas flow regimes, slip at the boundaries is modeled by fractional derivative for the whole Knudsen number range. For this purpose a version of Caputo derivative is defined, with the order α defined as a function of the local value of the Knudsen number in the microbearing. For no-slip boundary conditions i.e. for continuum flow regime , while for free molecular flow when the Knudsen number approaches infinity . The correlation between and Kn is derived in the following way. The flow rate coefficient of Poiseuille flow is calculated for various Knudsen numbers by utilizing the numerical solution of the Boltzmann equation obtained by Fukui and Kaneko (1988). The obtained values of for specified Kn numbers are used for the derivation of the analytical relation between and Kn. Such a universal boundary condition that defines velocity at the wall for an arbitrary Knudsen number value is incorporated in the system of continuity and momentum equation, which leads to the general slip-corrected Reynolds lubrication equation. It is shown that it possesses the analytical solution which is obtained by a suitable transformation of the independent variable (Stevanovic and Djordjevic, 2012). It provides the mass flow rate as well as the pressure distribution in the microbearing for a specified bearing number , the reference Knudsen number at the exit cross section and the ratio of the inlet and exit microbearing height. The results for a wide range of the Knudsen number and the continuum flow conditions, obtained by the general analytical solution from this paper, are in excellent agreement with Fukui and Kaneko’s (1988) numerical solution of the Boltzmann equation.