An exact analytical solution to the extended Navier - Stokes equations using the Lambert W function

Micro channel gas flows are of importance in a range Micro Electro Mechanical Systems (MEMS). Due to the confined geometry of the channel, the mean free path of the gas can be comparable to the characteristic length of the micro channel, leading to slip-like flow behavior and strong diffusion-enhanced transport of mass and momentum. The Extended Navier Stokes Equations (ENSE) have successfully been used to model the slip behavior in rarefied gas flows as well as quantitatively predict the mass flow rate through the channel, without introducing any ad hoc parameters. The model involves accounting for self-diffusion due to local pressure gradients, and decomposing the total transport as the additive sum of individual convective and diffusive terms. In this paper, we show that it is possible to obtain exact analytical solutions to the ENSE equation set for the pressure and velocity field using the Lambert W function. We find that diffusive contributions to the total transport are only dominant for low average pressures and low pressure drops across the micro channel. For large inlet pressures, we show that the expressions involving the Lambert function predict steep gradients in the pressure and velocity localized near the channel exit. Using a limit analysis, we extract a characteristic length for this boundary layer. Our analytical results are validated by numerical calculations as well as experimental results available in the literature.