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

Abstract

Microchannel gas flows are of importance in a wide range of microelectro mechanical devices. In these flows, the mean free path of the gas can be comparable to the characteristic length of the microchannel, leading to strong diffusion-enhanced transport of momentum. Numerical solutions to the extended Navier–Stokes equations (ENSE) have successfully modeled such microchannel flows. Analytical solutions to the ENSE for the pressure and velocity fields using the Lambert W function are derived. We find that diffusive contributions to the total transport are only dominant for low average pressures and low pressure drops across the microchannel. For large inlet pressures, we show that the expressions involving the Lambert W function predict steep gradients in the pressure and velocity localized near the channel exit. We extract a characteristic length for this boundary layer. Our analytical results are validated by numerical and experimental results available in the literature.