An analytical formula and FEM simulations for the viscous damping of a periodic perforated MEMS microstructure outside the lubrication approximation

The article presents an axi-symmetrical model for determining the total damping coefficient of a periodic perforated microelectromechanical systems (MEMS) microstructure for cases where the lubrication approximation yielding the Reynolds' equation is no longer appropriate. Damping is modeled by solving for the viscous flow in an approximation of the periodic cell geometry using an equivalent axi-symmetrical cylindrical flow domain. Using the Stokes approximation of the incompressible Navier- Stokes (N-S) equations with harmonic time behavior, an analytic solution is obtained yielding an explicit formula for the damping coefficient. Additionally, two numerical models were implemented based on the finite element method. Numerical solutions were obtained for the linearized, viscous acoustic equations (time harmonic) and the steady, incompressible N-S equations. The results given by the analytical formula compare well with those obtained from the FEM solutions and with measured values for MEMs microstructures found in the literature.