Time-scale identification and stretched-exponential modeling of flow-energy transients in a suction-type air bearing under gap-height steps

Levitation stages equipped with suction orifices enable contactless transport of fragile substrates; however, variations in the gap height render the thin-film flow unsteady, and transient variations in the bearing pressure and shear fields critically affect attitude stability. In this study, we perform direct numerical simulations of the incompressible Navier–Stokes equations with the convective terms omitted (unsteady Stokes formulation) to quantify the transient response induced by stepwise increases and decreases in the gap height. A periodic unit cell comprising one centrally located jet inlet and four corner suction outlets is analyzed at Re = 200 and 300, and the cell-averaged kinetic energy and the root-mean-square spatial variation of static pressure are tracked. The relaxation component of the kinetic energy is described by a rate kernel f eff ( t ) = ( 1 / τ ) ( τ / t ) m with non-dimensional time t , and a stretched-exponential response with m = 0 . 38 provides a good approximation across all conditions. The time constant τ collapses when normalized by the momentum-diffusion time scale, obtained by dividing the square of the levitation height by the kinematic viscosity. The pressure-fluctuation RMS falls below its steady-state value during transients associated with step increases, indicating an asymmetric response of pressure fluctuations to gap-height variations. These results provide concise and practical metrics for specifying admissible operating bandwidths.