It has been shown recently [A. Siria, A. Drezet, F. Marchi, F. Comin, S. Huant, and J. Chevrier, Phys. Rev. Lett. 102, 254503 (2009)] that in the plane-plane configuration, a mechanical resonator vibrating close to a rigid wall in a simple fluid can be overdamped to a frozen regime. Here, by solving analytically the Navier-Stokes equations with partial slip boundary conditions at the solid-fluid interface, we develop a theoretical approach justifying and extending these earlier findings. We show in particular that in the perfect-slip regime, the abovementioned results are, in the plane-plane configuration, very general and robust with respect to lever geometry considerations. We compare the results to those obtained previously for the sphere moving perpendicularly and close to a plane in a simple fluid and discuss in more details the differences concerning the dependence of the friction forces with the gap distance separating the moving object (i.e., plane or sphere) from the fixed plane. We show that the plane-plane geometry is more sensitive than the sphere-plane geometry for the measurement of slippage coefficients. Finally, we show that the submicron fluidic effect reported in the reference above, and discussed further in the present work, can have dramatic implications in the design of nanoelectromechanical systems.