Squeezing flow between rigid tilted surfaces: A general solution and case study for MEMS

Abstract

Squeeze film flow occurs when two surfaces move in a normal direction relative to each other and is a phenomenon of importance to many engineering systems, from macro to microscale. Squeeze film damping is widely used in large‐scale rotating machinery but even more so presently in microsystems. In the latter case, for modelling purposes, the two surfaces producing the squeeze film flow are typically assumed perfectly parallel, which is often not the case in practice. This paper presents a general formula for squeezing flow between two rigid surfaces for both parallel and tilted configurations in the 1‐dimensional case (2‐dimensional flow). The solution is derived from the Reynolds equation. The results in the parallel case compare favorably to previous literature data. A case study is presented for plates with dimensions characteristic of microelectromechanical systems. The important contribution of this paper is to isolate and study this “tilt effect” which can contribute to discrepancies and confusion in interpreting squeeze film behaviour, particularly at the microscale.

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“…The solution to the air flow problem is derived from Reynolds equation in literature. A general formula is available in [23] to calculate the squeeze film damping force between two rigid surfaces with tilt configuration for incompressible fluid. The damping force between the left electrode pair is shown in equation ( 23), where η is the air viscosity, T f is the device layer thickness, L f is the electrode length, α is the tilt angle and g b and g t are the nominal gaps between electrodes at bottom and top, respectively.…”

“…The damping force between the left electrode pair is shown in equation ( 23), where η is the air viscosity, T f is the device layer thickness, L f is the electrode length, α is the tilt angle and g b and g t are the nominal gaps between electrodes at bottom and top, respectively. The dimensional parameters A 1_l and A 2_l are given by equations ( 24) and ( 25) and are used to compact the force expression equations (23). The damping force for the right electrode pair can be derived as shown in equations ( 26)- (28).…”

“…The governing equations for the system are shifted between the two modes based on the relative displacement between the movable and fixed electrodes. The governing equations for the mechanical domain are solved simultaneously with the governing equation for the electrical domain, being linked by the varying charge flowing through the capacitor, which depends on the electrode velocity and displacement [14,23]. Because the electrodes considered have tilted walls, this model also considers the effect of nonuniform gap for both the squeeze film damping [23] and the electrostatic forces acting between the electrodes.…”