# Interdisciplinary Applied Mathematics

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Navier-Stokes equations, e.g., the Reynolds squeezed film equation, which requires considerably less computational cost. Reynolds squeezed film equation (see Section 6.1) is typically applicable when a small gap between the two plates/structures opens and closes with time. This assumption holds for structures where the seismic mass moves perpendicular to a fixed wall, for plates with tilt around horizontal axes, and for clamped beams where the flexible    part    moves    against    a fixed    wall.    Some    examples    of    MEM    de

vices where the Reynolds equation is valid are fixed-fixed beams, cantilever beams, and micromirrors.

The nonlinear isothermal compressible Reynolds squeezed film equation for air damping with slip flow is (see Section 6.1)

V- [(1 + 6Kn)h3pVp]

###### 12p

where h(x, y, t) is the variable gap between the movable part and the ground electrode of the MEM device, p(x, y, t) is the air pressure under the beam, Kn(x,y,t) = X/h is the Knudsen number, where A is the mean free path of air. Figure 18.22 shows a typical MEM device, a deformable beam/plate at a height h(x, y, t) over a ground plane, which in the undeformed state is the initial gap between beam and the ground plane. The shaded region (on the xy plane) is the domain where the Reynolds equation is solved with the boundaries indicated in Figure 18.22. Depending on the example considered, the boundary conditions can change. For a mirror, the fluid system is assumed to be open (ambient pressure) on all sides, whereas for a fixed-fixed beam, the fluid system is open along the sides of the beam and closed (no flow) at the ends of the beam. Squeezed film damping in MEMS is a coupled phenomenon (mechanics, electrostatics, and fluidics). In order to obtain a self-consistent solution at any time instant, an iterative scheme has to be followed (e.g., a relaxation scheme) among the three domains. Considerable amount of work has been done in the reduced-order modeling of squeezed film damping in MEMS. They fall into the categories already discussed in Chapter 17, namely, equivalent circuit models, Galerkin methods, description language models/mixed-level simulation, and black box models.

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