# Interdisciplinary Applied Mathematics

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(Section 3.3) and the corresponding viscous drag (damping) can be approximately analyzed using the oscillatory Couette flow model. In its simplest form, the governing equation is reduced to

du

~dt

d2u

r-—

dy

(6.1)

where v is the kinematic viscosity, and u is the streamwise velocity. For oscillatory flows with a specified frequency ш0 and corresponding boundary conditions u = Uo exp (iu0t) and u = 0 on the oscillating and stationary surfaces, respectively, an analytic solution of the above equation can be found in (Panton, 1984; Sherman, 1990). The results can be summarized as a function of the normalized penetration distance given by

Y = —r=-

V v/ш

If the normalized penetration distance between the two walls is Y > 7, then the plates are sufficiently away from each other, and thus they do not interact. Equation (6.1) can also be solved using various slip conditions. A study of damping models for laterally moving microstructures with gas rarefaction effects can be found in Section 3.3 and in (Veijola, 2000; Park et al., 2004).

###### 6.1.1 Reynolds Equation

Background material for thin-film lubrication theory can be found in (Sz-eri, 1998). For squeezed film or slider-type applications, the Navier-Stokes equations are reduced to the Reynolds equation (Sherman, 1990):

where p and p are the local gas density and pressure, respectively. The local film thickness is h, p is the dynamic viscosity, t is time, and Uo is the lateral velocity of the moving plate. The first and second terms on the right-hand side correspond to the normal and lateral plate motions, respectively.

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