Interdisciplinary Applied Mathematics

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Derivation of the Reynolds Equation

The Reynolds equation is derived starting from the Navier-Stokes equations presented in Chapter 2. For simplicity, we analyze two plates with a small gap between them. The upper plate is placed at a slight angle with respect to the lower plate, while the latter is moving from left to right with velocity Uo, as shown in Figure 6.3. For simplicity, we assume that the plate length L is much larger than the plate separation distance hand that the angle of inclination a is small. We also assume that the plate width W in the spanwise direction is larger than ho, and that the plate motion is solely in the x direction, as shown in Figure 6.3. Therefore, the flow is two-dimensional in the (x, y) plane. Details of the derivation of the Reynolds equation can be found in (Panton, 1984). Here, we briefly outline this derivation and discuss the underlying assumptions and limitations.

The ratio of the inertial forces to the viscous forces in the Navier-Stokes equations is given by

pudu/dx ^ pYl/L _ pl0L / hQ _ pd2u/dy2    pJQ/h2 p у L )

Therefore, the inertial forces can be neglected with respect to the viscous forces if R* = Re • <C 1. In most MEMS applications, due to the very small Reynolds number (Re = pU*L) of the flow and the small h0/L ratio, this condition is satisfied. A comparison of the streamwise and the crossflow momentum equations reveals that pressure variation across the channel is very small compared to the pressure variation along the streamwise direction. Similarly, the cross-flow velocity (v) is negligible compared to the streamwise velocity component (u). Hence, the leading-order steady-state solution is determined from the streamwise momentum equation simplified in the following form:

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