CHAPTER 6

FLUID DYNAMICS

As discussed in Chapter 5, the dynamic response of MEMS is affected by energy dissipation involved in the system (i.e., viscous damping due to surrounding gases or structural damping dissipates the mechanical energy into heat). If microstructures operate in high vacuum where the energy dissipation due to fluid is much less than that due to structural damping, the viscous damping may be neglected. However, if the pressure of surrounding gas is not extremely low, the damping force of the gas must be taken into account. In this chapter we describe governing equations for gas and liquid flows and derive a damping coefficient from the governing equations.

6.1 VISCOUS FLOW

Let us begin with a simple problem, illustrated in Fig. 6.1, to understand the viscous flow of fluid (liquid and gas). In Fig. 6.1, a plate of area A is placed on top of an oil film of thickness h, and a force F is applied to move the plate at a constant velocity of U in the x direction. Experiments show that the driving force F is proportional to the velocity and the area and inversely proportional to the thickness; mathematically, this relation may be expressed by F ∝ UA/h or, more conveniently, the expression is written as

(6.1)

where μ represents the viscosity, which is a property of fluid. The shear stress acting on the plate is then given by

which was given by Isaac Newton. If a fluid ...