The Reynolds Equation

The following two equations are the time dependent forms of the Reynolds equation and the modified Reynolds equation (Ref. 1, Ref. 2, and Ref. 3) written using the mean film velocity u. Both equations solve for the film-pressure variation, pf, in the gap:

In these equations, h h0 h( t) is the air gap’s height consisting of the initial gap and the deformation in the normal direction of the boundary, denotes the fluid density, and p pa pf is the total fluid pressure consisting of the initial/ambient pressure and the variation. Also, hs is the location of the solid wall, hb the location of the channel base and us and ub define the tangential velocity of the solid wall and the channel base, respectively. The subscript t in t refers to the fact that this equation is solved on a boundary and that all gradients are computed using the tangential derivatives along the boundary. The mean film velocity u is given by

where denotes the fluid viscosity at normal conditions, and the term Qch is the relative flow rate function that accounts for the rarefied gas effects (see Rarefaction and Slip Effects).

For frequency response or eigenfrequency analysis of a microsystem, the nonlinear equations Equation 5-1 and Equation 5-2 are linearized by the general linearization functionality in the corresponding solvers. The linearization is calculated at a linearization point which may be a zero solution or any previous static or transient solution of the coupled Solid Mechanics—Film Damping/Lubrication problem.

For static analysis, the terms containing time derivatives disappear from Equation 5-1 and Equation 5-2. Static analysis is mostly of interest in lubricating applications.

Equation 5-1 and Equation 5-2 give two representation of the full form of the Reynolds’ equation with both normal and tangential motions. For squeezed-film damping (Ref. 4, Ref. 5, and Ref. 6), a common assumption is that the tangential

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velocity components are zero. Thus, the equations simplify by setting any appearance of us and ub to zero.

An important measure of the validity of the Reynolds equation to is the squeeze number (Ref. 1, Ref. 2):

12 a2

= ---------------------

pah02

where a is a typical horizontal dimension of the moving structure. In order for the Reynolds equation to properly describe the flow problem in the thin gas film, must be much smaller the one: 1.

Structural Loads

The boundary adjacent to the fluid film experiences forces in normal and tangential directions. The normal load results directly from the pressure load pf (assuming that the ambient pressure surrounds the body elsewhere) where Fn npf where n is the outward pointing normal of the structure.

The tangential load results from the viscosity of the fluid and the velocity distribution over the film thickness. The validity of the Reynolds equation assumes that the fluid has parabolic velocity profile over the film thickness. Thus, rarefied gas the load expression is:

Here us and ub are the tangential velocities of the boundary, and channel base, respectively.

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