Gas Outflow Conditions

When the system’s aspect ratio is very large, a valid assumption is that the film-pressure

variation vanishes at the end of the channel. Thus, the corresponding end condition is pf 0.

If the aspect ratio is not very large, this condition is no longer valid, and the pressure variation continues outside of the gap. For these cases, a boundary flow condition (Ref. 5) can be used,

where the elongation L is the distance where pf goes to zero outside the boundary assuming that the pressure drop is linear. The elongation L can be a constant or it can be relative to the gap:

Here, nch is the outflow normal from the flow channel, as shown in Figure 5-1.

Table 5-2 summarizes the relative boundary elongation values for linear and torsional dampers from Ref. 5. This data especially concerns the squeezed-film damping.

TABLE 5-2: RELATIVE ELONGATION FOR DIFFERENT ASPECT RATIONS

Rarefaction and Slip Effects

M E A S U R E S O F R A R E F A C T I O N

There are several ways to identify the rarefaction level of the gas in the gap. The Knudsen number in the gap, Kn, is given by

Kn = -- h

178 | C H A P T E R 5 : T H I N - F I L M F L O W B R A N C H

where is the mean free path of the gas molecules at pressure pa pf, and h is the local gap height. is defined by

where 0 is the mean free path defined at constant pressure p 0 .

For liquids Kn can be seen as effective Knudsen number calculated using the slip length

Ls.

The scaled Knudsen number, Ks, is a product of the slip coefficient, p, and Knudsen number, Kn (Ref. 5):

Ks = P Kn

The slip coefficient, p, is defined by

where v is the tangential momentum accommodation coefficient (TMAC). For rough surfaces with diffuse molecular reflection v 1, but for polished surfaces v

1. In this example assume that both boundaries of the gap have the same v value or that v represents their average effect.

Another measure of the gas rarefaction is the scaled inverse Knudsen number, D (see Ref. 7):

D = -----------

2Kn

When modeling with time-dependent analysis, these parameters are always computed for the system’s instantaneous state, but for frequency-response analysis these parameters are constants and correspond to the initial conditions: h 0, pf 0.

R E L A T I V E F L O W R A T E

The relative flow rate function, Qch, describes how the viscous flow within the narrow gap changes when the continuum assumption is not valid or if the no-slip condition for the flow is not the best assumption. The following list gives four alternative formulations of the relative flow function.

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