Frequency-Domain Formulation

In the frequency domain it is necessary to make additional simplifications to the equation system to produce a fully linearized equation set. In the general case, the physical quantities in the Reynolds and flow equations take the form:

Here the tilde denotes a harmonically varying component. The components marked with the subscript 1 are static offsets to the harmonic terms. In order to linearize the equation system, it is necessary to assume that the offsets are much larger than the harmonic components. Practically speaking this means that the harmonically varying pressures induced by the flow should be significantly smaller than the ambient pressure, and the harmonic changes in the gap size due to the wall and base displacements should be significantly less than the gap height itself. Given these assumptions, the average fluid velocity can also be written on the form:

since in general both the Couette and Poiseuille terms vary harmonically to first-order accuracy.

Substituting these terms into the Reynolds equation gives:

In the frequency domain, the total absolute pressure pA is set to the reference pressure pref because the deviation from the reference pressure pf in this case represents an oscillating component and cannot be added to the pressure in a straightforward manner. ρ is defined as a function of pA and so contains no small harmonic component. The products of small harmonic terms result in second-order effects (at double the frequency of interest) and can be neglected provided the harmonic terms are much smaller than the static terms. The Reynolds equation therefore reduces to:

Note that:

(4-15)

So the following result holds:

(4-16)

Dropping the tildes leads to the form of the equation shown in the equation display:

The modified Reynolds Equation takes the following form:

(4-17)

Linearizing Equation 4-17 gives:

Equation 4-15 can be used to substitute for the time derivative of the harmonic component of h in the above equation, yielding:

(4-18)

Dropping the tildes and using complex notation to express the time derivative of pf gives:

Equation 4-16 and Equation 4-18 are used as the basis for the frequency domain formulation in the Thin-Film Flow interfaces. It is important to note that these equations are independent of the harmonic components of the displacement, since those terms lead to second-order contributions to the frequency-domain response (that is, they produce a response proportional to the square of the harmonic term, which results in a signal at twice the driving frequency).


	Since the frequency domain results do not depend on the variation in the displacement, the additional displacement setting has no effect on the solution in a frequency-domain problem (unless the velocity is computed from it by selecting Calculate from wall displacement or Calculate from base displacement in the wall and base velocity settings, respectively). This is reflected in the equations given above (and in the equation display in the physics interface) but can be nonintuitive, particularly when setting up a model that is coupled with a structural analysis. If the velocity of both the wall and the base is set to zero, then in the frequency domain there is no response.