Theory of the Greenspan viscometer

We present a detailed acoustic model of the Greenspan acoustic viscometer, a practical instrument for accurately measuring the viscosity η of gases. As conceived by Greenspan, the viscometer is a Helmholtz resonator composed of two chambers coupled by a duct of radius $r_d.$ In the lowest order, ${\mathrm{\eta =\pi f\rho (r}}_d{\mathrm{/Q)}}^2,$ where f and Q are the frequency and quality factor of the isolated Greenspan mode, and ρ is the gas density. In this level of approximation, the viscosity can be determined by measuring the duct radius and frequency response of the resonator. In the full acoustic model of the resonator, the duct is represented by a T-equivalent circuit, the chambers as lumped impedances, and the effects of the diverging fields at the duct ends by lumped end impedances with inertial and resistive components. The model accounts for contributions to $\text{1/Q}$ from thermal dissipation (primarily localized in the chambers) and from a capillary used for filling and evacuating the resonator. A robust, prototype instrument is being used for measuring the viscosity of reactive gases used in semiconductor processing. For well-characterized surrogate gases, the prototype viscometer generated values of η that were within ±0.8% of published reference values throughout the pressure range 0.2–3.2 MPa. Remarkably, we achieved this level of agreement by only slight adjustment of the numerically calculated inertial and resistive end effect parameters to improve the agreement with helium reference values. No other parameters were adjusted.