RESUMO
Vibrational modes of unrestrained elastic cylinders of trigonal crystals are studied using Ritz-based polynomial approximations for displacements formulated in rectangular Cartesian coordinates. The selected orientation of the threefold trigonal axis is perpendicular to the cylinder axis, corresponding to the configuration employed in torsional quartz viscometry (TQV) for characterizing Newtonian fluids. A revised working equation for TQV is derived, incorporating effects of crystalline anisotropy, and Ritz results are used to numerically quantify effects of acoustic radiation from surface-normal displacements and viscous loss from nontorsional surface-parallel displacements of resonant modes corresponding to the purely torsional modes of isotropic cylinders traditionally employed as an approximation in TQV analysis. For a cylinder typical of TQV, with 3 mm diameter and 50 mm length, the anisotropy-related correction to the extracted fluid viscosity is a positive shift of 36 ppm relative to the isotropic approximation, if radiative losses are neglected. This contribution is independent of fluid properties. Radiative losses depend on the properties of the fluid and reduce the extracted viscosity. The total magnitude of corrections varies between several tens of parts per million for low density gases to values on the order of 0.01% for normal liquids near atmospheric pressure and 0.06% for superfluid helium.
RESUMO
The viscosities of three pentaerythritol tetraalkanoate ester base oils and one fully formulated lubricant were measured with an oscillating piston viscometer in the overall temperature range from 275 K to 450 K with pressures up to 137 MPa. The alkanoates were pentanoate, heptanoate, and nonanoate. Three sensing cylinders covering the combined viscosity range from 1 mPa·s to 100 mPa·s were calibrated with squalane. This required a re-correlation of a squalane viscosity data set in the literature that was measured with a vibrating wire viscometer, with an estimated extended uncertainty of 2 %, because the squalane viscosity formulations in the literature did not represent this data set within its experimental uncertainty. In addition, a new formulation for the viscosity of squalane at atmospheric pressure was developed that represents experimental data from 169.5 K to 473 K within their estimated uncertainty over a viscosity range of more than eleven orders of magnitude. The viscosity of squalane was measured over the entire viscometer range, and the results were used together with the squalane correlations to develop accurate calibrating functions for the instrument. The throughput of the instrument was tripled by a custom-developed LabVIEW application. The measured viscosity data for the ester base oils and the fully formulated lubricant were tabulated and compared with literature data. An unpublished viscosity data set for pentaerythritol tetrapentanoate measured in this laboratory in 2006 at atmospheric pressure from 253 K to 373 K agrees with the new data within their experimental uncertainty and confirms the deviations from the literature data. The density data measured in this project for the three base oils deviate from the literature data in a way that is by sign and magnitude consistent with the deviations of the viscosity data. This points to differences in the sample compositions as the most likely cause for the deviations.
