Inertial contributions to the pressure in the truncated-cone-and-plate apparatus with application to dilute polymer solutions

New measurements of the pressure distribution generated by two Newtonian liquids in the Truncated Cone-and-Plate Apparatus are presented, in order to evaluate the exact form of the inertial contribution for a range of Reynolds numbers (Re) fromRe = 140 toRe = 36,000;Re = R ² /, where and are the liquid density and viscosity respectively,R is the plate radius, and is the angular velocity of the cone. The Walters equation for lowRe, p ^w = 0.15 ² (r² – R²), is shown to be in excellent agreement with the measurements up toRe = 1000, provided an appropriate correction for the Newtonian hole pressure is made. Up toRe = 1000, the measured slope is within 1% of the theoretical value of 0.15 given by the Walters equation; as the Reynolds number increases above 1000, the data become increasingly nonlinear inr ². Other theoretical predictions made especially for largeRe begin to disagree with the data even belowRe = 1000. The application of the experimentally determined additive inertial contribution to measurements of pressure distribution in four dilute polymer solutions is found to reproduce adequately the expected form of the viscoelastic pressure distribution, even at highRe where the Walters equation is not valid. Measurements of a combination of normal-stress differencesN ₁ + 2N ₂ for polymer solutions involving specific polymer/solvent interaction sites show a difference of 45% with change of solvent, while no difference is observed in solutions of polymers without the interaction sites. The normal-stress ratio —N ₂/N ₁ for a 5% solution of cis-polybutadiene is 0.24 at a shear rate of 100 s^–1, and it appears to approach the zero shear limit of 2/7 given by the Doi-Edwards theory. The Higashitani-Pritchard-Baird-Lodge equation relating the elastic hole pressure to the normal-stress differenceN ₁ –N ₂ gives a qualitative agreement betweenN ₁ –N ₂ from the TCP Apparatus and the hole pressure from the Stressmeter; the percent difference is 0 at shear stress < 25 Pa, 35% at = 45 Pa, and 18% at the highest = 63 Pa.