Higher-order theory of the Weissenberg effect

Abstract

The higher-order theory of the Weissenberg effect is developed as a perturbation of the state of rest. The perturbation is given in powers of the angular frequency Ω of the rod and the solution is carried out to O(Ω4). The perturbation induces a slow-motion expansion of the stress into Rivlin-Ericksen tensors in combinations which are completely characterized by five viscoelastic parameters. The effects of each of the material parameters may be computed separately and their overall effect by superposition. The values of the parameters may be determined by measurement of the torque, surface angular velocity and height of climb. Such measurements are reported here for several different sample fluids. Good agreement between the third-order theory and measured values of the velocity is reported. Secondary motions which appear at are computed using biorthogonal series. The analysis predicts the surprising fact that secondary motions run up the climbing bubble against gravity and intuition.