Transient Dynamics of the Rod-Climbing Effect in Oldroyd-B Fluids

The Weissenberg effect, or rod-climbing phenomenon, occurs in non-Newtonian fluids where the fluid interface ascends along a rotating rod. Despite its prominence, theoretical insights into this phenomenon remain limited. In earlier work, D.D. Joseph and R. L. Fosdick (Arch. Ration. Mech. Anal., 1973); see also More et al. (Soft Matter, 2023), employed domain perturbation methods for second-order fluids to determine the equilibrium interface height by expanding solutions based on the rotation speed. Here, we investigate the time-dependent interface height by examining the interplay between gravity and viscoelasticity, while neglecting surface tension and inertia effects. We conduct our theoretical analysis using dimensionless quantities and derive equations from the Oldroyd-B model. By focusing on the small-deformation scenario, we obtain an analytical solution for the time-varying height that depends on the Weissenberg number and a dimensionless gravity parameter.