The Weissenberg (rod-climbing) effect, i.e., the rise of a viscoelastic fluid along a thin rotating rod, has long served as a canonical demonstration of elasticity and normal–stress differences in complex fluids. The effect is most commonly attributed to the first normal–stress difference N1, which induces tensile hoop stresses that draw fluid upward along the rod. The second normal–stress difference N2, in contrast, is often presumed negligible or dynamically unimportant. However, many polymer solutions and industrial fluids, such as suspensions, exhibit N2 of appreciable magnitude, and modern constitutive models predict that it can significantly modify free-surface stresses and thereby the climbing behavior. In this work, we perform high-resolution axisymmetric simulations of the Linear Phan–Thien–Tanner model to systematically isolate the influence of N2 on rod climbing. We show that increasing the magnitude of N2 progressively weakens the climbing response and ultimately reverses it, producing rod-descending once the normal–stress ratio exceeds a critical value ψ0≈0.25. Larger N2 also destabilizes the flow, promoting early onset (in terms of the rotation speed) of bubble formation, subcritical Hopf oscillations, and fully asymmetric three-dimensional motion that culminates in rupture. By mapping these regimes in the (Wi,ψ0) parameter space, where Wi is the Weissenberg number, we reconcile discrepancies among perturbation theory, experiments, and numerical simulations. These results establish N2 as a crucial control parameter governing free-surface stability in viscoelastic liquids.
Rishabh V. More (Sun,) studied this question.