The role of the second normal stress difference in rod-climbing effect

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 $N_{1}$, which induces tensile hoop stresses that draw fluid upward along the rod. The second normal stress difference $N_{2}$, in contrast, is often presumed negligible or dynamically unimportant. However, many polymer solutions and industrial fluids, such as suspensions, exhibit $N_{2}$ of appreciable magnitude, and modern constitutive models predict that it can significantly modify free-surface stresses and thereby the climbing behaviour. In this work, we perform high-resolution axisymmetric simulations of the Linear Phan--Thien--Tanner (LPTT) model to systematically isolate the influence of $N_{2}$ on rod climbing. We show that increasing the magnitude of $N_{2}$ progressively weakens the climbing response and ultimately reverses it, producing rod-descending once the normal-stress ratio exceeds a critical value $\psi_{0}\approx 0.25$. Larger $N_{2}$ also destabilises 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 $(Wi,\psi_{0})$ parameter space, where $Wi$ is the Weissenberg number, we reconcile discrepancies among perturbation theory, experiments, and numerical simulations. These results establish $N_{2}$ as a crucial control parameter governing free-surface stability in viscoelastic liquids.