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

TL;DR

The paper addresses how the second normal stress difference $N_2$ influences the Weissenberg rod-climbing effect by independently varying the normal-stress ratio $\psi_0=-N_2/N_1$ within the Linear Phan–Thien–Tanner model. Using fully resolved axisymmetric simulations with log-conformation stabilization, it shows that increasing $|N_2|$ weakens climbing and can reverse it to rod-descending around $\psi_0\approx 0.25$, even when inertia is negligible. The results reveal a domain-wide redistribution of meridional stresses and the emergence of a secondary vortex, establishing $\psi_0$ as a critical parameter alongside $Wi$ and $Re$ in governing free-surface stability and flow transitions from steady climbing to unsteady, three-dimensional states. This reconciles discrepancies between classical theory and experiments and broadens the viscoelastic stability landscape for free-surface flows with non-negligible $N_2$.

Abstract

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 $ψ_{0}\approx 0.25$. 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 $(Wi,ψ_{0})$ 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.

Figures (8)

• Figure 1: A three-dimensional representation of the parameter space accessed in the instabilities in complex flows of complex fluids. The Weissenberg-Reynolds number space has been extensively studied and it is well known that complex flows become unstable due to 1) Elastic instabilities if $Wi > Wi_c$ when $Re \ll 1$, or 2) Inertial instabilities if $Re > Re_c$ when $Wi \ll 1$, or 3) a combination $f(Wi, Re)>M_c$ in the intermediate limit datta2022perspectivesmore2024elasto. However, the role of the second normal stress difference remains incompletely understood and is represented as the third dimension of this parameter space. This third dimension is explored using the popular rod-climbing problem in this study.
• Figure 2: Problem set up. We compute the flow of a Linear Phan-Thien-Tanner viscoelastic liquid phan1978tanner with varying second normal stress difference around a thin rod of radius $a=0.00635$ m rotating at a rotational speed of $\Omega \in [0 - 100]$ rad/s. The flow is assumed to be 2-D axisymmetric, and a non-uniform grid is used, finer near the rod and progressively coarser in the r-direction, shown at progressively higher zoom levels. No-slip boundary conditions are applied to all solid surfaces. Acceleration due to gravity $g=9.81$ m/s$^2$ acts in the negative $z$ direction.
• Figure 3: a) Comparison of the interface descending depth $-\Delta h(\Omega,a)$ scaled by the rod radius $a$ near the rod due to inertia for a Newtonian fluid from simulations (symbols) and theory predictions of Eq. \ref{['eq:eq2']} (line). b) Effect of systematically increasing the normal stress difference ratio $\psi_0 = -\Psi_{2}/\Psi_{1}$ on the change in the interface height $\Delta h(\Omega,a)$ due to the Weissenberg effect, keeping all other fluid properties the same. Symbols are mean interface heights near the rod from the simulations at a long time, while dashed lines are predictions of Eq. \ref{['eq:eq2']} using fluid properties in Table \ref{['tab:LPTTparams']}. The agreement is good at low $\Omega$, and the simulation results diverge at high $\Omega$ as expected due to the validity of the theory only on the low rotation speed limit. The rod-climbing effect weakens as the second normal stress difference increases so much so that it transitions to rod-descending at the critical value $\psi_{0} \geq 0.25$, which gives climbing constant $\hat{\beta}=0$. However, note that the interface descends further than the perturbation theory predictions obtained using $\hat{\beta}=0$, which are valid only to $\approx O(\Omega^2)$.
• Figure 4: Contours of the first (top row), second (middle row) and the normal stress difference ratio (bottom row) for increasing values of the second normal stress difference relative to the first normal stress difference (left to right columns). The first normal stress difference has almost a similar magnitude near the rod as well as a similar contour, irrespective of $\psi_0$. However, we observe weakening rod-climbing with increasing $\psi_0$, which can be explained by the increasing magnitude of the contours of $N_2$ with increasing $\psi_0$. So, while $N_1$ pushes the liquid up, $N_2$ works in opposition to it and is dominant enough to reverse the rod-climbing to rod-descending beyond a critical value $\psi_0=0.25$. The governing role of $\psi_0$ is also clear from its contours in the bottom row, which shows progressively increasing and almost uniform $\psi_0$ field in the entire liquid.
• Figure 5: Streamlines around the rotating rod for Newtonian and viscoelastic liquids. (a) In a Newtonian fluid, only a single primary recirculation cell forms. (b–e) In contrast, viscoelastic simulations exhibit a distinct secondary vortex generated by normal-stress differences, consistent with experimental observations saville1969secondary. The strength and extent of this secondary vortex increase systematically with the magnitude of the second normal stress difference $N_{2}$ or $\psi_0$ [(b) $\psi_0=0$, (c) $\psi_0=0.05$, (d) $\psi_0=0.15$, (e) $\psi_0=0.25$], analogous to behaviour reported in confined viscoelastic flows through non-circular geometries maklad2021review. (f) A comparison between rod-descending in a Newtonian fluid (driven purely by inertia) and in a viscoelastic fluid with $\psi_{0}=0.25$ (corresponding to $\hat{\beta}=0$) highlights the role of higher-order $O(\Omega^{4})$ contributions that are neglected in the perturbation expression given in Eq. \ref{['eq:eq2']}.