A thin rod is spun about its own axis in the centre of a wide dish

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Condensed Matter, Materials & Soft Matter · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Non-Newtonian and viscoelastic fluids

Problem

A thin rod is spun about its own axis in the centre of a wide dish, first in glycerine (a Newtonian liquid) and then, at the same rotation rate, in a dilute polymer solution. In the glycerine the free surface dips down near the rod; in the polymer solution it climbs up the rod. Someone argues: "The polymer solution is far more viscous than the glycerine, so as it spins it drags more liquid up with it; rod climbing is just a stronger version of ordinary viscous drag, and the higher viscosity is why it climbs while the thin glycerine barely changes shape." ((a)) Explain what is wrong with this reasoning. ((b)) Using the low-shear-rate result N1=2ητMγ˙2N_1 = 2\eta\tau_M\dot\gamma^2 quoted for the first normal stress difference, show that a very viscous silicone oil with η=1000 Pa s\eta = 1000\,\mathrm{Pa\,s} but no elastic relaxation (τM=0\tau_M = 0) cannot climb, however viscous it is, while a low-viscosity polymer solution with η=0.05 Pa s\eta = 0.05\,\mathrm{Pa\,s}, τM=0.2 s\tau_M = 0.2\,\mathrm{s}, sheared locally at γ˙=1 s−1\dot\gamma = 1\,\mathrm{s^{-1}}, has a positive N1N_1. Check first that this shear rate is low enough for the quoted result to apply.

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