Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 352 / 2

A plain material derivative of the conformation tensor is not an objective time derivative: an observer undergoing a time-dependent rigid rotation would infer a different constitutive response, and even rigid-body rotation could appear to change polymer deformation. The upper-convected derivative subtracts deformation and rotation carried by the velocity gradient and is frame indifferent.

The Oldroyd-B model is often inadequate because its Hookean dumbbells are infinitely extensible. It predicts constant shear viscosity rather than shear thinning, zero second normal-stress difference, and an unbounded extensional viscosity at a finite extension rate. Real polymer chains have finite extensibility and commonly exhibit shear thinning, bounded extensional stress, multiple relaxation times, and nonlinear solvent or concentration effects.

Let $a=\lambda \dot{\gamma }$. For simple shear, the steady conformation equation isIts nonzero components areSince $\mathbf{\tau }={\eta }_s\dot{\mathbf{\gamma }}+({\eta }_p/\lambda )(F\mathbf{C}-\mathbf{I})$,Thus the FENE-P model has

The trace isSubstituting it into $F=(L^2-3)/(L^2-T)$ givesFor $\beta \ll 1$, $F=1+\beta +O({\beta }^2)$. For $\beta \gg 1$, $F\sim {\beta }^{1/3}$. The effective shear viscosity is thereforeThe FENE-P curve decreases from ${\eta }_s+{\eta }_p$ toward the solvent plateau ${\eta }_s$, displaying shear thinning. Oldroyd-B has $F=1$ and remains at the constant value ${\eta }_s+{\eta }_p$.

For uniaxial extension,The diagonal steady conformation tensor iswhere physical solutions require $F>2a$. The extensional viscosity isThe implicit closure is

As $\dot{ϵ}\to 0$, $a\to 0$, $F\to 1$, andThe Trouton ratio is therefore three. This is also the small-rate Oldroyd-B result because finite extensibility is irrelevant while polymer deformation remains small.

For FENE-P at $a\gg 1$, write $F=2a+\delta$. The trace constraint approaches the finite maximum $T\to L^2$, so $C_{zz}=1/\delta \to L^2$ and $\delta \to L^{-2}$ to leading order. Henceup to corrections that vanish as $a^{-1}$. Its extensional viscosity rises from the Newtonian plateau and saturates at a finite-extensibility plateau.

For Oldroyd-B, $F=1$ andIt diverges as $\lambda \dot{ϵ}\uparrow 1/2$ and has no physical steady homogeneous branch beyond that point. This extensional catastrophe is removed by finite chain extensibility.