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

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.