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

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 ${\eta }_s+{\eta }_p$.