Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 352 / 2 / b

The convected terms are quadratic in the disturbance. Linear response requires small strain and small rate-based Weissenberg numbers, in particularThe constitutive equation then reduces toFor $\dot{\gamma }={\dot{\gamma }}_0\cos \omega t$, its periodic stress response isSince $\gamma =({\dot{\gamma }}_0/\omega )\sin \omega t$, comparison with $\tau ={\gamma }_0[G^{\prime }\sin \omega t+G^{\prime \prime }\cos \omega t]$ givesThe condition ${\lambda }_1\ge {\lambda }_2$ makes the storage modulus nonnegative.