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

An objective time derivative transforms as a tensor under every superposed time-dependent rigid translation and rotation of the observer. A constitutive prediction using it is therefore frame-indifferent. The upper-convected derivative, lower-convected derivative, and Jaumann derivative are objective; the componentwise partial time derivative, and generally the uncorrected material derivative of a tensor, are not objective under rotating observers.

The Johnson--Segalman--Oldroyd model uses the Gordon--Schowalter derivativeHere $\eta$ is the zero-shear viscosity, ${\lambda }_1$ is the stress-relaxation time, and ${\lambda }_2$ is the retardation time associated with the more rapidly responding viscous contribution. Passive Oldroyd-type materials normally have $0\le {\lambda }_2\le {\lambda }_1$.

A low-rate steady-shear experiment measures $\eta$. A step strain followed by stress relaxation determines ${\lambda }_1$ from the exponential decay. Creep and recovery, or a frequency sweep in small-amplitude oscillatory shear, then separates ${\lambda }_2$ from ${\lambda }_1$; fitting both storage modulus and loss modulus is especially direct.

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.

Take steady simple shear $u_x=\dot{\gamma }y$ and define $W_i={\lambda }_i\dot{\gamma }$. Substitution of the symmetric stress components into the full constitutive equation gives, after solving the three coupled algebraic equations,Thus the Steady shear viscosity of a Johnson--Segalman--Oldroyd fluid isFor the stated small positive $a$, so that $0<a<2$, this decreases from $\eta$ at zero rate to $\eta {\lambda }_2/{\lambda }_1$ at high rate exactly whenEquality gives constant viscosity, while ${\lambda }_2>{\lambda }_1$ gives shear thickening.