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

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.