Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 49 / 3 / a / Solution

For the specified affine linear PTT fluid, perturb about the relaxed state $A=0$, $\mathbf{v}=0$. Products of $A$ with the velocity gradient and the term $\alpha (\mathrm{tr}A)A$ are second order in the perturbation. The material-advection term is also second order. The linearized equation is thereforeWith vanishing stress in the remote past, an integrating factor givesSince ${\sigma }^{\prime }=G_{0}A$, comparison with the linear viscoelastic fluid convolution yields $G(s)=G_0{e}^{-s/\tau }$ and zero-shear viscosity $G_0\tau$. The nonlinear relaxation parameter does not appear at linear order; the response is that of a Maxwell fluid.