Uniaxial extension of an affine linear PTT fluid ($T(T+2q-1)(T+3q)=6\alpha q^2$)

For uniaxial extensional flow put $q=\tau \dot{\gamma }>0$, $f=1+\alpha \mathrm{tr}A$ and $T=f-2q$. The diagonal components are $a_{11}=2q/T$, $a_{22}=a_{33}=-q/(T+3q)$. The identity $f-1=\alpha \mathrm{tr}A$ gives the cubic equation in the title. For $\alpha >0$ its physical root has $T>\max (0,1-2q)$ and connects to $T=1$ at rest. Its extensional viscosity is ${\eta }_E=G_0\tau [2/T+1/(T+3q)]$, tending to $3G_0\tau$ at small $q$ and $2G_0\tau /\alpha$ at large $q$. At $\alpha =0$ the stable steady Maxwell extension instead ends at $q=1/2$; the finite high-rate limit must not be extended to that degenerate case.