= Uniaxial extension of an affine linear PTT fluid
{title2=$T(T+2q-1)(T+3q)=6\alpha q^2$}
For <uniaxial extensional flow> put $q=\tau\dot\gamma>0$, $f=1+\alpha\operatorname{tr}A$ and $T=f-2q$. The diagonal components are $a_{11}=2q/T$, $a_{22}=a_{33}=-q/(T+3q)$. The identity $f-1=\alpha\operatorname{tr}A$ 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.
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