Steady uniaxial extension of the Giesekus model ($b{s}_1^2+(1-2W)s_1-2W=0$)

For uniaxial extensional flow with $G=\mathrm{diag}(\gamma ,-\gamma /2,-\gamma /2)$, write $W=\lambda \gamma$, $b=\alpha p_T$, and the stress as $p_T\mathrm{diag}(s_1,s_2,s_2)$. The displayed longitudinal equation is supplemented by $b{s}_2^2+(1+W)s_2+W=0$. Choose the roots tending to zero at $W=0$. For $0<b<1$ these roots have $1+s_i>0$ at every nonnegative rate. The stretching stress is finite at $W=1/2$ and the extensional viscosity $({\sigma }_{11}-{\sigma }_{22})/\gamma$ tends to $2\lambda /\alpha$ as $\gamma \to \infty$. In the zero-$\alpha$ Upper-convected Maxwell model, the physical steady longitudinal branch terminates at $W=1/2$. Arbitrary positive $b$ is insufficient: $b=2,W=1$ gives the impossible real transverse equation $2s_2^2+2s_2+1=0$.