= Steady uniaxial extension of the Giesekus model
{title2=$b s_1^2+(1-2W)s_1-2W=0$}
For <uniaxial extensional flow> with $G=\operatorname{diag}(\gamma,-\gamma/2,-\gamma/2)$, write $W=\lambda\gamma$, $b=\alpha p_T$, and the <stress> as $p_T\operatorname{diag}(s_1,s_2,s_2)$. 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 $2s_2^2+2s_2+1=0$.
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