= Solution
For <steady uniaxial extension of the Giesekus model>, the component-first <velocity gradient> is <Diagonal matrix>[diagonal], $G=\operatorname{diag}(\gamma,-\gamma/2,-\gamma/2)$. For a <Steady state>[steady] spatially uniform <Cauchy stress tensor>[diagonal stress], the <material derivative> vanishes and the <Upper-convected derivative>[convected terms] are $-2\gamma\sigma_{11}$ and $+\gamma\sigma_{22}$. The off-diagonal equations vanish identically, and symmetry permits equal transverse components. The remaining equations are
$$
\boxed{\alpha\sigma_{11}^2+(1-2\lambda\gamma)\sigma_{11}-2\mu\gamma=0,\qquad
\alpha\sigma_{22}^2+(1+\lambda\gamma)\sigma_{22}+\mu\gamma=0.}
$$
Set $W=\lambda\gamma\ge0$, $b=\alpha p_T$ and $s_i=\sigma_{ii}/p_T$, with $\mu=\lambda p_T$. For $b>0$, the branches continuously connected to zero <stress> at zero rate are
$$
\boxed{s_1=\frac{2W-1+\sqrt{(1-2W)^2+8bW}}{2b},\qquad
s_2=\frac{-(1+W)+\sqrt{(1+W)^2-4bW}}{2b}.}
$$
A sufficient nondegenerate range for physical solutions at every nonnegative rate is $0<b<1$. Indeed $M=(1-b)I+bA$ with $A=HC/(k_BT)$, so positive <Conformation tensor>[conformation] implies <Hydrodynamic mobility matrix>[positive mobility]. The longitudinal <quadratic equation> is nonpositive at $s_1=0$ and has a nonnegative <root of a polynomial>. The transverse <quadratic equation> has values $b-1<0$ at $s_2=-1$ and $W\ge0$ at $s_2=0$, so its larger <root of a polynomial> lies in $(-1,0]$. Thus the displayed branches satisfy $s_i>-1$, giving $A_i=1+s_i>0$. The other <Root of a polynomial>[algebraic roots] do not provide a physical branch connected to the <Boltzmann distribution>[equilibrium distribution]. For constant positive <Diagonal matrix>[diagonal] $A$, the associated stationary configuration distribution is a <Multivariate normal distribution>[Gaussian with covariance] $C=(k_BT/H)A$.
For positive $\alpha$ on the physical branch, the <Stress>[stretching stress] remains finite through $W=1/2$. At that rate,
$$
\boxed{\sigma_{11}=\sqrt{\frac{p_T}{\alpha}}.}
$$
At large <Extensional strain rate>[extension rate],
$$
\boxed{\sigma_{11}\sim\frac{2\lambda\gamma}{\alpha},\qquad
\frac{\sigma_{11}}\gamma\longrightarrow\frac{2\lambda}{\alpha}.}
$$
Thus the <stress> continues to grow, but the <Extensional viscosity>[uniaxial extensional viscosity] $\eta_E=(\sigma_{11}-\sigma_{22})/\gamma$ approaches the finite plateau $2\lambda/\alpha$ rather than developing a finite-rate pole. The <Stress>[transverse stress] approaches $-p_T$ as its <conformation tensor> variance becomes small. At small rate the two <stresses> are $2\mu\gamma$ and $-\mu\gamma$, so $\eta_E\to3\mu$.
When $\alpha=0$, the physical <Steady state>[steady] branch instead is
$$
\boxed{\sigma_{11}=\frac{2\mu\gamma}{1-2\lambda\gamma},\qquad
\sigma_{22}=-\frac{\mu\gamma}{1+\lambda\gamma}.}
$$
The <Stress>[stretching stress] diverges as $W\to1/2$ from below. Above that threshold its algebraic continuation has $1+s_1=1/(1-2W)<0$ and cannot be a <Conformation tensor>[conformation] <Covariance matrix>[covariance]. There is no physical finite <Steady state>[steady] <Conformation tensor>[stretching] state. The transient equation
$$
\lambda\dot\sigma_{11}=(2W-1)\sigma_{11}+2p_TW
$$
gives linear growth at $W=1/2$ and exponential growth at larger rates, the <Hookean dumbbell model>[Hookean] <Coil-stretch transition>[coil-stretch] failure of the <Upper-convected Maxwell model>[upper-convected Maxwell limit].
\Image[/past-exam-of-the-advanced-fluid-dynamics-course-of-the-university-of-oxford/2017-giesekus-extensional-stress.png]
{title=Anisotropic mobility rounds the Hookean extensional-stress singularity}
{description=Dimensionless longitudinal and transverse stresses are plotted against the extension-rate parameter W. Positive dimensionless mobility factors 0.1 and 0.5 give regular equilibrium-connected branches. The isotropic-mobility longitudinal branch is shown only below its physical steady-state threshold W equal to one half.}
\b[Parameter qualification.] Positivity of $\alpha$ alone, without a restriction on $b=\alpha nk_BT$, does not ensure real physical <Steady state>[steady] solutions for the entire <tensor>. For example $b=2$, $W=1$ makes the transverse equation $2s_2^2+2s_2+1=0$, which has no real root. Its <diffusion>/<Hydrodynamic resistance matrix>[resistance] interpretation is already inadmissible for some conformations. The physically positive-mobility regime stated above avoids that counterexample; the source's positive-$\alpha$ case must not be taken as a guarantee for arbitrary parameters.
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