= Hookean dumbbell in planar extension
{c}
{title2=$\dot C=AC+CA^{\mathsf T}-(C-I)/\lambda$}
Represent one <polymer> by two beads joined by an ideal linear entropic spring, and stipulate a dimensionless <conformation tensor> $C$ with equilibrium value $I$. The displayed model uses <velocity gradient> $A=\operatorname{diag}(\dot\epsilon,-\dot\epsilon,0)$ and positive <polymer relaxation time> $\lambda$. It omits finite chain extensibility, interactions with other chains, and feedback from the <polymer> onto the flow. It is a teaching model, not a fitted description of Keller's apparatus.
Define the <Weissenberg number> for this model as $\mathrm{Wi}=\lambda\dot\epsilon$ and dimensionless <time> $u=t/\lambda$. With an initially isotropic $C=I$, the diagonal entries obey
$$
\frac{dC_{xx}}{du}=(2\mathrm{Wi}-1)C_{xx}+1,\qquad
\frac{dC_{yy}}{du}=-(2\mathrm{Wi}+1)C_{yy}+1,\qquad C_{zz}=1.
$$
Below $\mathrm{Wi}=1/2$, both transverse and longitudinal responses approach positive finite steady values:
$$
\boxed{C_{xx}=\frac1{1-2\mathrm{Wi}},\qquad C_{yy}=\frac1{1+2\mathrm{Wi}},\qquad C_{zz}=1.}
$$
For a chosen $\mathrm{Wi}=0.4$, $C_{xx}=5$ and $C_{yy}=5/9$: the longitudinal mean-square separation is five times its quiescent value, while the transverse separation contracts. A root-mean-square length grows as the square root of a variance, so the longitudinal length factor is $\sqrt5$, not five. The anisotropy $C_{xx}-C_{yy}$ is an orientational indicator; converting it to measured <birefringence> requires an additional calibrated optical relation.
\Image[/polymer-coil-stretch-relaxation-competition.png]
{title=Extension competes with relaxation in a chosen dumbbell model}
At $\mathrm{Wi}=1/2$, $C_{xx}=1+u$. Above that value, the growing homogeneous solution prevents a positive finite steady state. For instance, $\mathrm{Wi}=0.6$ gives $C_{xx}=6e^{u/5}-5$. Real chains have finite contour lengths, so this growth identifies the failure of the linear spring model. It does not establish a universal threshold for arbitrary flow histories, finite chains or other definitions of <polymer relaxation time>. Combining the growth equation with <residence time in planar extension> also explains why a large instantaneous <Weissenberg number> alone does not guarantee a strongly stretched exiting chain.
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