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A coil-stretch transition is a sharp change from a coiled to a substantially extended polymer conformation under sufficiently strong extensional flow. The driving extensional strain rate must compete with the chain's relaxation rate, and the available residence time matters. A long polymer relaxation time lets a weak but sustained flow produce a large effect. The Hookean dumbbell in planar extension below isolates this competition; its unbounded growth signals a model limitation, rather than a prediction of infinitely long molecules.

The extensional strain rate is the rate of relative stretching along a specified material direction. If a short material line of length $\ell$ follows a principal stretching direction, $\dot{ϵ}=d\log \ell /dt$ is the corresponding eigenvalue of the rate-of-strain tensor. For the planar model in extensional flow, $\dot{ϵ}=\partial u_x/\partial x$; the two transverse rates are $-\dot{ϵ}$ and zero. For uniaxial extensional flow, each transverse rate is $-\dot{ϵ}/2$.

For steady uniaxial extension of the Giesekus model, the component-first velocity gradient is diagonal, $G=\mathrm{diag}(\gamma ,-\gamma /2,-\gamma /2)$. For a steady spatially uniform diagonal stress, the material derivative vanishes and the 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 areSet $W=\lambda \gamma \ge 0$, $b=\alpha p_T$ 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 areA sufficient nondegenerate range for physical solutions at every nonnegative rate is $0<b<1$. Indeed $M=(1-b)I+bA$ with $A=HC/(k_{B}T)$, so positive conformation implies 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 $W\ge 0$ 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 algebraic roots do not provide a physical branch connected to the equilibrium distribution. For constant positive diagonal $A$, the associated stationary configuration distribution is a Gaussian with covariance $C=(k_{B}T/H)A$.

For positive $\alpha$ on the physical branch, the stretching stress remains finite through $W=1/2$. At that rate,At large extension rate,Thus the stress continues to grow, but the 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 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 ${\eta }_E\to 3\mu$.

When $\alpha =0$, the physical steady branch instead isThe stretching stress diverges as $W\to 1/2$ from below. Above that threshold its algebraic continuation has $1+s_1=1/(1-2W)<0$ and cannot be a conformation covariance. There is no physical finite steady stretching state. The transient equationgives linear growth at $W=1/2$ and exponential growth at larger rates, the Hookean coil-stretch failure of the upper-convected Maxwell limit.

Parameter qualification. Positivity of $\alpha$ alone, without a restriction on $b=\alpha n{k}_{B}T$, does not ensure real physical 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/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.

An original local flow model has velocity $\mathbf{v}=(\dot{ϵ}x,-\dot{ϵ}y,0)$ for constant extensional strain rate $\dot{ϵ}>0$. Its divergence is zero, and its trajectories satisfy the displayed equations. Material approaches along $y$ and leaves along $x$. If the idealized observation region ends at $|x|=L$, a trajectory starting at $0<|x_0|<L$ has residence timeFor chosen $\dot{ϵ}=1{\mathrm{s}}^{-1}$, $L=1\mathrm{mm}$ and $|x_0|=0.01\mathrm{mm}$, this is $\log 100\mathrm{s}\simeq 4.61\mathrm{s}$. Starting ten times farther from the symmetry line halves this particular residence time. The mathematical trajectory $x_0=0$ never exits in this model; diffusion, imperfect flow and finite apparatus prevent treating that ideal trajectory as a guaranteed infinite exposure in an experiment.