Appendix 1: One-dimensional model

Let w be the axial component of velocity and z the axial coordinate. The flow domain is $z\in (0,H(t))$, where H(t) is the position of the moving plate. We define a rescaled coordinate

$$\xi={z\over H(t)},$$

For exponential stretching, we have $H(t)=\exp(t)$. Furthermore, let $r=f(t,\xi)$ be the filament radius, $\tau_{11}$ and $\tau_{33}$ the radial and axial components of the stress tensor. For the Oldroyd-B model, the remaining stress component $\tau_{22}$ is equal to $\tau_{11}$, and the shear stresses are either zero by axial symmetry or negligible by the slender rod approximation. We define new variables W and $\phi$ as

$$w=H'(t)(W+\xi),\ \phi=\ln f.$$

For a slender viscoelastic filament the equations of motion can be approximated by the following equations [6].

   $$\begin{eqnarray} 3(1-\beta)\frac{\partial^2 W}{\partial \xi^{2}} &=&-[6(1-\beta... ...partial \xi} +\frac{\partial \tau_{11}}{\partial \xi} \nonumber \end{eqnarray}$$

 

$$\frac{\partial \phi}{\partial t}=-\frac{1}{2} -W\frac{\parti... ...i}{\partial \xi} -\frac{1}{2}\frac{\partial W}{\partial \xi}$$ (232)

 

$$\tau_{11}+{\rm De}(\frac{\partial \tau_{11}}{\partial t}+\ta... ...W}{\partial \xi})=-\beta (1+\frac{\partial W}{\partial \xi})$$ (233)

 

$$\tau_{33}+{\rm De}(\frac{\partial \tau_{33}}{\partial t}-2\t... ...W}{\partial \xi}) =2\beta(1+\frac{\partial W}{\partial \xi}).$$ (234)

The boundary conditions are  

$$W=0 \: \: \mbox{on} \: \: \xi =0, 1.$$ (235)