= Residence time in planar extension
{title2=$x(t)=x_0e^{\dot\epsilon t},\quad y(t)=y_0e^{-\dot\epsilon t}$}
An original local flow model has <velocity> $\mathbf v=(\dot\epsilon x,-\dot\epsilon y,0)$ for constant <extensional strain rate> $\dot\epsilon>0$. 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 <time>
$$
t_{\rm exit}=\frac{1}{\dot\epsilon}\log\frac{L}{|x_0|}.
$$
For chosen $\dot\epsilon=1\,\mathrm{s^{-1}}$, $L=1\,\mathrm{mm}$ and $|x_0|=0.01\,\mathrm{mm}$, this is $\log100\,\mathrm s\simeq4.61\,\mathrm s$. 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.
Back to article page
