Initial velocity profile of an annularly confined column ($W=\cosh (\Lambda -Z)/\cosh \Lambda -1$)

Using the extensional screening length of a slumping column, an initially uniform column of length $\Lambda$ obeys $W_{ZZ}=1+W$, $W(0)=0$ and $W_Z(\Lambda )=0$. Thus $W=\cosh (\Lambda -Z)/\cosh \Lambda -1$ and its initial area-fraction growth is ${\varphi }_T={\varphi }_0\sinh (\Lambda -Z)/\cosh \Lambda$. For $\Lambda \gg 1$, the velocity is $e^{-Z}-1$ outside exponentially small end corrections; for $\Lambda \ll 1$, it is $Z^2/2-\Lambda Z$. The first limit balances excess weight with annular hydraulic drag, the second with extensional viscous stress.