= Initial velocity profile of an annularly confined column
{title2=$W=\cosh(\Lambda-Z)/\cosh\Lambda-1$}
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 $\phi_T=\phi_0\sinh(\Lambda-Z)/\cosh\Lambda$. For $\Lambda\gg1$, the <velocity> is $e^{-Z}-1$ outside exponentially small end corrections; for $\Lambda\ll1$, it is $Z^2/2-\Lambda Z$. The first limit balances excess weight with annular hydraulic drag, the second with extensional viscous <stress>.
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