= Constant-flux floating extensional gravity current
{title2=$h(x)=h_0(1+x/(4\ell))^{-1/2},\quad\ell=\mu u_0/(\rho g'h_0)$}
With fixed inlet thickness $h_0$ and <velocity> $u_0$, and nose tension $4\mu h_Nu_x=\rho g'h_N^2/2$, integrating momentum gives $u_x=\rho g'h/(8\mu)$. For a slice injected at $t_0$, write $\tau=t-t_0$ and $T=8\mu/(\rho g'h_0)$. Its thickness, <velocity> and position are $h=h_0/(1+\tau/T)$, $u=u_0(1+\tau/T)$, and $x=u_0[\tau+\tau^2/(2T)]$. Eliminating its age gives the steady profile behind the nose, while the nose is the slice with the earliest injection time.
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