Floating extensional viscous gravity current ($h_t+(hu{)}_x=0,4\mu (h{u}_x{)}_x=\rho g^{\prime }h{h}_x$)

A thin layer of very viscous liquid floating on an effectively inviscid denser liquid has negligible shear traction on both faces. Its leading horizontal velocity is a plug, and incompressibility gives vertical strain $w_z=-u_x$. Normal-stress balance shifts the pressure by $-2\mu u_x$, so the longitudinal excess stress is $4\mu u_x$. Combining depth-integrated momentum balance with hydrostatic reduced gravity gives the displayed extensional equations. This is distinct from a no-slip, shear-driven thin-film gravity current.

With fixed inlet thickness $h_0$ and velocity $u_0$, and nose tension $4\mu h_N{u}_x=\rho g^{\prime }{h}_N^2/2$, integrating momentum gives $u_x=\rho g^{\prime }h/(8\mu )$. For a slice injected at $t_0$, write $\tau =t-t_0$ and $T=8\mu /(\rho g^{\prime }{h}_0)$. 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.