= Floating extensional viscous gravity current
{title2=$h_t+(hu)_x=0,\quad4\mu(hu_x)_x=\rho g' h h_x$}
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.
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