Line-source upstream reach under imposed shear ($x_N=2\mu \Delta \rho g{Q}_{2d}/(3{\tau }^2)$)

For a steady line-source shear-driven viscous gravity current with downstream constant thickness, $h_d^2=2\mu Q_{2d}/\tau$. Zero upstream volume flux per unit width gives $h{h}_x=3\tau /(2\Delta \rho g)$, so $h^2=h_d^2+3\tau x/(\Delta \rho g)$ until it vanishes at $x=-x_N$. Unlike the point-source scaling estimate, this reduced line-source model fixes the exact coefficient $2/3$. The square-root nose has an unresolved steep edge.