= Line-source upstream reach under imposed shear
{title2=$x_N=2\mu\Delta\rho gQ_{2d}/(3\tau^2)$}
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.
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