= Shear-driven viscous gravity current
{title2=$h_t+A(h^2)_x=D\nabla\cdot(h^3\nabla h)$}
A thin dense <gravity current> on a rigid no-slip substrate can be carried downstream by imposed upper <shear stress> $\tau$. For density contrast $\Delta\rho$, <lubrication theory> gives $\mathbf q=Ah^2\mathbf e_x-Dh^3\nabla h$, where $A=\tau/(2\mu)$ and $D=\Delta\rho g/(3\mu)$. The <continuity equation> gives the displayed equation. Besides small slopes and reduced inertia, the hydrostatic reduction requires $\tau/(\Delta\rho gL)\ll1$ to keep shear-related normal stress small. Finite zero-thickness fronts are formal outer profiles and need a separate local description when their slopes are large.
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