Shear-driven viscous gravity current ($h_t+A(h^2{)}_x=D\nabla \cdot (h^3\nabla h)$)

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 $\mathbf{q}=A{h}^2{\mathbf{e}}_x-D{h}^3\nabla h$, 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 $\tau /(\Delta \rho gL)\ll 1$ 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.

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.

For a steady point-source shear-driven viscous gravity current far downstream, $h\ll y_N\ll x$ permits dropping the downstream gravity flux. With $w=h^2$, the equation is $w_x=D(w^2{)}_{yy}/(4A)$, a quadratic porous medium equation. Conserving $A\int wdy=Q$ yields $y_N=(3\mu \Delta \rho gQx/{\tau }^2{)}^{1/3}$ and $h^2=\tau (y_N^2-y^2)/(2\Delta \rho gx)$ inside the current. The central height decays as $x^{-1/6}$. Near the source, balancing shear and gravity gives an upstream reach of order $\sqrt{\mu \Delta \rho gQ}/\tau$, whose numerical coefficient is not determined by scaling.