 Constant-settling runout in a triangular valley

For a dilute well-mixed particle current of conserved bulk volume $W$, triangular section $\lambda h^2$, constant particle settling speed $w_s$ and initial reduced gravity $g_0^{\prime }$, put $s=\sqrt{\varphi /{\varphi }_0}$. The balances give $\dot{X}=Bs{X}^{-1/4}$ and $\dot{\varphi }=-2w_s\varphi /h$, where $B=\mathrm{Fr}\sqrt{g_0^{\prime }}(W/\lambda {)}^{1/4}$. Eliminating time gives $s=1-(X/R{)}^{7/4}$ and the displayed limiting runout. It is approached at infinite time. Early spreading has the inertial $t^{4/5}$ exponent, while settling limits the eventual reach.