= Constant-settling runout in a triangular valley
{title2=$R=[7\operatorname{Fr}\sqrt{g'_0}(W/\lambda)^{3/4}/(4w_s)]^{4/7}$}
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 $g'_0$, put $s=\sqrt{\phi/\phi_0}$. The balances give $\dot X=B sX^{-1/4}$ and $\dot\phi=-2w_s\phi/h$, where $B=\operatorname{Fr}\sqrt{g'_0}(W/\lambda)^{1/4}$. 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.
Back to article page
