= Hindered-settling runout invariant in a prismatic triangular channel
{title2=$\operatorname{artanh}\sqrt\phi+KX^{7/4}=\mathrm{constant}$}
In the <prismatic triangular-channel gravity-current box model>, take $h=h_0\sqrt{L/X}$ and $w(\phi)=w_0(1-\phi)$. Eliminating time gives $d\phi/[\sqrt\phi(1-\phi)]=-2w_0X^{3/4}dX/(\operatorname{Fr}\sqrt G h_0^{3/2}L^{3/4})$. Integration yields the displayed invariant with $K=4w_0/(7\operatorname{Fr}\sqrt G h_0^{3/2}L^{3/4})$. The positive-concentration branch approaches a finite <runout length of a gravity current> as time tends to infinity.
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