Quadratic hindered-settling flux
= Quadratic hindered-settling flux
{title2=$j(\phi)=-u_s\phi(1-\phi)$}
For downward settling speed $u_s(1-\phi)$ in upward coordinate $z$, particle flux is $j=-u_s\phi(1-\phi)$. The concentration-characteristic speed is $j^{\prime}=u_s(2\phi-1)$ and differs from particle speed. Since $j^{\prime\prime}=2u_s>0$, decreasing concentration with height creates a compressive <shock>, while increasing concentration creates a <rarefaction wave>.