 Quadratic hindered-settling flux

For downward settling speed $u_s(1-\varphi )$ in upward coordinate $z$, particle flux is $j=-u_s\varphi (1-\varphi )$. The concentration-characteristic speed is $j^{\prime }=u_s(2\varphi -1)$ 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.