= Hindered-settling flux inflection
{title2=$f''(s)=V_0\alpha(1-s)^{\alpha-2}[2-(\alpha+1)s]$}
For <hindered settling> with normalized <particle volume fraction> $s$, upward coordinate and downward isolated <settling velocity> magnitude $V_0$, the <scalar conservation law> has <conservation law flux> $f(s)=-V_0s(1-s)^\alpha$. For $\alpha>0$, its <convexity> changes at $s=2/(\alpha+1)$ when this lies in the physical interval. This change distinguishes spreading <rarefaction waves> from compressing <characteristic curves>; a flux with an inflection can produce both a <shock wave> and a <rarefaction wave> in one transition. At $\alpha=0$ the flux is linear and the profile simply translates.
Back to article page
