Parabolic-profile sedimentation shock

Let $c=\varphi /{\varphi }_{\max }$ be particle volume fraction normalized by the packed value, and use the quadratic hindered-settling flux $j(c)=-u_{s}c(1-c)$. Takewith clear fluid above $h$. To isolate breaking of this smooth profile, do not impose an additional packed-bed discontinuity. A characteristic curve labelled by its initial height $\xi$ carries constant concentration and followsThe mapping first loses invertibility whenThe earliest time occurs at $\xi =h$, giving $t_c=3h/(4u_s)$ and $z_c=h/4$. This explicitly shows why the time uses normalized concentration and the isolated settling-speed scale $u_s$; the physical flux for $\varphi$ is $-u_s\varphi (1-\varphi /{\varphi }_{\max })$.

The new sedimentation shock joins concentrated fluid below to clear fluid above. Its signed speed is $u_s(c_L-1)$, initially $-u_s$ as $c_L\to 0$. As the state behind it becomes more concentrated, its downward speed decreases until a boundary interaction changes the problem. This is the onset of characteristic breaking, not the onset of a separately imposed deposition front.