= Parabolic-profile sedimentation shock
Let $c=\phi/\phi_{\max}$ be <particle volume fraction> normalized by the packed value, and use the <quadratic hindered-settling flux> $j(c)=-u_sc(1-c)$. Take
$$
c(z,0)=\frac13\left(1-\frac{z^2}{h^2}\right),\qquad 0\leq z\leq h,
$$
with 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 follows
$$
z(\xi,t)=\xi+u_s[2c(\xi,0)-1]t
=\xi-u_s\left(\frac13+\frac{2\xi^2}{3h^2}\right)t.
$$
The mapping first loses invertibility when
$$
\frac{\partial z}{\partial\xi}=1-\frac{4u_s\xi t}{3h^2}=0.
$$
The 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 $\phi$ is $-u_s\phi(1-\phi/\phi_{\max})$.
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 $c_L\to0$. 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.
Back to article page
