= Solution
<Hindered settling> is the reduction of the particle <settling velocity> by displaced-fluid backflow and interactions with other particles. In the specified model, the downward speed is $V_s$ at vanishing <concentration> and falls to zero at the maximum packing fraction $\phi_{\max}$. The <characteristic speed> of a <concentration> disturbance will differ from this individual-particle speed.
Take $z$ positive upwards. For a locally uniform horizontal suspension without diffusion or bulk fluid motion, the upward particle flux is
$$
j(\phi)=-\phi V(\phi)=-V_s\phi+\frac{V_s}{\phi_{\max}}\phi^2.
$$
Particle <volume conservation> gives the <scalar conservation law>
$$
\boxed{\phi_t+\partial_zj(\phi)=0,\qquad
\phi_t+\left(-V_s+\frac{2V_s\phi}{\phi_{\max}}\right)\phi_z=0.}
$$
This is the <quadratic hindered-settling flux>. Along its <characteristic curves>,
$$
\boxed{\frac{d\phi}{dt}=0,\qquad\frac{dz}{dt}=c(\phi)=-V_s+\frac{2V_s\phi}{\phi_{\max}}.}
$$
For a smooth initial profile $\phi_0(\zeta)$, the map is $z=\zeta+c(\phi_0(\zeta))t$. Its Jacobian is $1+(2V_s/\phi_{\max})\phi_0'(\zeta)t$. A <shock> develops where this first reaches zero: the compressive condition is $\phi_0'<0$, and
$$
t_s=-\frac{\phi_{\max}}{2V_s\min_\zeta\phi_0'(\zeta)}
$$
when the minimum is negative and no earlier boundary discontinuity is imposed. A discontinuous compressive initial state can instead contain a <shock> from $t=0$.
Let $z=s(t)$ separate states $\phi_L$ below the <shock> and $\phi_R$ above it. Integrating the <conservation law> across a moving thin interval gives the <Rankine-Hugoniot condition>
$$
\dot s(\phi_R-\phi_L)=j(\phi_R)-j(\phi_L).
$$
Equivalently, the flux relative to the <shock> agrees on both sides. For this quadratic flux,
$$
\boxed{\dot s=-V_s+\frac{V_s}{\phi_{\max}}(\phi_L+\phi_R)
=\frac{c(\phi_L)+c(\phi_R)}2.}
$$
Since $j''=2V_s/\phi_{\max}>0$, a physical compression <shock> has $\phi_L>\phi_R$ and satisfies $c(\phi_L)>\dot s>c(\phi_R)$: <characteristic curves> enter it from both sides. These conditions select the <shock> over a nonphysical expansive discontinuity.
Back to article page
