= Solution
Take $z$ increasing upwards and write $s=\phi/\phi_{\max}$. The particles have downward <settling velocity>; their upward <volume flux> is $\phi V$. Conservation of particle volume in any interval gives
$$
\frac{d}{dt}\int_{z_1}^{z_2}\phi\,dz=\phi V(z_1,t)-\phi V(z_2,t).
$$
Thus the <kinematic sedimentation> equation and its <characteristic speed> are
$$
\boxed{s_t+\partial_zf(s)=0,\qquad f(s)=-V_0s(1-s)^\alpha,}
$$
$$
a(s)=f'(s)=-V_0(1-s)^{\alpha-1}\bigl[1-(\alpha+1)s\bigr].
$$
For $0<s<1$ this is a scalar <hyperbolic partial differential equation>: its one <characteristic speed> is real. The <method of characteristics> gives $ds/dt=0$ and $dz/dt=a(s)$. A <concentration> disturbance need not descend at the particle <settling velocity>; $a(s)$ is the <derivative> of the transported flux, not $V(s)$. In particular $a(s)$ can be positive even though every particle settles downwards. At $\alpha=0$, $a=-V_0$ and the equation is ordinary translation. At the packing endpoint $s=1$ the differentiability of the chosen constitutive law depends on $\alpha$; none of the initial values used below is at that endpoint.
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