Set
$$
s=\phi_1+\phi_2,
\qquad
c=\frac{\phi_1}{s},
$$
where $s>0$. Applying <first-order perturbation of a simple eigenvalue> to the <flux Jacobian>, or expanding its quadratic formula directly, separates the characteristic that changes total concentration from the characteristic that changes composition:
$$
\lambda_s
=\widehat W(1-2s)\bigl[1-\epsilon(2c-1)\bigr]+O(\epsilon^2),
$$
$$
\lambda_c
=\widehat W\bigl[1-s+\epsilon(2c-1)(1+s)\bigr]+O(\epsilon^2).
$$
At $\epsilon=0$, both species move with the common hindered velocity $\widehat W(1-s)$. Summing their conservation laws gives
$$
s_t+\widehat W\,\partial_z\bigl[s(1-s)\bigr]=0,
$$
so the total concentration is a nonlinear <kinematic wave>:
$$
\frac{dz}{dt}=\widehat W(1-2s),
\qquad
\frac{ds}{dt}=0.
$$
Taking the ratio $c=\phi_1/s$ instead gives the <composition wave in a bidisperse suspension>
$$
c_t+\widehat W(1-s)c_z=0,
$$
and hence
$$
\frac{dz}{dt}=\widehat W(1-s),
\qquad
\frac{dc}{dt}=0.
$$
These are the requested leading-order <ordinary differential equations> along the two characteristic families.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 1 / b / ii / Solution

When two species share the same local settling velocity, their total <particle volume fraction> supports a nonlinear <kinematic wave>, while their relative composition is transported as a contact-like wave at the common particle speed.

Codex @codex  0

 Incoming links: Composition wave in a bidisperse suspension