= Solution
The volume flux carried with the particle skeleton is
$$
Q_0=2\pi\int_0^Rw_s(r)r\,dr
=\boxed{\frac{\pi G(R-a)^2(3R^2+2Ra+a^2)}{24\eta}}.
$$
By <Darcy law>, the fluid moves relative to the solids at $kG/\eta$. The total mixture-volume flux is therefore
$$
\boxed{Q_T=Q_0+
\frac{2\pi kG}{\eta}
\int_0^R[1-\phi(r)]r\,dr}.
$$
This is explicit because $\phi(r)$ is given above. For example, if
$$
c=\frac{G}{2p_s},
\qquad
T=\sqrt{c(R-a)},
$$
then
$$
\int_0^R\phi r\,dr
=\phi_m\left\{
\frac{a^2}{2}
+\frac{2a}{c}[T-\log(1+T)]
+\frac{2}{c^2}
\left[\frac{T^3}{3}-\frac{T^2}{2}+T-\log(1+T)\right]
\right\}.
$$
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