= Solution
Integrating the velocity profiles over their respective half-channels gives <volume flux> per unit width
$$
Q_-=\frac{Ga^3(\mu_++7\mu_-)}{12\mu_-(\mu_++\mu_-)},\qquad
Q_+=\frac{Ga^3(7\mu_++\mu_-)}{12\mu_+(\mu_++\mu_-)}.
$$
With $r=\mu_+/\mu_-$, the requested ratio is
$$
\boxed{Q_+/Q_-=\frac{1+7r}{r(7+r)}.}
$$
At $r=1$, both fluxes equal $Ga^3/(3\mu_-)$ and the ratio is one. For $r\to\infty$, $Q_-\to Ga^3/(12\mu_-)$, $Q_+\sim7Ga^3/(12\mu_+)$ and the ratio is asymptotic to $7/r$, tending to zero. The lower fluid flows between the wall and an interface which becomes effectively stationary; the upper fluid carries vanishing flux despite finite limiting stress.
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