= Solution
The signed tangential stress component is $\sigma_{xy}=\mu u'=C-Gy$. Thus
$$
\boxed{\sigma_{xy}(-a)=\frac{Ga(\mu_++3\mu_-)}{2(\mu_++\mu_-)},\qquad
\sigma_{xy}(a)=-\frac{Ga(3\mu_++\mu_-)}{2(\mu_++\mu_-)}.}
$$
When $\mu_+/\mu_-\to1$, these approach $+Ga$ and $-Ga$, the usual symmetric <plane Poiseuille flow> stresses. When $\mu_+/\mu_-\to\infty$, they approach $+Ga/2$ and $-3Ga/2$. The upper fluid's velocity tends to zero, but its viscosity times velocity gradient has a finite limit. These are tensor components; traction on a chosen wall normal includes that normal's sign.
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