= Solution
The shear stress is $\tau_{rz}=\mu u'(r)$, where
$$
u'(r)=-\frac{\rho g}{2\mu}r+\frac Ar.
$$
Thus the axial forces exerted by the fluid per unit length on the inner and outer cylinders are
$$
\boxed{F_{\rm in}=2\pi\alpha R\mu u'(\alpha R)
=2\pi\mu A-\pi\rho g\alpha^2R^2},
$$
$$
\boxed{F_{\rm out}=-2\pi R\mu u'(R)
=\pi\rho gR^2-2\pi\mu A}.
$$
Their sum is
$$
F_{\rm in}+F_{\rm out}=\pi\rho gR^2(1-\alpha^2),
$$
the weight per unit length of the fluid in the annulus. This is the expected overall vertical force balance.
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