= Solution
Put
$$
G=\frac{\rho g}{4\mu},\qquad
A=\frac{V-GR^2(1-\alpha^2)}{\log\alpha}.
$$
Integrating the equation in part (b) and imposing both no-slip conditions gives
$$
\boxed{u(r)=G(R^2-r^2)+A\log\frac rR}.
$$
For $V=0$, gravity drives downward annular Poiseuille flow, positive in the interior and zero at both walls. Positive $V$ raises the inner-wall endpoint to $V$ and strengthens the downward flow; sufficiently negative $V$ creates a region of upward flow near the inner cylinder. In each case the graph is the displayed quadratic-plus-logarithmic profile.
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