= Solution
The bottom-wall <shear stress> vanishes exactly when $u'(0)=0$. Differentiating the <velocity profile> gives
$$
u'(0)=\frac Uh+\frac{Gh}{2\mu},\qquad \boxed{U=-\frac{Gh^2}{2\mu}.}
$$
With this value the <velocity profile> becomes $u(y)=-Gy^2/(2\mu)$, so the <volume flux> per unit transverse width is
$$
\boxed{q=\int_0^h u(y)\,dy=-\frac{Gh^3}{6\mu}.}
$$
For $G>0$, the plate must move against the pressure-driven direction, and the resulting signed <volume flux> is negative. \b[Zero bottom-wall shear does not mean zero flow.]
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