= Solution
The <volume flux> per unit transverse width is
$$
Q=\int_0^h u(z)\,dz=\frac{\rho g\sin\alpha}{3\mu}h^3-\frac{S}{2\mu}h^2.
$$
Thus
$$
\boxed{Q=0\iff S=\frac23\rho gh\sin\alpha}.
$$
Assume a genuinely inclined plane, $0<\alpha<\pi/2$. With $Z=z/h$ and $U=\rho gh^2\sin\alpha/\mu$, the zero-flux <velocity profile> is $u/U=Z/3-Z^2/2$. It is positive for $0<Z<2/3$, zero at $Z=2/3$, and negative above that level. Its maximum is $U/18$ at $Z=1/3$, and its surface value is $-U/6$. Thus equal integrated downslope and upslope transports cancel, rather than the fluid being motionless.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ib/paper-1-zero-flux.png]
{title=Zero net flux with downslope flow near the wall and upslope flow near the surface}
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