Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 1 / 17D / c

The volume flux per unit transverse width isThusAssume a genuinely inclined plane, $0<\alpha <\pi /2$. With $Z=z/h$ and $U=\rho g{h}^2\sin \alpha /\mu$, 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.