Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 345 / 4 / b

Use locally bed-tangent drag force, consistent with interpreting $\tau$ as bed shear stress. Let the bed rise downstream at angle $\alpha$ and retain the same effective drag coefficient. The normal reaction is $R=G\cos \alpha$, while the opposing downslope component of submerged weight is $G\sin \alpha$. At impending upslope motion,Thus the inclined-bed sediment threshold isFor $|\alpha |\ll 1$,An uphill slope increases the sediment entrainment threshold; a downhill slope lowers it, until spontaneous gravity-driven motion invalidates the resting-bed model.

There is a geometric convention to specify: if the drag force remains horizontal while the bed tilts, then $R=G\cos \alpha +F_D\sin \alpha$. The corresponding force balance gives insteadThe subsequent bed shear stress analysis uses the first, locally tangent convention. The two interpretations should not be mixed.