Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2010 / A / 3 / iii / Solution

Set $G=\rho g(\sin \alpha -\cos \alpha H_x)$. For downstream shear $G\ge 0$, Glen's flow law and the no-slip boundary condition giveHere $A$ is the ice rate factor, distinct from the cross-sectional area in the river problem. Integrating this velocity profile through the thickness gives the volume flux per unit valley width:The factor $H^{n+2}$ combines the depth-dependent shear stress, integration to obtain velocity, and integration to obtain flux. If reverse shear is admitted for noninteger $n$, the signed version of Glen's flow law is $u_z=A|\tau {|}^{n-1}\tau$; the displayed unsigned power presumes the downstream regime.