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

Measure pressure relative to atmospheric pressure. At leading order the stress-free upper surface gives $p(H)=0$, $\tau (H)=0$. Integrating the normal balance first yieldsAt fixed $z$, $p_x=\rho g\cos \alpha H_x$. Therefore ${\tau }_z=\rho g(\cos \alpha H_x-\sin \alpha )$, independent of $z$. Integrating from $z$ to $H$ givesThis is linear in $z$, is zero at the surface, and reaches its basal value at $z=0$. In particular the surface slope modifies the shear driving; a flat upper surface relative to the horizontal eliminates it in this approximation.